https://wiki.math.wisc.edu/api.php?action=feedcontributions&user=Mehrotra&feedformat=atomUW-Math Wiki - User contributions [en]2022-10-03T23:56:08ZUser contributionsMediaWiki 1.35.6https://wiki.math.wisc.edu/index.php?title=Algebraic_Geometry_Seminar_Spring_2012&diff=3571Algebraic Geometry Seminar Spring 20122012-02-27T12:56:19Z<p>Mehrotra: /* Abstracts */</p>
<hr />
<div>The seminar meets on Fridays at 2:25 pm in Van Vleck B215.<br />
<br />
The schedule for the previous semester is [https://www.math.wisc.edu/wiki/index.php/Algebraic_Geometry_Seminar_Fall_2011 here].<br />
<br />
== Spring 2012 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s) <br />
|-<br />
|January 27<br />
|[http://www.math.wisc.edu/~maehrotra/ Sukhendu Mehrotra] (Madison)<br />
|''Generalized deformations of K3 surfaces''<br />
|(local)<br />
|-<br />
|February 3<br />
|[http://math.mit.edu/~trasched/cv.html Travis Schedler] (MIT)<br />
|''Symplectic resolutions and Poisson-de Rham homology''<br />
|Andrei<br />
|-<br />
|February 10<br />
|[http://www.math.wisc.edu/~ballard/ Matthew Ballard] (UW-Madison)<br />
|''Variation of GIT for gauged Landau-Ginzburg models''<br />
|(local)<br />
|-<br />
|February 17<br />
|[http://www.math.uconn.edu/~bayer/ Arend Bayer] (UConn)<br />
|''Projectivity and birational geometry of Bridgeland moduli spaces''<br />
|Andrei<br />
|-<br />
|February 24<br />
|[http://www.math.wisc.edu/~maxim/ Laurentiu Maxim] (UW-Madison)<br />
|''Characteristic classes of Hilbert schemes of points via symmetric products''<br />
|local<br />
|-<br />
|March 2<br />
|[http://www.hcm.uni-bonn.de/people/postdocs/profile/marti-lahoz-vilalta/ Marti Lahoz] (Bonn)<br />
|''Effective Iitaka fibrations of varieties of maximal Albanese dimension''<br />
|Sukhendu<br />
|-<br />
|March 9<br />
|[http://www.math.psu.edu/yu/ Shilin Yu] (Penn State)<br />
|''TBD''<br />
|Andrei<br />
|-<br />
|March 16<br />
|[http://www.math.columbia.edu/~zheng/ Weizhe Zheng] (Columbia)<br />
|''TBD''<br />
|Tonghai<br />
|-<br />
|March 23<br />
|[http://www.nd.edu/~rgrady/cv.pdf Ryan Grady] (Notre Dame)<br />
|''Twisted differential operators as observables in QFT.''<br />
|Andrei<br />
|-<br />
|April 27<br />
|[http://people.uwec.edu/whitchua/ Ursula Whitcher] (UW-Eau Claire)<br />
|''TBA''<br />
|Matt<br />
|-<br />
|May 4<br />
|[http://www.math.sunysb.edu/~mde/ Mark Andrea de Cataldo] (Stony Brook)<br />
|''TBA''<br />
|Laurentiu<br />
<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Sukhendu Mehrotra=== <br />
''Generalized deformations of K3 surfaces''<br />
<br />
===Travis Schedler===<br />
''Symplectic resolutions and Poisson-de Rham homology''<br />
<br />
Abstract: A symplectic resolution is a resolution of singularities of a singular variety by a symplectic algebraic variety. Examples include symmetric powers of Kleinian (or du Val) singularities, resolved by Hilbert schemes of the minimal resolutions of Kleinian singularities, and the Springer resolution of the nilpotent cone of semisimple Lie algebras. Based on joint work with P. Etingof, I define a new homology theory on the singular variety, called Poisson-de Rham homology, which conjecturally coincides with the de Rham cohomology of the symplectic resolution. Its definition is based on "derived solutions" of Hamiltonian flow, using the algebraic theory of D-modules. I will give applications to the representation theory of noncommutative deformations of the algebra of functions of the singular variety. In the examples above, these are the spherical symplectic reflection algebras and finite W-algebras (modulo their center).<br />
<br />
===Matthew Ballard=== <br />
''Variation of GIT for gauged Landau-Ginzburg models''<br />
<br />
Abstract: Let X be a variety equipped with a G-action and G-invariant regular function, w. The GIT quotient X//G depends on the additional data of a G-linearized line bundle. As one varies the G-linearized line bundle, X//G changes in a very controlled manner. We will discuss how the category of matrix factorizations, mf(X//G,w), changes as the G-linearized line bundle varies. We will focus on the case where G is toroidal. In this case, we show that, as one travels through a wall in the GIT cone, semi-orthogonal components coming from the wall are either added or subtracted.<br />
<br />
===Arend Bayer===<br />
''Projectivity and birational geometry of Bridgeland moduli spaces ''<br />
<br />
I will present a construction of a nef divisor for every moduli space of<br />
Bridgeland stable complexes on an algebraic variety. In the case of K3<br />
surfaces, we can use it to prove projectivity of the moduli space,<br />
generalizing a result of Minamide, Yanagida and Yoshioka. It's<br />
dependence on the stability condition gives a systematic explanation for<br />
the compatibility of wall-crossing of the moduli space with its<br />
birational transformations; this had first been observed in examples by<br />
Arcara-Bertram. This is based on joint work with Emanuele Macrì.<br />
<br />
===Laurentiu Maxim=== <br />
''Characteristic classes of Hilbert schemes of points via symmetric products''<br />
<br />
I will explain a formula for the generating series of (the push-forward under the Hilbert-Chow morphism of) homology characteristic classes of the Hilbert schemes of points for a smooth quasi-projective variety of arbitrary pure dimension. The result is based on a geometric construction of a motivic exponentiation generalizing the notion of motivic power structure, as well as a formula for the generating series of homology characteristic classes of symmetric products.<br />
<br />
===Marti Lahoz===<br />
''Effective Iitaka fibrations of varieties of maximal Albanese dimension''<br />
<br />
Let X be a variety of maximal Albanese dimension.<br />
Chen and Hacon proved that if X has positive holomorphic Euler-characteristic, then its <br />
tricanonical map is birational onto its image; in particular, X is of general type.<br />
When the Euler characteristic is not positive, we use generic vanishing techniques to <br />
prove that the tetracanonical map of X induces the Iitaka fibration.<br />
Moreover, if X is of general type, then the tricanonical map is already birational.<br />
If time permits, I will also construct examples showing that these results are optimal.<br />
<br />
This is a joint work with Zhi Jiang and Sofia Tirabassi.<br />
<br />
===Ryan Grady===<br />
<br />
''Twisted differential operators as observables in QFT''<br />
<br />
We discuss Chern-Simons type theories in perturbative quantum field theory. The observables of such a theory has the structure of a factorization algebra. We recover the Rees algebra of differential operators on a space X from the one-dimensional theory with target T*X. We also consider twists of these theories which lead to twisted differential operators. If time allows, we will sketch dimensional reduction from 3 to 2 real dimensions. We will not assume any familiarity with quantum field theory in this talk.</div>Mehrotrahttps://wiki.math.wisc.edu/index.php?title=Algebraic_Geometry_Seminar_Spring_2012&diff=3570Algebraic Geometry Seminar Spring 20122012-02-27T12:55:11Z<p>Mehrotra: /* Spring 2012 */</p>
<hr />
<div>The seminar meets on Fridays at 2:25 pm in Van Vleck B215.<br />
<br />
The schedule for the previous semester is [https://www.math.wisc.edu/wiki/index.php/Algebraic_Geometry_Seminar_Fall_2011 here].<br />
<br />
== Spring 2012 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s) <br />
|-<br />
|January 27<br />
|[http://www.math.wisc.edu/~maehrotra/ Sukhendu Mehrotra] (Madison)<br />
|''Generalized deformations of K3 surfaces''<br />
|(local)<br />
|-<br />
|February 3<br />
|[http://math.mit.edu/~trasched/cv.html Travis Schedler] (MIT)<br />
|''Symplectic resolutions and Poisson-de Rham homology''<br />
|Andrei<br />
|-<br />
|February 10<br />
|[http://www.math.wisc.edu/~ballard/ Matthew Ballard] (UW-Madison)<br />
|''Variation of GIT for gauged Landau-Ginzburg models''<br />
|(local)<br />
|-<br />
|February 17<br />
|[http://www.math.uconn.edu/~bayer/ Arend Bayer] (UConn)<br />
|''Projectivity and birational geometry of Bridgeland moduli spaces''<br />
|Andrei<br />
|-<br />
|February 24<br />
|[http://www.math.wisc.edu/~maxim/ Laurentiu Maxim] (UW-Madison)<br />
|''Characteristic classes of Hilbert schemes of points via symmetric products''<br />
|local<br />
|-<br />
|March 2<br />
|[http://www.hcm.uni-bonn.de/people/postdocs/profile/marti-lahoz-vilalta/ Marti Lahoz] (Bonn)<br />
|''Effective Iitaka fibrations of varieties of maximal Albanese dimension''<br />
|Sukhendu<br />
|-<br />
|March 9<br />
|[http://www.math.psu.edu/yu/ Shilin Yu] (Penn State)<br />
|''TBD''<br />
|Andrei<br />
|-<br />
|March 16<br />
|[http://www.math.columbia.edu/~zheng/ Weizhe Zheng] (Columbia)<br />
|''TBD''<br />
|Tonghai<br />
|-<br />
|March 23<br />
|[http://www.nd.edu/~rgrady/cv.pdf Ryan Grady] (Notre Dame)<br />
|''Twisted differential operators as observables in QFT.''<br />
|Andrei<br />
|-<br />
|April 27<br />
|[http://people.uwec.edu/whitchua/ Ursula Whitcher] (UW-Eau Claire)<br />
|''TBA''<br />
|Matt<br />
|-<br />
|May 4<br />
|[http://www.math.sunysb.edu/~mde/ Mark Andrea de Cataldo] (Stony Brook)<br />
|''TBA''<br />
|Laurentiu<br />
<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Sukhendu Mehrotra=== <br />
''Generalized deformations of K3 surfaces''<br />
<br />
===Travis Schedler===<br />
''Symplectic resolutions and Poisson-de Rham homology''<br />
<br />
Abstract: A symplectic resolution is a resolution of singularities of a singular variety by a symplectic algebraic variety. Examples include symmetric powers of Kleinian (or du Val) singularities, resolved by Hilbert schemes of the minimal resolutions of Kleinian singularities, and the Springer resolution of the nilpotent cone of semisimple Lie algebras. Based on joint work with P. Etingof, I define a new homology theory on the singular variety, called Poisson-de Rham homology, which conjecturally coincides with the de Rham cohomology of the symplectic resolution. Its definition is based on "derived solutions" of Hamiltonian flow, using the algebraic theory of D-modules. I will give applications to the representation theory of noncommutative deformations of the algebra of functions of the singular variety. In the examples above, these are the spherical symplectic reflection algebras and finite W-algebras (modulo their center).<br />
<br />
===Matthew Ballard=== <br />
''Variation of GIT for gauged Landau-Ginzburg models''<br />
<br />
Abstract: Let X be a variety equipped with a G-action and G-invariant regular function, w. The GIT quotient X//G depends on the additional data of a G-linearized line bundle. As one varies the G-linearized line bundle, X//G changes in a very controlled manner. We will discuss how the category of matrix factorizations, mf(X//G,w), changes as the G-linearized line bundle varies. We will focus on the case where G is toroidal. In this case, we show that, as one travels through a wall in the GIT cone, semi-orthogonal components coming from the wall are either added or subtracted.<br />
<br />
===Arend Bayer===<br />
''Projectivity and birational geometry of Bridgeland moduli spaces ''<br />
<br />
I will present a construction of a nef divisor for every moduli space of<br />
Bridgeland stable complexes on an algebraic variety. In the case of K3<br />
surfaces, we can use it to prove projectivity of the moduli space,<br />
generalizing a result of Minamide, Yanagida and Yoshioka. It's<br />
dependence on the stability condition gives a systematic explanation for<br />
the compatibility of wall-crossing of the moduli space with its<br />
birational transformations; this had first been observed in examples by<br />
Arcara-Bertram. This is based on joint work with Emanuele Macrì.<br />
<br />
===Laurentiu Maxim=== <br />
''Characteristic classes of Hilbert schemes of points via symmetric products''<br />
<br />
I will explain a formula for the generating series of (the push-forward under the Hilbert-Chow morphism of) homology characteristic classes of the Hilbert schemes of points for a smooth quasi-projective variety of arbitrary pure dimension. The result is based on a geometric construction of a motivic exponentiation generalizing the notion of motivic power structure, as well as a formula for the generating series of homology characteristic classes of symmetric products.<br />
<br />
===Ryan Grady===<br />
<br />
''Twisted differential operators as observables in QFT''<br />
<br />
We discuss Chern-Simons type theories in perturbative quantum field theory. The observables of such a theory has the structure of a factorization algebra. We recover the Rees algebra of differential operators on a space X from the one-dimensional theory with target T*X. We also consider twists of these theories which lead to twisted differential operators. If time allows, we will sketch dimensional reduction from 3 to 2 real dimensions. We will not assume any familiarity with quantum field theory in this talk.</div>Mehrotrahttps://wiki.math.wisc.edu/index.php?title=Algebraic_Geometry_Seminar_Fall_2011&diff=3211Algebraic Geometry Seminar Fall 20112011-12-23T19:42:56Z<p>Mehrotra: /* Spring 2012 */</p>
<hr />
<div>The seminar meets on Fridays at 2:25 pm in Van Vleck B215.<br />
<br />
The schedule for the previous semester is [https://www.math.wisc.edu/wiki/index.php/Algebraic_Geometry_Seminar_Spring_2011 here].<br />
<br />
== Fall 2011 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|Sep. 23<br />
|Yifeng Liu (Columbia)<br />
|Enhanced Grothendieck's operations and base change theorem for<br />
sheaves on Artin stacks<br />
|Tonghai Yang<br />
|-<br />
|Sep. 30<br />
|Matthew Ballard (UW-Madison)<br />
|''You got your Hodge Conjecture in my matrix factorizations''<br />
|(local)<br />
|-<br />
|Oct. 7<br />
|[http://math.mit.edu/~zyun Zhiwei Yun] (MIT)<br />
|''Cohomology of Hilbert schemes of singular curves''<br />
|Shamgar Gurevich<br />
|-<br />
|Oct. 14<br />
|[http://www.icmat.es/research/international-grants/bobadilla Javier Fernández de Bobadilla] (Instituto de Ciencias Matematicas, Madrid)<br />
|''Nash problem for surfaces''<br />
| L. Maxim<br />
|-<br />
|Oct. 21<br />
|Andrei Caldararu (UW-Madison)<br />
|''The Hodge theorem as a derived self-intersection''<br />
|(local)<br />
|-<br />
|Nov. 11<br />
|John Francis (Northwestern)<br />
|''Integral transforms and Drinfeld centers in derived algebraic geometry''<br />
|Andrei Caldararu<br />
|-<br />
|Dec. 2<br />
|Shamgar Gurevich (Madison)<br />
|''Canonical Hilbert Space: Why? How? and its Categorification''<br />
|-<br />
|Dec. 9<br />
|Sean Paul (Madison)<br />
|''Hyperdiscriminants and Semistable pairs''<br />
|(local)<br />
|-<br />
<br />
|}<br />
<br />
== Spring 2012 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s) <br />
|-<br />
|January 27<br />
|[http://www.math.wisc.edu/~maehrotra/ Sukhendu Mehrotra] (Madison)<br />
|''Generalized deformations of K3 surfaces''<br />
|(local)<br />
|-<br />
|March 2<br />
|[http://www.hcm.uni-bonn.de/people/postdocs/profile/marti-lahoz-vilalta/ Marti Lahoz] (Bonn)<br />
|''TBD''<br />
|Mehrotra<br />
|-<br />
|March 16<br />
|[http://www.math.columbia.edu/~zheng/ Weizhe Zheng] (Columbia)<br />
|''TBD''<br />
|Tonghai Yang<br />
|-<br />
|March 23<br />
|[http://www.nd.edu/~rgrady/cv.pdf Ryan Grady] (Notre Dame)<br />
|''Twisted differential operators as observables in QFT.''<br />
|Caldararu<br />
|-<br />
|May 4<br />
|[http://www.math.sunysb.edu/~mde/ Mark Andrea de Cataldo] (Stony Brook)<br />
|''TBA''<br />
|Maxim<br />
<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Yifeng Liu=== <br />
''TBA''<br />
<br />
===Matthew Ballard=== <br />
''You got your Hodge Conjecture in my matrix factorizations''<br />
<br />
Abstract: I will describe how to prove some new cases of Hodge conjecture<br />
using the following tools: categories of graded matrix factorizations, <br />
the homotopy category of dg-categories, Orlov's Calabi-Yau/Landau-Ginzburg<br />
correspondence, Kuznetsov's relationship between the derived categories<br />
of a certain K3 surface and the Fermat cubic fourfold, and Hochschild homology.<br />
This is joint work with David Favero (Wien) and Ludmil Katzarkov (Miami/Wien).<br />
<br />
===Zhiwei Yun=== <br />
''Cohomology of Hilbert schemes of singular curves''<br />
<br />
Abstract: For a smooth curve, the Hilbert schemes are just symmetric<br />
powers of the curve, and their cohomology is easily computed by the<br />
H^1 of the curve. This is known as Macdonald's formula. In joint work<br />
with Davesh Maulik, we generalize this formula to curves with planar<br />
singularities (which was conjectured by L.Migliorini). In the singular<br />
case, the compactified Jacobian will play an important role in the<br />
formula, and we make use of Ngo's technique in his celebrated proof of<br />
the fundamental lemma.<br />
<br />
===Javier Fernández de Bobadilla=== <br />
''Nash problem for surfaces''<br />
<br />
The space of arcs through the singular set of an algebraic variety has a infinite dimensional scheme structure. In the late sixties Nash proved that it has finitely many irreducible components. He defined a natural mapping from this set of irreducible components to the set of essential divisors of a resolution of singularities. Roughly speaking the set of essential divisors is the set of irreducible components of the exceptional divisor of a resolution whose birational transform is an irreducible component of the exceptional divisor of any other resolution. <br />
<br />
Nash proved that this mapping is injective and proposed to study its bijectivity. In 2003 S. Ishii and J. Kollar gave a counterexample to the surjectivity in dimension at least 4. Recently, in a joint work with M. Pe Pereira, the speaker has settled the bijectivity for surfaces. In this talk I will explain the proof.<br />
<br />
===Andrei Caldararu===<br />
<br />
''The Hodge theorem as a derived self-intersection''<br />
<br />
The Hodge theorem is one of the most important results in complex geometry. It asserts that for a complex projective variety X the topological invariants H^*(X, C) can be refined to new ones that reflect the complex structure. The traditional statement and proof of the Hodge theorem are analytic. Given the multiple applications of the Hodge theorem in algebraic geometry, for many years it has been a major challenge to eliminate this analytic aspect and to obtain a purely algebraic proof of the Hodge theorem.<br />
An algebraic formulation of the Hodge theorem has been known since Grothendieck's work in the early 1970's. However, the first purely algebraic (and very surprising) proof was obtained only in 1991 by Deligne and Illusie, using methods involving reduction to characteristic p. In my talk I shall try to explain their ideas, and how recent developments in the field of derived algebraic geometry make their proof more geometric.<br />
<br />
===John Francis===<br />
<br />
''Integral transforms and Drinfeld centers in derived algebraic geometry''<br />
<br />
For a finite group G, conjugation invariant vector bundles on G have a<br />
universal property with respect to Rep(G): they form its Drinfeld<br />
center. Joint work with David Ben-Zvi and David Nadler generalizes<br />
this result, extending work of Hinich, in the setting of derived<br />
algebraic geometry. We describe a generalization of the Drinfeld<br />
center for a monoidal stable infinity category as a Hochschild<br />
cohomology category. For quasi-coherent sheaves on a perfect stack X,<br />
we prove that its center is equivalent to sheaves on the derived loop<br />
space LX. The structure of this category of sheaves defines an<br />
extended 2-dimensional topological quantum field theory.<br />
<br />
<br />
<br />
===Shamgar Gurevich===<br />
<br />
''Canonical Hilbert space: Why? How? and its Categorification''<br />
<br />
There is an idea in the mathematical physics community that quantization should be a functorial procedure. <br />
Our motivation in this talk is to show an example of such a procedure, answering a question of Kazhdan. I will describe a natural construction of an explicit quantization functor from the category SYMP of finite-dimensional symplectic vector spaces over a finite field to the category VECT of finite-dimensional complex vector spaces. In particular, for a fixed symplectic vector space V we obtain a canonical Hilbert space H(V), acted upon by the symplectic group Sp(V). This is called the canonical model of the Weil representation.<br />
<br />
The main idea in the construction of our functor is to overcome the traditional choice of a Lagrangian that appears in the classical constructions in the field of geometric quantization. For doing this we will explain the Grothendieck geometrization procedure, replacing sets by algebraic varieties, and function theoretic constructions, by sheaf theoretic analogues. In particular, I will explain the use of "Perverse Extension" to improve on the standard constructions that appear in the literature.<br />
<br />
Time permits, I will explain the categorification, or sign problem, which appears naturally in our setting. This categorification problem was formulated by Bernstein and Deligne, and was solved recently with the help of Ofer Gabber (IHES). I will speak on it in the future.<br />
<br />
Joint work with Ronny Hadani (Austin).<br />
<br />
I will assume only knowledge of very elementary representation theory.<br />
<br />
<br />
===Sean Timothy Paul===<br />
''Hyperdiscriminants and Semistable Pairs''<br />
<br />
One of the main problems in complex geometry is to detect the existence of "canonical" Kahler metrics in a given Kahler class on a compact complex (Kahler) manifold. In particular one seeks necessary and sufficient conditions for the existence of a Kahler Einstein metric on a Fano manifold. In this case the presence of positive curvature makes this problem extremely difficult and has led to a striking series of conjectures--the "standard conjectures"-- which relate the existence of these special metrics (which are solutions to the complex Monge-Ampere equation, a fully non-linear elliptic p.d.e . ) to the algebraic geometry of the pluri-anticanonical images of the manifold. Yau speculated that the relevant algebraic geometry would be related (somehow) to Mumford's Geometric Invariant Theory. Eventually it was conjectured that K-energy bounds along Bergman potentials could be deduced from an appropriate notion of "semi-stability". Recently this conjecture has been completely justified by the speaker, building upon work of Gang Tian and Gelfand-Kapranov-Zelevinsky and Weyman-Zelevinsky. It is the aim of this talk to outline progress on the standard conjectures and to discuss the entire theory in the context of complex algebraic groups and dominance of rational representations of such groups.</div>Mehrotrahttps://wiki.math.wisc.edu/index.php?title=Algebraic_Geometry_Seminar_Fall_2011&diff=3208Algebraic Geometry Seminar Fall 20112011-12-13T04:02:01Z<p>Mehrotra: /* Spring 2012 */</p>
<hr />
<div>The seminar meets on Fridays at 2:25 pm in Van Vleck B215.<br />
<br />
The schedule for the previous semester is [https://www.math.wisc.edu/wiki/index.php/Algebraic_Geometry_Seminar_Spring_2011 here].<br />
<br />
== Fall 2011 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|Sep. 23<br />
|Yifeng Liu (Columbia)<br />
|Enhanced Grothendieck's operations and base change theorem for<br />
sheaves on Artin stacks<br />
|Tonghai Yang<br />
|-<br />
|Sep. 30<br />
|Matthew Ballard (UW-Madison)<br />
|''You got your Hodge Conjecture in my matrix factorizations''<br />
|(local)<br />
|-<br />
|Oct. 7<br />
|[http://math.mit.edu/~zyun Zhiwei Yun] (MIT)<br />
|''Cohomology of Hilbert schemes of singular curves''<br />
|Shamgar Gurevich<br />
|-<br />
|Oct. 14<br />
|[http://www.icmat.es/research/international-grants/bobadilla Javier Fernández de Bobadilla] (Instituto de Ciencias Matematicas, Madrid)<br />
|''Nash problem for surfaces''<br />
| L. Maxim<br />
|-<br />
|Oct. 21<br />
|Andrei Caldararu (UW-Madison)<br />
|''The Hodge theorem as a derived self-intersection''<br />
|(local)<br />
|-<br />
|Nov. 11<br />
|John Francis (Northwestern)<br />
|''Integral transforms and Drinfeld centers in derived algebraic geometry''<br />
|Andrei Caldararu<br />
|-<br />
|Dec. 2<br />
|Shamgar Gurevich (Madison)<br />
|''Canonical Hilbert Space: Why? How? and its Categorification''<br />
|-<br />
|Dec. 9<br />
|Sean Paul (Madison)<br />
|''Hyperdiscriminants and Semistable pairs''<br />
|(local)<br />
|-<br />
<br />
|}<br />
<br />
== Spring 2012 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s) <br />
|-<br />
|March 2<br />
|[http://www.hcm.uni-bonn.de/people/postdocs/profile/marti-lahoz-vilalta/ Marti Lahoz] (Bonn)<br />
|''TBD''<br />
|Mehrotra<br />
|-<br />
|March 16<br />
|[http://www.math.columbia.edu/~zheng/ Weizhe Zheng] (Columbia)<br />
|''TBD''<br />
|Tonghai Yang<br />
|-<br />
|March 23<br />
|[http://www.nd.edu/~rgrady/cv.pdf Ryan Grady] (Notre Dame)<br />
|''Twisted differential operators as observables in QFT.''<br />
|Caldararu<br />
|-<br />
|May 4<br />
|[http://www.math.sunysb.edu/~mde/ Mark Andrea de Cataldo] (Stony Brook)<br />
|''TBA''<br />
|Maxim<br />
<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Yifeng Liu=== <br />
''TBA''<br />
<br />
===Matthew Ballard=== <br />
''You got your Hodge Conjecture in my matrix factorizations''<br />
<br />
Abstract: I will describe how to prove some new cases of Hodge conjecture<br />
using the following tools: categories of graded matrix factorizations, <br />
the homotopy category of dg-categories, Orlov's Calabi-Yau/Landau-Ginzburg<br />
correspondence, Kuznetsov's relationship between the derived categories<br />
of a certain K3 surface and the Fermat cubic fourfold, and Hochschild homology.<br />
This is joint work with David Favero (Wien) and Ludmil Katzarkov (Miami/Wien).<br />
<br />
===Zhiwei Yun=== <br />
''Cohomology of Hilbert schemes of singular curves''<br />
<br />
Abstract: For a smooth curve, the Hilbert schemes are just symmetric<br />
powers of the curve, and their cohomology is easily computed by the<br />
H^1 of the curve. This is known as Macdonald's formula. In joint work<br />
with Davesh Maulik, we generalize this formula to curves with planar<br />
singularities (which was conjectured by L.Migliorini). In the singular<br />
case, the compactified Jacobian will play an important role in the<br />
formula, and we make use of Ngo's technique in his celebrated proof of<br />
the fundamental lemma.<br />
<br />
===Javier Fernández de Bobadilla=== <br />
''Nash problem for surfaces''<br />
<br />
The space of arcs through the singular set of an algebraic variety has a infinite dimensional scheme structure. In the late sixties Nash proved that it has finitely many irreducible components. He defined a natural mapping from this set of irreducible components to the set of essential divisors of a resolution of singularities. Roughly speaking the set of essential divisors is the set of irreducible components of the exceptional divisor of a resolution whose birational transform is an irreducible component of the exceptional divisor of any other resolution. <br />
<br />
Nash proved that this mapping is injective and proposed to study its bijectivity. In 2003 S. Ishii and J. Kollar gave a counterexample to the surjectivity in dimension at least 4. Recently, in a joint work with M. Pe Pereira, the speaker has settled the bijectivity for surfaces. In this talk I will explain the proof.<br />
<br />
===Andrei Caldararu===<br />
<br />
''The Hodge theorem as a derived self-intersection''<br />
<br />
The Hodge theorem is one of the most important results in complex geometry. It asserts that for a complex projective variety X the topological invariants H^*(X, C) can be refined to new ones that reflect the complex structure. The traditional statement and proof of the Hodge theorem are analytic. Given the multiple applications of the Hodge theorem in algebraic geometry, for many years it has been a major challenge to eliminate this analytic aspect and to obtain a purely algebraic proof of the Hodge theorem.<br />
An algebraic formulation of the Hodge theorem has been known since Grothendieck's work in the early 1970's. However, the first purely algebraic (and very surprising) proof was obtained only in 1991 by Deligne and Illusie, using methods involving reduction to characteristic p. In my talk I shall try to explain their ideas, and how recent developments in the field of derived algebraic geometry make their proof more geometric.<br />
<br />
===John Francis===<br />
<br />
''Integral transforms and Drinfeld centers in derived algebraic geometry''<br />
<br />
For a finite group G, conjugation invariant vector bundles on G have a<br />
universal property with respect to Rep(G): they form its Drinfeld<br />
center. Joint work with David Ben-Zvi and David Nadler generalizes<br />
this result, extending work of Hinich, in the setting of derived<br />
algebraic geometry. We describe a generalization of the Drinfeld<br />
center for a monoidal stable infinity category as a Hochschild<br />
cohomology category. For quasi-coherent sheaves on a perfect stack X,<br />
we prove that its center is equivalent to sheaves on the derived loop<br />
space LX. The structure of this category of sheaves defines an<br />
extended 2-dimensional topological quantum field theory.<br />
<br />
<br />
<br />
===Shamgar Gurevich===<br />
<br />
''Canonical Hilbert space: Why? How? and its Categorification''<br />
<br />
There is an idea in the mathematical physics community that quantization should be a functorial procedure. <br />
Our motivation in this talk is to show an example of such a procedure, answering a question of Kazhdan. I will describe a natural construction of an explicit quantization functor from the category SYMP of finite-dimensional symplectic vector spaces over a finite field to the category VECT of finite-dimensional complex vector spaces. In particular, for a fixed symplectic vector space V we obtain a canonical Hilbert space H(V), acted upon by the symplectic group Sp(V). This is called the canonical model of the Weil representation.<br />
<br />
The main idea in the construction of our functor is to overcome the traditional choice of a Lagrangian that appears in the classical constructions in the field of geometric quantization. For doing this we will explain the Grothendieck geometrization procedure, replacing sets by algebraic varieties, and function theoretic constructions, by sheaf theoretic analogues. In particular, I will explain the use of "Perverse Extension" to improve on the standard constructions that appear in the literature.<br />
<br />
Time permits, I will explain the categorification, or sign problem, which appears naturally in our setting. This categorification problem was formulated by Bernstein and Deligne, and was solved recently with the help of Ofer Gabber (IHES). I will speak on it in the future.<br />
<br />
Joint work with Ronny Hadani (Austin).<br />
<br />
I will assume only knowledge of very elementary representation theory.<br />
<br />
<br />
===Sean Timothy Paul===<br />
''Hyperdiscriminants and Semistable Pairs''<br />
<br />
One of the main problems in complex geometry is to detect the existence of "canonical" Kahler metrics in a given Kahler class on a compact complex (Kahler) manifold. In particular one seeks necessary and sufficient conditions for the existence of a Kahler Einstein metric on a Fano manifold. In this case the presence of positive curvature makes this problem extremely difficult and has led to a striking series of conjectures--the "standard conjectures"-- which relate the existence of these special metrics (which are solutions to the complex Monge-Ampere equation, a fully non-linear elliptic p.d.e . ) to the algebraic geometry of the pluri-anticanonical images of the manifold. Yau speculated that the relevant algebraic geometry would be related (somehow) to Mumford's Geometric Invariant Theory. Eventually it was conjectured that K-energy bounds along Bergman potentials could be deduced from an appropriate notion of "semi-stability". Recently this conjecture has been completely justified by the speaker, building upon work of Gang Tian and Gelfand-Kapranov-Zelevinsky and Weyman-Zelevinsky. It is the aim of this talk to outline progress on the standard conjectures and to discuss the entire theory in the context of complex algebraic groups and dominance of rational representations of such groups.</div>Mehrotrahttps://wiki.math.wisc.edu/index.php?title=Algebraic_Geometry_Seminar_Fall_2011&diff=3207Algebraic Geometry Seminar Fall 20112011-12-13T03:59:00Z<p>Mehrotra: /* Spring 2012 */</p>
<hr />
<div>The seminar meets on Fridays at 2:25 pm in Van Vleck B215.<br />
<br />
The schedule for the previous semester is [https://www.math.wisc.edu/wiki/index.php/Algebraic_Geometry_Seminar_Spring_2011 here].<br />
<br />
== Fall 2011 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|Sep. 23<br />
|Yifeng Liu (Columbia)<br />
|Enhanced Grothendieck's operations and base change theorem for<br />
sheaves on Artin stacks<br />
|Tonghai Yang<br />
|-<br />
|Sep. 30<br />
|Matthew Ballard (UW-Madison)<br />
|''You got your Hodge Conjecture in my matrix factorizations''<br />
|(local)<br />
|-<br />
|Oct. 7<br />
|[http://math.mit.edu/~zyun Zhiwei Yun] (MIT)<br />
|''Cohomology of Hilbert schemes of singular curves''<br />
|Shamgar Gurevich<br />
|-<br />
|Oct. 14<br />
|[http://www.icmat.es/research/international-grants/bobadilla Javier Fernández de Bobadilla] (Instituto de Ciencias Matematicas, Madrid)<br />
|''Nash problem for surfaces''<br />
| L. Maxim<br />
|-<br />
|Oct. 21<br />
|Andrei Caldararu (UW-Madison)<br />
|''The Hodge theorem as a derived self-intersection''<br />
|(local)<br />
|-<br />
|Nov. 11<br />
|John Francis (Northwestern)<br />
|''Integral transforms and Drinfeld centers in derived algebraic geometry''<br />
|Andrei Caldararu<br />
|-<br />
|Dec. 2<br />
|Shamgar Gurevich (Madison)<br />
|''Canonical Hilbert Space: Why? How? and its Categorification''<br />
|-<br />
|Dec. 9<br />
|Sean Paul (Madison)<br />
|''Hyperdiscriminants and Semistable pairs''<br />
|(local)<br />
|-<br />
<br />
|}<br />
<br />
== Spring 2012 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s) <br />
|-<br />
|March 2<br />
|[http://www.hcm.uni-bonn.de/people/postdocs/profile/marti-lahoz-vilalta/ Marti Lahoz] (Bonn)<br />
|''TBD''<br />
|Sukhendu Mehrotra<br />
|-<br />
|March 16<br />
|[http://www.math.columbia.edu/~zheng/ Weizhe Zheng] (Columbia)<br />
|''TBD''<br />
|Tonghai Yang<br />
|-<br />
|March 23<br />
|[http://www.nd.edu/~rgrady/cv.pdf Ryan Grady] (Notre Dame)<br />
|''Twisted differential operators as observables in QFT.''<br />
|Caldararu<br />
|-<br />
|May 4<br />
|[http://www.math.sunysb.edu/~mde/ Mark Andrea de Cataldo] (Stony Brook)<br />
|''TBA''<br />
|Maxim<br />
<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Yifeng Liu=== <br />
''TBA''<br />
<br />
===Matthew Ballard=== <br />
''You got your Hodge Conjecture in my matrix factorizations''<br />
<br />
Abstract: I will describe how to prove some new cases of Hodge conjecture<br />
using the following tools: categories of graded matrix factorizations, <br />
the homotopy category of dg-categories, Orlov's Calabi-Yau/Landau-Ginzburg<br />
correspondence, Kuznetsov's relationship between the derived categories<br />
of a certain K3 surface and the Fermat cubic fourfold, and Hochschild homology.<br />
This is joint work with David Favero (Wien) and Ludmil Katzarkov (Miami/Wien).<br />
<br />
===Zhiwei Yun=== <br />
''Cohomology of Hilbert schemes of singular curves''<br />
<br />
Abstract: For a smooth curve, the Hilbert schemes are just symmetric<br />
powers of the curve, and their cohomology is easily computed by the<br />
H^1 of the curve. This is known as Macdonald's formula. In joint work<br />
with Davesh Maulik, we generalize this formula to curves with planar<br />
singularities (which was conjectured by L.Migliorini). In the singular<br />
case, the compactified Jacobian will play an important role in the<br />
formula, and we make use of Ngo's technique in his celebrated proof of<br />
the fundamental lemma.<br />
<br />
===Javier Fernández de Bobadilla=== <br />
''Nash problem for surfaces''<br />
<br />
The space of arcs through the singular set of an algebraic variety has a infinite dimensional scheme structure. In the late sixties Nash proved that it has finitely many irreducible components. He defined a natural mapping from this set of irreducible components to the set of essential divisors of a resolution of singularities. Roughly speaking the set of essential divisors is the set of irreducible components of the exceptional divisor of a resolution whose birational transform is an irreducible component of the exceptional divisor of any other resolution. <br />
<br />
Nash proved that this mapping is injective and proposed to study its bijectivity. In 2003 S. Ishii and J. Kollar gave a counterexample to the surjectivity in dimension at least 4. Recently, in a joint work with M. Pe Pereira, the speaker has settled the bijectivity for surfaces. In this talk I will explain the proof.<br />
<br />
===Andrei Caldararu===<br />
<br />
''The Hodge theorem as a derived self-intersection''<br />
<br />
The Hodge theorem is one of the most important results in complex geometry. It asserts that for a complex projective variety X the topological invariants H^*(X, C) can be refined to new ones that reflect the complex structure. The traditional statement and proof of the Hodge theorem are analytic. Given the multiple applications of the Hodge theorem in algebraic geometry, for many years it has been a major challenge to eliminate this analytic aspect and to obtain a purely algebraic proof of the Hodge theorem.<br />
An algebraic formulation of the Hodge theorem has been known since Grothendieck's work in the early 1970's. However, the first purely algebraic (and very surprising) proof was obtained only in 1991 by Deligne and Illusie, using methods involving reduction to characteristic p. In my talk I shall try to explain their ideas, and how recent developments in the field of derived algebraic geometry make their proof more geometric.<br />
<br />
===John Francis===<br />
<br />
''Integral transforms and Drinfeld centers in derived algebraic geometry''<br />
<br />
For a finite group G, conjugation invariant vector bundles on G have a<br />
universal property with respect to Rep(G): they form its Drinfeld<br />
center. Joint work with David Ben-Zvi and David Nadler generalizes<br />
this result, extending work of Hinich, in the setting of derived<br />
algebraic geometry. We describe a generalization of the Drinfeld<br />
center for a monoidal stable infinity category as a Hochschild<br />
cohomology category. For quasi-coherent sheaves on a perfect stack X,<br />
we prove that its center is equivalent to sheaves on the derived loop<br />
space LX. The structure of this category of sheaves defines an<br />
extended 2-dimensional topological quantum field theory.<br />
<br />
<br />
<br />
===Shamgar Gurevich===<br />
<br />
''Canonical Hilbert space: Why? How? and its Categorification''<br />
<br />
There is an idea in the mathematical physics community that quantization should be a functorial procedure. <br />
Our motivation in this talk is to show an example of such a procedure, answering a question of Kazhdan. I will describe a natural construction of an explicit quantization functor from the category SYMP of finite-dimensional symplectic vector spaces over a finite field to the category VECT of finite-dimensional complex vector spaces. In particular, for a fixed symplectic vector space V we obtain a canonical Hilbert space H(V), acted upon by the symplectic group Sp(V). This is called the canonical model of the Weil representation.<br />
<br />
The main idea in the construction of our functor is to overcome the traditional choice of a Lagrangian that appears in the classical constructions in the field of geometric quantization. For doing this we will explain the Grothendieck geometrization procedure, replacing sets by algebraic varieties, and function theoretic constructions, by sheaf theoretic analogues. In particular, I will explain the use of "Perverse Extension" to improve on the standard constructions that appear in the literature.<br />
<br />
Time permits, I will explain the categorification, or sign problem, which appears naturally in our setting. This categorification problem was formulated by Bernstein and Deligne, and was solved recently with the help of Ofer Gabber (IHES). I will speak on it in the future.<br />
<br />
Joint work with Ronny Hadani (Austin).<br />
<br />
I will assume only knowledge of very elementary representation theory.<br />
<br />
<br />
===Sean Timothy Paul===<br />
''Hyperdiscriminants and Semistable Pairs''<br />
<br />
One of the main problems in complex geometry is to detect the existence of "canonical" Kahler metrics in a given Kahler class on a compact complex (Kahler) manifold. In particular one seeks necessary and sufficient conditions for the existence of a Kahler Einstein metric on a Fano manifold. In this case the presence of positive curvature makes this problem extremely difficult and has led to a striking series of conjectures--the "standard conjectures"-- which relate the existence of these special metrics (which are solutions to the complex Monge-Ampere equation, a fully non-linear elliptic p.d.e . ) to the algebraic geometry of the pluri-anticanonical images of the manifold. Yau speculated that the relevant algebraic geometry would be related (somehow) to Mumford's Geometric Invariant Theory. Eventually it was conjectured that K-energy bounds along Bergman potentials could be deduced from an appropriate notion of "semi-stability". Recently this conjecture has been completely justified by the speaker, building upon work of Gang Tian and Gelfand-Kapranov-Zelevinsky and Weyman-Zelevinsky. It is the aim of this talk to outline progress on the standard conjectures and to discuss the entire theory in the context of complex algebraic groups and dominance of rational representations of such groups.</div>Mehrotrahttps://wiki.math.wisc.edu/index.php?title=Algebraic_Geometry_Seminar_Fall_2011&diff=3104Algebraic Geometry Seminar Fall 20112011-11-16T22:16:45Z<p>Mehrotra: /* Sukhendu Mehrotra */</p>
<hr />
<div>The seminar meets on Fridays at 2:25 pm in Van Vleck B215.<br />
<br />
The schedule for the previous semester is [https://www.math.wisc.edu/wiki/index.php/Algebraic_Geometry_Seminar_Spring_2011 here].<br />
<br />
== Fall 2011 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|Sep. 23<br />
|Yifeng Liu (Columbia)<br />
|Enhanced Grothendieck's operations and base change theorem for<br />
sheaves on Artin stacks<br />
|Tonghai Yang<br />
|-<br />
|Sep. 30<br />
|Matthew Ballard (UW-Madison)<br />
|''You got your Hodge Conjecture in my matrix factorizations''<br />
|(local)<br />
|-<br />
|Oct. 7<br />
|[http://math.mit.edu/~zyun Zhiwei Yun] (MIT)<br />
|''Cohomology of Hilbert schemes of singular curves''<br />
|Shamgar Gurevich<br />
|-<br />
|Oct. 14<br />
|[http://www.icmat.es/research/international-grants/bobadilla Javier Fernández de Bobadilla] (Instituto de Ciencias Matematicas, Madrid)<br />
|''Nash problem for surfaces''<br />
| L. Maxim<br />
|-<br />
|Oct. 21<br />
|Andrei Caldararu (UW-Madison)<br />
|''The Hodge theorem as a derived self-intersection''<br />
|(local)<br />
|-<br />
|Nov. 11<br />
|John Francis (Northwestern)<br />
|''Integral transforms and Drinfeld centers in derived algebraic geometry''<br />
|Andrei Caldararu<br />
|-<br />
|Dec. 2<br />
|Shamgar Gurevich (Madison)<br />
|''Canonical Hilbert Space: Why? How? and its Categorification''<br />
|-<br />
|Dec. 9<br />
|Sean Paul (Madison)<br />
|''Semistable pairs and quasi-closed orbits''<br />
|-<br />
<br />
|}<br />
<br />
== Spring 2012 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s) <br />
|-<br />
|March 16<br />
|[http://www.math.columbia.edu/~zheng/ Weizhe Zheng] (Columbia)<br />
|''TBD''<br />
|Tonghai Yang<br />
|-<br />
|March 23<br />
|[http://www.nd.edu/~rgrady/cv.pdf Ryan Grady] (Notre Dame)<br />
|''Twisted differential operators as observables in QFT.''<br />
|Caldararu<br />
|-<br />
|May 4<br />
|[http://www.math.sunysb.edu/~mde/ Mark Andrea de Cataldo] (Stony Brook)<br />
|''TBA''<br />
|Maxim<br />
<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Yifeng Liu=== <br />
''TBA''<br />
<br />
===Matthew Ballard=== <br />
''You got your Hodge Conjecture in my matrix factorizations''<br />
<br />
Abstract: I will describe how to prove some new cases of Hodge conjecture<br />
using the following tools: categories of graded matrix factorizations, <br />
the homotopy category of dg-categories, Orlov's Calabi-Yau/Landau-Ginzburg<br />
correspondence, Kuznetsov's relationship between the derived categories<br />
of a certain K3 surface and the Fermat cubic fourfold, and Hochschild homology.<br />
This is joint work with David Favero (Wien) and Ludmil Katzarkov (Miami/Wien).<br />
<br />
===Zhiwei Yun=== <br />
''Cohomology of Hilbert schemes of singular curves''<br />
<br />
Abstract: For a smooth curve, the Hilbert schemes are just symmetric<br />
powers of the curve, and their cohomology is easily computed by the<br />
H^1 of the curve. This is known as Macdonald's formula. In joint work<br />
with Davesh Maulik, we generalize this formula to curves with planar<br />
singularities (which was conjectured by L.Migliorini). In the singular<br />
case, the compactified Jacobian will play an important role in the<br />
formula, and we make use of Ngo's technique in his celebrated proof of<br />
the fundamental lemma.<br />
<br />
===Javier Fernández de Bobadilla=== <br />
''Nash problem for surfaces''<br />
<br />
The space of arcs through the singular set of an algebraic variety has a infinite dimensional scheme structure. In the late sixties Nash proved that it has finitely many irreducible components. He defined a natural mapping from this set of irreducible components to the set of essential divisors of a resolution of singularities. Roughly speaking the set of essential divisors is the set of irreducible components of the exceptional divisor of a resolution whose birational transform is an irreducible component of the exceptional divisor of any other resolution. <br />
<br />
Nash proved that this mapping is injective and proposed to study its bijectivity. In 2003 S. Ishii and J. Kollar gave a counterexample to the surjectivity in dimension at least 4. Recently, in a joint work with M. Pe Pereira, the speaker has settled the bijectivity for surfaces. In this talk I will explain the proof.<br />
<br />
===Andrei Caldararu===<br />
<br />
''The Hodge theorem as a derived self-intersection''<br />
<br />
The Hodge theorem is one of the most important results in complex geometry. It asserts that for a complex projective variety X the topological invariants H^*(X, C) can be refined to new ones that reflect the complex structure. The traditional statement and proof of the Hodge theorem are analytic. Given the multiple applications of the Hodge theorem in algebraic geometry, for many years it has been a major challenge to eliminate this analytic aspect and to obtain a purely algebraic proof of the Hodge theorem.<br />
An algebraic formulation of the Hodge theorem has been known since Grothendieck's work in the early 1970's. However, the first purely algebraic (and very surprising) proof was obtained only in 1991 by Deligne and Illusie, using methods involving reduction to characteristic p. In my talk I shall try to explain their ideas, and how recent developments in the field of derived algebraic geometry make their proof more geometric.<br />
<br />
===John Francis===<br />
<br />
''Integral transforms and Drinfeld centers in derived algebraic geometry''<br />
<br />
For a finite group G, conjugation invariant vector bundles on G have a<br />
universal property with respect to Rep(G): they form its Drinfeld<br />
center. Joint work with David Ben-Zvi and David Nadler generalizes<br />
this result, extending work of Hinich, in the setting of derived<br />
algebraic geometry. We describe a generalization of the Drinfeld<br />
center for a monoidal stable infinity category as a Hochschild<br />
cohomology category. For quasi-coherent sheaves on a perfect stack X,<br />
we prove that its center is equivalent to sheaves on the derived loop<br />
space LX. The structure of this category of sheaves defines an<br />
extended 2-dimensional topological quantum field theory.</div>Mehrotrahttps://wiki.math.wisc.edu/index.php?title=Algebraic_Geometry_Seminar_Fall_2011&diff=3103Algebraic Geometry Seminar Fall 20112011-11-16T22:16:33Z<p>Mehrotra: /* Fall 2011 */</p>
<hr />
<div>The seminar meets on Fridays at 2:25 pm in Van Vleck B215.<br />
<br />
The schedule for the previous semester is [https://www.math.wisc.edu/wiki/index.php/Algebraic_Geometry_Seminar_Spring_2011 here].<br />
<br />
== Fall 2011 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|Sep. 23<br />
|Yifeng Liu (Columbia)<br />
|Enhanced Grothendieck's operations and base change theorem for<br />
sheaves on Artin stacks<br />
|Tonghai Yang<br />
|-<br />
|Sep. 30<br />
|Matthew Ballard (UW-Madison)<br />
|''You got your Hodge Conjecture in my matrix factorizations''<br />
|(local)<br />
|-<br />
|Oct. 7<br />
|[http://math.mit.edu/~zyun Zhiwei Yun] (MIT)<br />
|''Cohomology of Hilbert schemes of singular curves''<br />
|Shamgar Gurevich<br />
|-<br />
|Oct. 14<br />
|[http://www.icmat.es/research/international-grants/bobadilla Javier Fernández de Bobadilla] (Instituto de Ciencias Matematicas, Madrid)<br />
|''Nash problem for surfaces''<br />
| L. Maxim<br />
|-<br />
|Oct. 21<br />
|Andrei Caldararu (UW-Madison)<br />
|''The Hodge theorem as a derived self-intersection''<br />
|(local)<br />
|-<br />
|Nov. 11<br />
|John Francis (Northwestern)<br />
|''Integral transforms and Drinfeld centers in derived algebraic geometry''<br />
|Andrei Caldararu<br />
|-<br />
|Dec. 2<br />
|Shamgar Gurevich (Madison)<br />
|''Canonical Hilbert Space: Why? How? and its Categorification''<br />
|-<br />
|Dec. 9<br />
|Sean Paul (Madison)<br />
|''Semistable pairs and quasi-closed orbits''<br />
|-<br />
<br />
|}<br />
<br />
== Spring 2012 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s) <br />
|-<br />
|March 16<br />
|[http://www.math.columbia.edu/~zheng/ Weizhe Zheng] (Columbia)<br />
|''TBD''<br />
|Tonghai Yang<br />
|-<br />
|March 23<br />
|[http://www.nd.edu/~rgrady/cv.pdf Ryan Grady] (Notre Dame)<br />
|''Twisted differential operators as observables in QFT.''<br />
|Caldararu<br />
|-<br />
|May 4<br />
|[http://www.math.sunysb.edu/~mde/ Mark Andrea de Cataldo] (Stony Brook)<br />
|''TBA''<br />
|Maxim<br />
<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Yifeng Liu=== <br />
''TBA''<br />
<br />
===Matthew Ballard=== <br />
''You got your Hodge Conjecture in my matrix factorizations''<br />
<br />
Abstract: I will describe how to prove some new cases of Hodge conjecture<br />
using the following tools: categories of graded matrix factorizations, <br />
the homotopy category of dg-categories, Orlov's Calabi-Yau/Landau-Ginzburg<br />
correspondence, Kuznetsov's relationship between the derived categories<br />
of a certain K3 surface and the Fermat cubic fourfold, and Hochschild homology.<br />
This is joint work with David Favero (Wien) and Ludmil Katzarkov (Miami/Wien).<br />
<br />
===Zhiwei Yun=== <br />
''Cohomology of Hilbert schemes of singular curves''<br />
<br />
Abstract: For a smooth curve, the Hilbert schemes are just symmetric<br />
powers of the curve, and their cohomology is easily computed by the<br />
H^1 of the curve. This is known as Macdonald's formula. In joint work<br />
with Davesh Maulik, we generalize this formula to curves with planar<br />
singularities (which was conjectured by L.Migliorini). In the singular<br />
case, the compactified Jacobian will play an important role in the<br />
formula, and we make use of Ngo's technique in his celebrated proof of<br />
the fundamental lemma.<br />
<br />
===Javier Fernández de Bobadilla=== <br />
''Nash problem for surfaces''<br />
<br />
The space of arcs through the singular set of an algebraic variety has a infinite dimensional scheme structure. In the late sixties Nash proved that it has finitely many irreducible components. He defined a natural mapping from this set of irreducible components to the set of essential divisors of a resolution of singularities. Roughly speaking the set of essential divisors is the set of irreducible components of the exceptional divisor of a resolution whose birational transform is an irreducible component of the exceptional divisor of any other resolution. <br />
<br />
Nash proved that this mapping is injective and proposed to study its bijectivity. In 2003 S. Ishii and J. Kollar gave a counterexample to the surjectivity in dimension at least 4. Recently, in a joint work with M. Pe Pereira, the speaker has settled the bijectivity for surfaces. In this talk I will explain the proof.<br />
<br />
===Andrei Caldararu===<br />
<br />
''The Hodge theorem as a derived self-intersection''<br />
<br />
The Hodge theorem is one of the most important results in complex geometry. It asserts that for a complex projective variety X the topological invariants H^*(X, C) can be refined to new ones that reflect the complex structure. The traditional statement and proof of the Hodge theorem are analytic. Given the multiple applications of the Hodge theorem in algebraic geometry, for many years it has been a major challenge to eliminate this analytic aspect and to obtain a purely algebraic proof of the Hodge theorem.<br />
An algebraic formulation of the Hodge theorem has been known since Grothendieck's work in the early 1970's. However, the first purely algebraic (and very surprising) proof was obtained only in 1991 by Deligne and Illusie, using methods involving reduction to characteristic p. In my talk I shall try to explain their ideas, and how recent developments in the field of derived algebraic geometry make their proof more geometric.<br />
<br />
===John Francis===<br />
<br />
''Integral transforms and Drinfeld centers in derived algebraic geometry''<br />
<br />
For a finite group G, conjugation invariant vector bundles on G have a<br />
universal property with respect to Rep(G): they form its Drinfeld<br />
center. Joint work with David Ben-Zvi and David Nadler generalizes<br />
this result, extending work of Hinich, in the setting of derived<br />
algebraic geometry. We describe a generalization of the Drinfeld<br />
center for a monoidal stable infinity category as a Hochschild<br />
cohomology category. For quasi-coherent sheaves on a perfect stack X,<br />
we prove that its center is equivalent to sheaves on the derived loop<br />
space LX. The structure of this category of sheaves defines an<br />
extended 2-dimensional topological quantum field theory.<br />
<br />
===Sukhendu Mehrotra===<br />
<br />
''Moduli spaces of sheaves on K3 surfaces and generalized deformations''<br />
<br />
It is a result of Mukai that connected components of the moduli space of stable sheaves on<br />
a K3 surface X are holomorphic symplectic varities. As any such component Y deforms in a<br />
21 dimensional family, while the moduli space of K3 surfaces is 20 dimensional, the general<br />
deformation <math>\hat{Y}</math> of Y will not be be a moduli space of sheaves on a K3. This talk presents an<br />
attempt to associate to such a <math>\hat{Y}</math> a "non-commutative K3 surface" <math>\hat{X}</math><br />
for which the modular description carries over. This joint work with Eyal Markman.</div>Mehrotrahttps://wiki.math.wisc.edu/index.php?title=Algebraic_Geometry_Seminar_Fall_2011&diff=3089Algebraic Geometry Seminar Fall 20112011-11-15T05:09:52Z<p>Mehrotra: /* Sukhendu Mehrotra */</p>
<hr />
<div>The seminar meets on Fridays at 2:25 pm in Van Vleck B215.<br />
<br />
The schedule for the previous semester is [https://www.math.wisc.edu/wiki/index.php/Algebraic_Geometry_Seminar_Spring_2011 here].<br />
<br />
== Fall 2011 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|Sep. 23<br />
|Yifeng Liu (Columbia)<br />
|Enhanced Grothendieck's operations and base change theorem for<br />
sheaves on Artin stacks<br />
|Tonghai Yang<br />
|-<br />
|Sep. 30<br />
|Matthew Ballard (UW-Madison)<br />
|''You got your Hodge Conjecture in my matrix factorizations''<br />
|(local)<br />
|-<br />
|Oct. 7<br />
|[http://math.mit.edu/~zyun Zhiwei Yun] (MIT)<br />
|''Cohomology of Hilbert schemes of singular curves''<br />
|Shamgar Gurevich<br />
|-<br />
|Oct. 14<br />
|[http://www.icmat.es/research/international-grants/bobadilla Javier Fernández de Bobadilla] (Instituto de Ciencias Matematicas, Madrid)<br />
|''Nash problem for surfaces''<br />
| L. Maxim<br />
|-<br />
|Oct. 21<br />
|Andrei Caldararu (UW-Madison)<br />
|''The Hodge theorem as a derived self-intersection''<br />
|(local)<br />
|-<br />
|Nov. 11<br />
|John Francis (Northwestern)<br />
|''Integral transforms and Drinfeld centers in derived algebraic geometry''<br />
|Andrei Caldararu<br />
|-<br />
|Nov. 18<br />
|Sukhendu Mehrotra (Madison)<br />
|''Moduli spaces of sheaves on K3 surfaces and generalized deformations''<br />
|(local)<br />
|-<br />
|Dec. 2<br />
|Shamgar Gurevich (Madison)<br />
|''Canonical Hilbert Space: Why? How? and its Categorification''<br />
|-<br />
|Dec. 9<br />
|Sean Paul (Madison)<br />
|''Semistable pairs and quasi-closed orbits''<br />
|-<br />
<br />
|}<br />
<br />
== Spring 2012 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s) <br />
|-<br />
|March 16<br />
|[http://www.math.columbia.edu/~zheng/ Weizhe Zheng] (Columbia)<br />
|''TBD''<br />
|Tonghai Yang<br />
|-<br />
|March 23<br />
|[http://www.nd.edu/~rgrady/cv.pdf Ryan Grady] (Notre Dame)<br />
|''Twisted differential operators as observables in QFT.''<br />
|Caldararu<br />
|-<br />
|May 4<br />
|[http://www.math.sunysb.edu/~mde/ Mark Andrea de Cataldo] (Stony Brook)<br />
|''TBA''<br />
|Maxim<br />
<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Yifeng Liu=== <br />
''TBA''<br />
<br />
===Matthew Ballard=== <br />
''You got your Hodge Conjecture in my matrix factorizations''<br />
<br />
Abstract: I will describe how to prove some new cases of Hodge conjecture<br />
using the following tools: categories of graded matrix factorizations, <br />
the homotopy category of dg-categories, Orlov's Calabi-Yau/Landau-Ginzburg<br />
correspondence, Kuznetsov's relationship between the derived categories<br />
of a certain K3 surface and the Fermat cubic fourfold, and Hochschild homology.<br />
This is joint work with David Favero (Wien) and Ludmil Katzarkov (Miami/Wien).<br />
<br />
===Zhiwei Yun=== <br />
''Cohomology of Hilbert schemes of singular curves''<br />
<br />
Abstract: For a smooth curve, the Hilbert schemes are just symmetric<br />
powers of the curve, and their cohomology is easily computed by the<br />
H^1 of the curve. This is known as Macdonald's formula. In joint work<br />
with Davesh Maulik, we generalize this formula to curves with planar<br />
singularities (which was conjectured by L.Migliorini). In the singular<br />
case, the compactified Jacobian will play an important role in the<br />
formula, and we make use of Ngo's technique in his celebrated proof of<br />
the fundamental lemma.<br />
<br />
===Javier Fernández de Bobadilla=== <br />
''Nash problem for surfaces''<br />
<br />
The space of arcs through the singular set of an algebraic variety has a infinite dimensional scheme structure. In the late sixties Nash proved that it has finitely many irreducible components. He defined a natural mapping from this set of irreducible components to the set of essential divisors of a resolution of singularities. Roughly speaking the set of essential divisors is the set of irreducible components of the exceptional divisor of a resolution whose birational transform is an irreducible component of the exceptional divisor of any other resolution. <br />
<br />
Nash proved that this mapping is injective and proposed to study its bijectivity. In 2003 S. Ishii and J. Kollar gave a counterexample to the surjectivity in dimension at least 4. Recently, in a joint work with M. Pe Pereira, the speaker has settled the bijectivity for surfaces. In this talk I will explain the proof.<br />
<br />
===Andrei Caldararu===<br />
<br />
''The Hodge theorem as a derived self-intersection''<br />
<br />
The Hodge theorem is one of the most important results in complex geometry. It asserts that for a complex projective variety X the topological invariants H^*(X, C) can be refined to new ones that reflect the complex structure. The traditional statement and proof of the Hodge theorem are analytic. Given the multiple applications of the Hodge theorem in algebraic geometry, for many years it has been a major challenge to eliminate this analytic aspect and to obtain a purely algebraic proof of the Hodge theorem.<br />
An algebraic formulation of the Hodge theorem has been known since Grothendieck's work in the early 1970's. However, the first purely algebraic (and very surprising) proof was obtained only in 1991 by Deligne and Illusie, using methods involving reduction to characteristic p. In my talk I shall try to explain their ideas, and how recent developments in the field of derived algebraic geometry make their proof more geometric.<br />
<br />
===John Francis===<br />
<br />
''Integral transforms and Drinfeld centers in derived algebraic geometry''<br />
<br />
For a finite group G, conjugation invariant vector bundles on G have a<br />
universal property with respect to Rep(G): they form its Drinfeld<br />
center. Joint work with David Ben-Zvi and David Nadler generalizes<br />
this result, extending work of Hinich, in the setting of derived<br />
algebraic geometry. We describe a generalization of the Drinfeld<br />
center for a monoidal stable infinity category as a Hochschild<br />
cohomology category. For quasi-coherent sheaves on a perfect stack X,<br />
we prove that its center is equivalent to sheaves on the derived loop<br />
space LX. The structure of this category of sheaves defines an<br />
extended 2-dimensional topological quantum field theory.<br />
<br />
===Sukhendu Mehrotra===<br />
<br />
''Moduli spaces of sheaves on K3 surfaces and generalized deformations''<br />
<br />
It is a result of Mukai that connected components of the moduli space of stable sheaves on<br />
a K3 surface X are holomorphic symplectic varities. As any such component Y deforms in a<br />
21 dimensional family, while the moduli space of K3 surfaces is 20 dimensional, the general<br />
deformation <math>\hat{Y}</math> of Y will not be be a moduli space of sheaves on a K3. This talk presents an<br />
attempt to associate to such a <math>\hat{Y}</math> a "non-commutative K3 surface" <math>\hat{X}</math><br />
for which the modular description carries over. This joint work with Eyal Markman.</div>Mehrotrahttps://wiki.math.wisc.edu/index.php?title=Algebraic_Geometry_Seminar_Fall_2011&diff=3088Algebraic Geometry Seminar Fall 20112011-11-15T05:09:12Z<p>Mehrotra: /* Sukhendu Mehrotra */</p>
<hr />
<div>The seminar meets on Fridays at 2:25 pm in Van Vleck B215.<br />
<br />
The schedule for the previous semester is [https://www.math.wisc.edu/wiki/index.php/Algebraic_Geometry_Seminar_Spring_2011 here].<br />
<br />
== Fall 2011 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|Sep. 23<br />
|Yifeng Liu (Columbia)<br />
|Enhanced Grothendieck's operations and base change theorem for<br />
sheaves on Artin stacks<br />
|Tonghai Yang<br />
|-<br />
|Sep. 30<br />
|Matthew Ballard (UW-Madison)<br />
|''You got your Hodge Conjecture in my matrix factorizations''<br />
|(local)<br />
|-<br />
|Oct. 7<br />
|[http://math.mit.edu/~zyun Zhiwei Yun] (MIT)<br />
|''Cohomology of Hilbert schemes of singular curves''<br />
|Shamgar Gurevich<br />
|-<br />
|Oct. 14<br />
|[http://www.icmat.es/research/international-grants/bobadilla Javier Fernández de Bobadilla] (Instituto de Ciencias Matematicas, Madrid)<br />
|''Nash problem for surfaces''<br />
| L. Maxim<br />
|-<br />
|Oct. 21<br />
|Andrei Caldararu (UW-Madison)<br />
|''The Hodge theorem as a derived self-intersection''<br />
|(local)<br />
|-<br />
|Nov. 11<br />
|John Francis (Northwestern)<br />
|''Integral transforms and Drinfeld centers in derived algebraic geometry''<br />
|Andrei Caldararu<br />
|-<br />
|Nov. 18<br />
|Sukhendu Mehrotra (Madison)<br />
|''Moduli spaces of sheaves on K3 surfaces and generalized deformations''<br />
|(local)<br />
|-<br />
|Dec. 2<br />
|Shamgar Gurevich (Madison)<br />
|''Canonical Hilbert Space: Why? How? and its Categorification''<br />
|-<br />
|Dec. 9<br />
|Sean Paul (Madison)<br />
|''Semistable pairs and quasi-closed orbits''<br />
|-<br />
<br />
|}<br />
<br />
== Spring 2012 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s) <br />
|-<br />
|March 16<br />
|[http://www.math.columbia.edu/~zheng/ Weizhe Zheng] (Columbia)<br />
|''TBD''<br />
|Tonghai Yang<br />
|-<br />
|March 23<br />
|[http://www.nd.edu/~rgrady/cv.pdf Ryan Grady] (Notre Dame)<br />
|''Twisted differential operators as observables in QFT.''<br />
|Caldararu<br />
|-<br />
|May 4<br />
|[http://www.math.sunysb.edu/~mde/ Mark Andrea de Cataldo] (Stony Brook)<br />
|''TBA''<br />
|Maxim<br />
<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Yifeng Liu=== <br />
''TBA''<br />
<br />
===Matthew Ballard=== <br />
''You got your Hodge Conjecture in my matrix factorizations''<br />
<br />
Abstract: I will describe how to prove some new cases of Hodge conjecture<br />
using the following tools: categories of graded matrix factorizations, <br />
the homotopy category of dg-categories, Orlov's Calabi-Yau/Landau-Ginzburg<br />
correspondence, Kuznetsov's relationship between the derived categories<br />
of a certain K3 surface and the Fermat cubic fourfold, and Hochschild homology.<br />
This is joint work with David Favero (Wien) and Ludmil Katzarkov (Miami/Wien).<br />
<br />
===Zhiwei Yun=== <br />
''Cohomology of Hilbert schemes of singular curves''<br />
<br />
Abstract: For a smooth curve, the Hilbert schemes are just symmetric<br />
powers of the curve, and their cohomology is easily computed by the<br />
H^1 of the curve. This is known as Macdonald's formula. In joint work<br />
with Davesh Maulik, we generalize this formula to curves with planar<br />
singularities (which was conjectured by L.Migliorini). In the singular<br />
case, the compactified Jacobian will play an important role in the<br />
formula, and we make use of Ngo's technique in his celebrated proof of<br />
the fundamental lemma.<br />
<br />
===Javier Fernández de Bobadilla=== <br />
''Nash problem for surfaces''<br />
<br />
The space of arcs through the singular set of an algebraic variety has a infinite dimensional scheme structure. In the late sixties Nash proved that it has finitely many irreducible components. He defined a natural mapping from this set of irreducible components to the set of essential divisors of a resolution of singularities. Roughly speaking the set of essential divisors is the set of irreducible components of the exceptional divisor of a resolution whose birational transform is an irreducible component of the exceptional divisor of any other resolution. <br />
<br />
Nash proved that this mapping is injective and proposed to study its bijectivity. In 2003 S. Ishii and J. Kollar gave a counterexample to the surjectivity in dimension at least 4. Recently, in a joint work with M. Pe Pereira, the speaker has settled the bijectivity for surfaces. In this talk I will explain the proof.<br />
<br />
===Andrei Caldararu===<br />
<br />
''The Hodge theorem as a derived self-intersection''<br />
<br />
The Hodge theorem is one of the most important results in complex geometry. It asserts that for a complex projective variety X the topological invariants H^*(X, C) can be refined to new ones that reflect the complex structure. The traditional statement and proof of the Hodge theorem are analytic. Given the multiple applications of the Hodge theorem in algebraic geometry, for many years it has been a major challenge to eliminate this analytic aspect and to obtain a purely algebraic proof of the Hodge theorem.<br />
An algebraic formulation of the Hodge theorem has been known since Grothendieck's work in the early 1970's. However, the first purely algebraic (and very surprising) proof was obtained only in 1991 by Deligne and Illusie, using methods involving reduction to characteristic p. In my talk I shall try to explain their ideas, and how recent developments in the field of derived algebraic geometry make their proof more geometric.<br />
<br />
===John Francis===<br />
<br />
''Integral transforms and Drinfeld centers in derived algebraic geometry''<br />
<br />
For a finite group G, conjugation invariant vector bundles on G have a<br />
universal property with respect to Rep(G): they form its Drinfeld<br />
center. Joint work with David Ben-Zvi and David Nadler generalizes<br />
this result, extending work of Hinich, in the setting of derived<br />
algebraic geometry. We describe a generalization of the Drinfeld<br />
center for a monoidal stable infinity category as a Hochschild<br />
cohomology category. For quasi-coherent sheaves on a perfect stack X,<br />
we prove that its center is equivalent to sheaves on the derived loop<br />
space LX. The structure of this category of sheaves defines an<br />
extended 2-dimensional topological quantum field theory.<br />
<br />
===Sukhendu Mehrotra===<br />
<br />
''Moduli spaces of sheaves on K3 surfaces and generalized deformations''<br />
<br />
It is a result of Mukai that connected components of the moduli space of stable sheaves on<br />
a K3 surface X are holomorphic symplectic varities. As any such component Y deforms in a<br />
21 dimensional family, while the moduli space of K3 surfaces is 20 dimensional, the general<br />
deformation Y' of Y will not be be a moduli space of sheaves on a K3. This talk presents an<br />
attempt to associate to such a Y' a "non-commutative K3 surface" <math>\hat{X}</math><br />
for which the modular description carries over. This joint work with Eyal Markman.</div>Mehrotrahttps://wiki.math.wisc.edu/index.php?title=Algebraic_Geometry_Seminar_Fall_2011&diff=3087Algebraic Geometry Seminar Fall 20112011-11-15T05:08:13Z<p>Mehrotra: /* Sukhendu Mehrotra */</p>
<hr />
<div>The seminar meets on Fridays at 2:25 pm in Van Vleck B215.<br />
<br />
The schedule for the previous semester is [https://www.math.wisc.edu/wiki/index.php/Algebraic_Geometry_Seminar_Spring_2011 here].<br />
<br />
== Fall 2011 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|Sep. 23<br />
|Yifeng Liu (Columbia)<br />
|Enhanced Grothendieck's operations and base change theorem for<br />
sheaves on Artin stacks<br />
|Tonghai Yang<br />
|-<br />
|Sep. 30<br />
|Matthew Ballard (UW-Madison)<br />
|''You got your Hodge Conjecture in my matrix factorizations''<br />
|(local)<br />
|-<br />
|Oct. 7<br />
|[http://math.mit.edu/~zyun Zhiwei Yun] (MIT)<br />
|''Cohomology of Hilbert schemes of singular curves''<br />
|Shamgar Gurevich<br />
|-<br />
|Oct. 14<br />
|[http://www.icmat.es/research/international-grants/bobadilla Javier Fernández de Bobadilla] (Instituto de Ciencias Matematicas, Madrid)<br />
|''Nash problem for surfaces''<br />
| L. Maxim<br />
|-<br />
|Oct. 21<br />
|Andrei Caldararu (UW-Madison)<br />
|''The Hodge theorem as a derived self-intersection''<br />
|(local)<br />
|-<br />
|Nov. 11<br />
|John Francis (Northwestern)<br />
|''Integral transforms and Drinfeld centers in derived algebraic geometry''<br />
|Andrei Caldararu<br />
|-<br />
|Nov. 18<br />
|Sukhendu Mehrotra (Madison)<br />
|''Moduli spaces of sheaves on K3 surfaces and generalized deformations''<br />
|(local)<br />
|-<br />
|Dec. 2<br />
|Shamgar Gurevich (Madison)<br />
|''Canonical Hilbert Space: Why? How? and its Categorification''<br />
|-<br />
|Dec. 9<br />
|Sean Paul (Madison)<br />
|''Semistable pairs and quasi-closed orbits''<br />
|-<br />
<br />
|}<br />
<br />
== Spring 2012 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s) <br />
|-<br />
|March 16<br />
|[http://www.math.columbia.edu/~zheng/ Weizhe Zheng] (Columbia)<br />
|''TBD''<br />
|Tonghai Yang<br />
|-<br />
|March 23<br />
|[http://www.nd.edu/~rgrady/cv.pdf Ryan Grady] (Notre Dame)<br />
|''Twisted differential operators as observables in QFT.''<br />
|Caldararu<br />
|-<br />
|May 4<br />
|[http://www.math.sunysb.edu/~mde/ Mark Andrea de Cataldo] (Stony Brook)<br />
|''TBA''<br />
|Maxim<br />
<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Yifeng Liu=== <br />
''TBA''<br />
<br />
===Matthew Ballard=== <br />
''You got your Hodge Conjecture in my matrix factorizations''<br />
<br />
Abstract: I will describe how to prove some new cases of Hodge conjecture<br />
using the following tools: categories of graded matrix factorizations, <br />
the homotopy category of dg-categories, Orlov's Calabi-Yau/Landau-Ginzburg<br />
correspondence, Kuznetsov's relationship between the derived categories<br />
of a certain K3 surface and the Fermat cubic fourfold, and Hochschild homology.<br />
This is joint work with David Favero (Wien) and Ludmil Katzarkov (Miami/Wien).<br />
<br />
===Zhiwei Yun=== <br />
''Cohomology of Hilbert schemes of singular curves''<br />
<br />
Abstract: For a smooth curve, the Hilbert schemes are just symmetric<br />
powers of the curve, and their cohomology is easily computed by the<br />
H^1 of the curve. This is known as Macdonald's formula. In joint work<br />
with Davesh Maulik, we generalize this formula to curves with planar<br />
singularities (which was conjectured by L.Migliorini). In the singular<br />
case, the compactified Jacobian will play an important role in the<br />
formula, and we make use of Ngo's technique in his celebrated proof of<br />
the fundamental lemma.<br />
<br />
===Javier Fernández de Bobadilla=== <br />
''Nash problem for surfaces''<br />
<br />
The space of arcs through the singular set of an algebraic variety has a infinite dimensional scheme structure. In the late sixties Nash proved that it has finitely many irreducible components. He defined a natural mapping from this set of irreducible components to the set of essential divisors of a resolution of singularities. Roughly speaking the set of essential divisors is the set of irreducible components of the exceptional divisor of a resolution whose birational transform is an irreducible component of the exceptional divisor of any other resolution. <br />
<br />
Nash proved that this mapping is injective and proposed to study its bijectivity. In 2003 S. Ishii and J. Kollar gave a counterexample to the surjectivity in dimension at least 4. Recently, in a joint work with M. Pe Pereira, the speaker has settled the bijectivity for surfaces. In this talk I will explain the proof.<br />
<br />
===Andrei Caldararu===<br />
<br />
''The Hodge theorem as a derived self-intersection''<br />
<br />
The Hodge theorem is one of the most important results in complex geometry. It asserts that for a complex projective variety X the topological invariants H^*(X, C) can be refined to new ones that reflect the complex structure. The traditional statement and proof of the Hodge theorem are analytic. Given the multiple applications of the Hodge theorem in algebraic geometry, for many years it has been a major challenge to eliminate this analytic aspect and to obtain a purely algebraic proof of the Hodge theorem.<br />
An algebraic formulation of the Hodge theorem has been known since Grothendieck's work in the early 1970's. However, the first purely algebraic (and very surprising) proof was obtained only in 1991 by Deligne and Illusie, using methods involving reduction to characteristic p. In my talk I shall try to explain their ideas, and how recent developments in the field of derived algebraic geometry make their proof more geometric.<br />
<br />
===John Francis===<br />
<br />
''Integral transforms and Drinfeld centers in derived algebraic geometry''<br />
<br />
For a finite group G, conjugation invariant vector bundles on G have a<br />
universal property with respect to Rep(G): they form its Drinfeld<br />
center. Joint work with David Ben-Zvi and David Nadler generalizes<br />
this result, extending work of Hinich, in the setting of derived<br />
algebraic geometry. We describe a generalization of the Drinfeld<br />
center for a monoidal stable infinity category as a Hochschild<br />
cohomology category. For quasi-coherent sheaves on a perfect stack X,<br />
we prove that its center is equivalent to sheaves on the derived loop<br />
space LX. The structure of this category of sheaves defines an<br />
extended 2-dimensional topological quantum field theory.<br />
<br />
===Sukhendu Mehrotra===<br />
<br />
''Moduli spaces of sheaves on K3 surfaces and generalized deformations''<br />
<br />
It is a result of Mukai that connected components of the moduli space of stable sheaves on<br />
a K3 surface X are holomorphic symplectic varities. As any such component Y deforms in a<br />
21 dimensional family, while the moduli space of K3 surfaces is 20 dimensional, the general<br />
deformation Y' of Y will not be be a moduli space of sheaves on a K3. This talk presents an<br />
attempt to associate to such a Y' a "non-commutative K3 surface" <math>\widetilde{X}</math><br />
for which the modular description carries over. This joint work with Eyal Markman.</div>Mehrotrahttps://wiki.math.wisc.edu/index.php?title=Algebraic_Geometry_Seminar_Fall_2011&diff=3086Algebraic Geometry Seminar Fall 20112011-11-15T05:07:28Z<p>Mehrotra: /* Sukhendu Mehrotra */</p>
<hr />
<div>The seminar meets on Fridays at 2:25 pm in Van Vleck B215.<br />
<br />
The schedule for the previous semester is [https://www.math.wisc.edu/wiki/index.php/Algebraic_Geometry_Seminar_Spring_2011 here].<br />
<br />
== Fall 2011 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|Sep. 23<br />
|Yifeng Liu (Columbia)<br />
|Enhanced Grothendieck's operations and base change theorem for<br />
sheaves on Artin stacks<br />
|Tonghai Yang<br />
|-<br />
|Sep. 30<br />
|Matthew Ballard (UW-Madison)<br />
|''You got your Hodge Conjecture in my matrix factorizations''<br />
|(local)<br />
|-<br />
|Oct. 7<br />
|[http://math.mit.edu/~zyun Zhiwei Yun] (MIT)<br />
|''Cohomology of Hilbert schemes of singular curves''<br />
|Shamgar Gurevich<br />
|-<br />
|Oct. 14<br />
|[http://www.icmat.es/research/international-grants/bobadilla Javier Fernández de Bobadilla] (Instituto de Ciencias Matematicas, Madrid)<br />
|''Nash problem for surfaces''<br />
| L. Maxim<br />
|-<br />
|Oct. 21<br />
|Andrei Caldararu (UW-Madison)<br />
|''The Hodge theorem as a derived self-intersection''<br />
|(local)<br />
|-<br />
|Nov. 11<br />
|John Francis (Northwestern)<br />
|''Integral transforms and Drinfeld centers in derived algebraic geometry''<br />
|Andrei Caldararu<br />
|-<br />
|Nov. 18<br />
|Sukhendu Mehrotra (Madison)<br />
|''Moduli spaces of sheaves on K3 surfaces and generalized deformations''<br />
|(local)<br />
|-<br />
|Dec. 2<br />
|Shamgar Gurevich (Madison)<br />
|''Canonical Hilbert Space: Why? How? and its Categorification''<br />
|-<br />
|Dec. 9<br />
|Sean Paul (Madison)<br />
|''Semistable pairs and quasi-closed orbits''<br />
|-<br />
<br />
|}<br />
<br />
== Spring 2012 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s) <br />
|-<br />
|March 16<br />
|[http://www.math.columbia.edu/~zheng/ Weizhe Zheng] (Columbia)<br />
|''TBD''<br />
|Tonghai Yang<br />
|-<br />
|March 23<br />
|[http://www.nd.edu/~rgrady/cv.pdf Ryan Grady] (Notre Dame)<br />
|''Twisted differential operators as observables in QFT.''<br />
|Caldararu<br />
|-<br />
|May 4<br />
|[http://www.math.sunysb.edu/~mde/ Mark Andrea de Cataldo] (Stony Brook)<br />
|''TBA''<br />
|Maxim<br />
<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Yifeng Liu=== <br />
''TBA''<br />
<br />
===Matthew Ballard=== <br />
''You got your Hodge Conjecture in my matrix factorizations''<br />
<br />
Abstract: I will describe how to prove some new cases of Hodge conjecture<br />
using the following tools: categories of graded matrix factorizations, <br />
the homotopy category of dg-categories, Orlov's Calabi-Yau/Landau-Ginzburg<br />
correspondence, Kuznetsov's relationship between the derived categories<br />
of a certain K3 surface and the Fermat cubic fourfold, and Hochschild homology.<br />
This is joint work with David Favero (Wien) and Ludmil Katzarkov (Miami/Wien).<br />
<br />
===Zhiwei Yun=== <br />
''Cohomology of Hilbert schemes of singular curves''<br />
<br />
Abstract: For a smooth curve, the Hilbert schemes are just symmetric<br />
powers of the curve, and their cohomology is easily computed by the<br />
H^1 of the curve. This is known as Macdonald's formula. In joint work<br />
with Davesh Maulik, we generalize this formula to curves with planar<br />
singularities (which was conjectured by L.Migliorini). In the singular<br />
case, the compactified Jacobian will play an important role in the<br />
formula, and we make use of Ngo's technique in his celebrated proof of<br />
the fundamental lemma.<br />
<br />
===Javier Fernández de Bobadilla=== <br />
''Nash problem for surfaces''<br />
<br />
The space of arcs through the singular set of an algebraic variety has a infinite dimensional scheme structure. In the late sixties Nash proved that it has finitely many irreducible components. He defined a natural mapping from this set of irreducible components to the set of essential divisors of a resolution of singularities. Roughly speaking the set of essential divisors is the set of irreducible components of the exceptional divisor of a resolution whose birational transform is an irreducible component of the exceptional divisor of any other resolution. <br />
<br />
Nash proved that this mapping is injective and proposed to study its bijectivity. In 2003 S. Ishii and J. Kollar gave a counterexample to the surjectivity in dimension at least 4. Recently, in a joint work with M. Pe Pereira, the speaker has settled the bijectivity for surfaces. In this talk I will explain the proof.<br />
<br />
===Andrei Caldararu===<br />
<br />
''The Hodge theorem as a derived self-intersection''<br />
<br />
The Hodge theorem is one of the most important results in complex geometry. It asserts that for a complex projective variety X the topological invariants H^*(X, C) can be refined to new ones that reflect the complex structure. The traditional statement and proof of the Hodge theorem are analytic. Given the multiple applications of the Hodge theorem in algebraic geometry, for many years it has been a major challenge to eliminate this analytic aspect and to obtain a purely algebraic proof of the Hodge theorem.<br />
An algebraic formulation of the Hodge theorem has been known since Grothendieck's work in the early 1970's. However, the first purely algebraic (and very surprising) proof was obtained only in 1991 by Deligne and Illusie, using methods involving reduction to characteristic p. In my talk I shall try to explain their ideas, and how recent developments in the field of derived algebraic geometry make their proof more geometric.<br />
<br />
===John Francis===<br />
<br />
''Integral transforms and Drinfeld centers in derived algebraic geometry''<br />
<br />
For a finite group G, conjugation invariant vector bundles on G have a<br />
universal property with respect to Rep(G): they form its Drinfeld<br />
center. Joint work with David Ben-Zvi and David Nadler generalizes<br />
this result, extending work of Hinich, in the setting of derived<br />
algebraic geometry. We describe a generalization of the Drinfeld<br />
center for a monoidal stable infinity category as a Hochschild<br />
cohomology category. For quasi-coherent sheaves on a perfect stack X,<br />
we prove that its center is equivalent to sheaves on the derived loop<br />
space LX. The structure of this category of sheaves defines an<br />
extended 2-dimensional topological quantum field theory.<br />
<br />
===Sukhendu Mehrotra===<br />
<br />
''Moduli spaces of sheaves on K3 surfaces and generalized deformations''<br />
<br />
It is a result of Mukai that connected components of the moduli space of stable sheaves on<br />
a K3 surface X are holomorphic symplectic varities. As any such component Y deforms in a<br />
21 dimensional family, while the moduli space of K3 surfaces is 20 dimensional, the general<br />
deformation Y' of Y will not be be a moduli space of sheaves on a K3. This talk presents an<br />
attempt to associate to such a Y' a "non-commutative K3 surface" X^~<br />
for which the modular description carries over. This joint work with Eyal Markman.</div>Mehrotrahttps://wiki.math.wisc.edu/index.php?title=Algebraic_Geometry_Seminar_Fall_2011&diff=3085Algebraic Geometry Seminar Fall 20112011-11-15T05:06:39Z<p>Mehrotra: /* Sukhendu Mehrotra */</p>
<hr />
<div>The seminar meets on Fridays at 2:25 pm in Van Vleck B215.<br />
<br />
The schedule for the previous semester is [https://www.math.wisc.edu/wiki/index.php/Algebraic_Geometry_Seminar_Spring_2011 here].<br />
<br />
== Fall 2011 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|Sep. 23<br />
|Yifeng Liu (Columbia)<br />
|Enhanced Grothendieck's operations and base change theorem for<br />
sheaves on Artin stacks<br />
|Tonghai Yang<br />
|-<br />
|Sep. 30<br />
|Matthew Ballard (UW-Madison)<br />
|''You got your Hodge Conjecture in my matrix factorizations''<br />
|(local)<br />
|-<br />
|Oct. 7<br />
|[http://math.mit.edu/~zyun Zhiwei Yun] (MIT)<br />
|''Cohomology of Hilbert schemes of singular curves''<br />
|Shamgar Gurevich<br />
|-<br />
|Oct. 14<br />
|[http://www.icmat.es/research/international-grants/bobadilla Javier Fernández de Bobadilla] (Instituto de Ciencias Matematicas, Madrid)<br />
|''Nash problem for surfaces''<br />
| L. Maxim<br />
|-<br />
|Oct. 21<br />
|Andrei Caldararu (UW-Madison)<br />
|''The Hodge theorem as a derived self-intersection''<br />
|(local)<br />
|-<br />
|Nov. 11<br />
|John Francis (Northwestern)<br />
|''Integral transforms and Drinfeld centers in derived algebraic geometry''<br />
|Andrei Caldararu<br />
|-<br />
|Nov. 18<br />
|Sukhendu Mehrotra (Madison)<br />
|''Moduli spaces of sheaves on K3 surfaces and generalized deformations''<br />
|(local)<br />
|-<br />
|Dec. 2<br />
|Shamgar Gurevich (Madison)<br />
|''Canonical Hilbert Space: Why? How? and its Categorification''<br />
|-<br />
|Dec. 9<br />
|Sean Paul (Madison)<br />
|''Semistable pairs and quasi-closed orbits''<br />
|-<br />
<br />
|}<br />
<br />
== Spring 2012 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s) <br />
|-<br />
|March 16<br />
|[http://www.math.columbia.edu/~zheng/ Weizhe Zheng] (Columbia)<br />
|''TBD''<br />
|Tonghai Yang<br />
|-<br />
|March 23<br />
|[http://www.nd.edu/~rgrady/cv.pdf Ryan Grady] (Notre Dame)<br />
|''Twisted differential operators as observables in QFT.''<br />
|Caldararu<br />
|-<br />
|May 4<br />
|[http://www.math.sunysb.edu/~mde/ Mark Andrea de Cataldo] (Stony Brook)<br />
|''TBA''<br />
|Maxim<br />
<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Yifeng Liu=== <br />
''TBA''<br />
<br />
===Matthew Ballard=== <br />
''You got your Hodge Conjecture in my matrix factorizations''<br />
<br />
Abstract: I will describe how to prove some new cases of Hodge conjecture<br />
using the following tools: categories of graded matrix factorizations, <br />
the homotopy category of dg-categories, Orlov's Calabi-Yau/Landau-Ginzburg<br />
correspondence, Kuznetsov's relationship between the derived categories<br />
of a certain K3 surface and the Fermat cubic fourfold, and Hochschild homology.<br />
This is joint work with David Favero (Wien) and Ludmil Katzarkov (Miami/Wien).<br />
<br />
===Zhiwei Yun=== <br />
''Cohomology of Hilbert schemes of singular curves''<br />
<br />
Abstract: For a smooth curve, the Hilbert schemes are just symmetric<br />
powers of the curve, and their cohomology is easily computed by the<br />
H^1 of the curve. This is known as Macdonald's formula. In joint work<br />
with Davesh Maulik, we generalize this formula to curves with planar<br />
singularities (which was conjectured by L.Migliorini). In the singular<br />
case, the compactified Jacobian will play an important role in the<br />
formula, and we make use of Ngo's technique in his celebrated proof of<br />
the fundamental lemma.<br />
<br />
===Javier Fernández de Bobadilla=== <br />
''Nash problem for surfaces''<br />
<br />
The space of arcs through the singular set of an algebraic variety has a infinite dimensional scheme structure. In the late sixties Nash proved that it has finitely many irreducible components. He defined a natural mapping from this set of irreducible components to the set of essential divisors of a resolution of singularities. Roughly speaking the set of essential divisors is the set of irreducible components of the exceptional divisor of a resolution whose birational transform is an irreducible component of the exceptional divisor of any other resolution. <br />
<br />
Nash proved that this mapping is injective and proposed to study its bijectivity. In 2003 S. Ishii and J. Kollar gave a counterexample to the surjectivity in dimension at least 4. Recently, in a joint work with M. Pe Pereira, the speaker has settled the bijectivity for surfaces. In this talk I will explain the proof.<br />
<br />
===Andrei Caldararu===<br />
<br />
''The Hodge theorem as a derived self-intersection''<br />
<br />
The Hodge theorem is one of the most important results in complex geometry. It asserts that for a complex projective variety X the topological invariants H^*(X, C) can be refined to new ones that reflect the complex structure. The traditional statement and proof of the Hodge theorem are analytic. Given the multiple applications of the Hodge theorem in algebraic geometry, for many years it has been a major challenge to eliminate this analytic aspect and to obtain a purely algebraic proof of the Hodge theorem.<br />
An algebraic formulation of the Hodge theorem has been known since Grothendieck's work in the early 1970's. However, the first purely algebraic (and very surprising) proof was obtained only in 1991 by Deligne and Illusie, using methods involving reduction to characteristic p. In my talk I shall try to explain their ideas, and how recent developments in the field of derived algebraic geometry make their proof more geometric.<br />
<br />
===John Francis===<br />
<br />
''Integral transforms and Drinfeld centers in derived algebraic geometry''<br />
<br />
For a finite group G, conjugation invariant vector bundles on G have a<br />
universal property with respect to Rep(G): they form its Drinfeld<br />
center. Joint work with David Ben-Zvi and David Nadler generalizes<br />
this result, extending work of Hinich, in the setting of derived<br />
algebraic geometry. We describe a generalization of the Drinfeld<br />
center for a monoidal stable infinity category as a Hochschild<br />
cohomology category. For quasi-coherent sheaves on a perfect stack X,<br />
we prove that its center is equivalent to sheaves on the derived loop<br />
space LX. The structure of this category of sheaves defines an<br />
extended 2-dimensional topological quantum field theory.<br />
<br />
===Sukhendu Mehrotra===<br />
<br />
''Moduli spaces of sheaves on K3 surfaces and generalized deformations''<br />
<br />
It is a result of Mukai that connected components of the moduli space of stable sheaves on<br />
a K3 surface X are holomorphic symplectic varities. As any such component Y deforms in a<br />
21 dimensional family, while the moduli space of K3 surfaces is 20 dimensional, the general<br />
deformation Y' of Y will not be be a moduli space of sheaves on a K3. This talk presents an<br />
attempt to associate to such a Y' a "non-commutative K3 surface" X<br />
for which the modular description carries over. This joint work with Eyal Markman.</div>Mehrotrahttps://wiki.math.wisc.edu/index.php?title=Algebraic_Geometry_Seminar_Fall_2011&diff=3084Algebraic Geometry Seminar Fall 20112011-11-15T05:05:54Z<p>Mehrotra: /* Sukhendu Mehrotra */</p>
<hr />
<div>The seminar meets on Fridays at 2:25 pm in Van Vleck B215.<br />
<br />
The schedule for the previous semester is [https://www.math.wisc.edu/wiki/index.php/Algebraic_Geometry_Seminar_Spring_2011 here].<br />
<br />
== Fall 2011 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|Sep. 23<br />
|Yifeng Liu (Columbia)<br />
|Enhanced Grothendieck's operations and base change theorem for<br />
sheaves on Artin stacks<br />
|Tonghai Yang<br />
|-<br />
|Sep. 30<br />
|Matthew Ballard (UW-Madison)<br />
|''You got your Hodge Conjecture in my matrix factorizations''<br />
|(local)<br />
|-<br />
|Oct. 7<br />
|[http://math.mit.edu/~zyun Zhiwei Yun] (MIT)<br />
|''Cohomology of Hilbert schemes of singular curves''<br />
|Shamgar Gurevich<br />
|-<br />
|Oct. 14<br />
|[http://www.icmat.es/research/international-grants/bobadilla Javier Fernández de Bobadilla] (Instituto de Ciencias Matematicas, Madrid)<br />
|''Nash problem for surfaces''<br />
| L. Maxim<br />
|-<br />
|Oct. 21<br />
|Andrei Caldararu (UW-Madison)<br />
|''The Hodge theorem as a derived self-intersection''<br />
|(local)<br />
|-<br />
|Nov. 11<br />
|John Francis (Northwestern)<br />
|''Integral transforms and Drinfeld centers in derived algebraic geometry''<br />
|Andrei Caldararu<br />
|-<br />
|Nov. 18<br />
|Sukhendu Mehrotra (Madison)<br />
|''Moduli spaces of sheaves on K3 surfaces and generalized deformations''<br />
|(local)<br />
|-<br />
|Dec. 2<br />
|Shamgar Gurevich (Madison)<br />
|''Canonical Hilbert Space: Why? How? and its Categorification''<br />
|-<br />
|Dec. 9<br />
|Sean Paul (Madison)<br />
|''Semistable pairs and quasi-closed orbits''<br />
|-<br />
<br />
|}<br />
<br />
== Spring 2012 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s) <br />
|-<br />
|March 16<br />
|[http://www.math.columbia.edu/~zheng/ Weizhe Zheng] (Columbia)<br />
|''TBD''<br />
|Tonghai Yang<br />
|-<br />
|March 23<br />
|[http://www.nd.edu/~rgrady/cv.pdf Ryan Grady] (Notre Dame)<br />
|''Twisted differential operators as observables in QFT.''<br />
|Caldararu<br />
|-<br />
|May 4<br />
|[http://www.math.sunysb.edu/~mde/ Mark Andrea de Cataldo] (Stony Brook)<br />
|''TBA''<br />
|Maxim<br />
<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Yifeng Liu=== <br />
''TBA''<br />
<br />
===Matthew Ballard=== <br />
''You got your Hodge Conjecture in my matrix factorizations''<br />
<br />
Abstract: I will describe how to prove some new cases of Hodge conjecture<br />
using the following tools: categories of graded matrix factorizations, <br />
the homotopy category of dg-categories, Orlov's Calabi-Yau/Landau-Ginzburg<br />
correspondence, Kuznetsov's relationship between the derived categories<br />
of a certain K3 surface and the Fermat cubic fourfold, and Hochschild homology.<br />
This is joint work with David Favero (Wien) and Ludmil Katzarkov (Miami/Wien).<br />
<br />
===Zhiwei Yun=== <br />
''Cohomology of Hilbert schemes of singular curves''<br />
<br />
Abstract: For a smooth curve, the Hilbert schemes are just symmetric<br />
powers of the curve, and their cohomology is easily computed by the<br />
H^1 of the curve. This is known as Macdonald's formula. In joint work<br />
with Davesh Maulik, we generalize this formula to curves with planar<br />
singularities (which was conjectured by L.Migliorini). In the singular<br />
case, the compactified Jacobian will play an important role in the<br />
formula, and we make use of Ngo's technique in his celebrated proof of<br />
the fundamental lemma.<br />
<br />
===Javier Fernández de Bobadilla=== <br />
''Nash problem for surfaces''<br />
<br />
The space of arcs through the singular set of an algebraic variety has a infinite dimensional scheme structure. In the late sixties Nash proved that it has finitely many irreducible components. He defined a natural mapping from this set of irreducible components to the set of essential divisors of a resolution of singularities. Roughly speaking the set of essential divisors is the set of irreducible components of the exceptional divisor of a resolution whose birational transform is an irreducible component of the exceptional divisor of any other resolution. <br />
<br />
Nash proved that this mapping is injective and proposed to study its bijectivity. In 2003 S. Ishii and J. Kollar gave a counterexample to the surjectivity in dimension at least 4. Recently, in a joint work with M. Pe Pereira, the speaker has settled the bijectivity for surfaces. In this talk I will explain the proof.<br />
<br />
===Andrei Caldararu===<br />
<br />
''The Hodge theorem as a derived self-intersection''<br />
<br />
The Hodge theorem is one of the most important results in complex geometry. It asserts that for a complex projective variety X the topological invariants H^*(X, C) can be refined to new ones that reflect the complex structure. The traditional statement and proof of the Hodge theorem are analytic. Given the multiple applications of the Hodge theorem in algebraic geometry, for many years it has been a major challenge to eliminate this analytic aspect and to obtain a purely algebraic proof of the Hodge theorem.<br />
An algebraic formulation of the Hodge theorem has been known since Grothendieck's work in the early 1970's. However, the first purely algebraic (and very surprising) proof was obtained only in 1991 by Deligne and Illusie, using methods involving reduction to characteristic p. In my talk I shall try to explain their ideas, and how recent developments in the field of derived algebraic geometry make their proof more geometric.<br />
<br />
===John Francis===<br />
<br />
''Integral transforms and Drinfeld centers in derived algebraic geometry''<br />
<br />
For a finite group G, conjugation invariant vector bundles on G have a<br />
universal property with respect to Rep(G): they form its Drinfeld<br />
center. Joint work with David Ben-Zvi and David Nadler generalizes<br />
this result, extending work of Hinich, in the setting of derived<br />
algebraic geometry. We describe a generalization of the Drinfeld<br />
center for a monoidal stable infinity category as a Hochschild<br />
cohomology category. For quasi-coherent sheaves on a perfect stack X,<br />
we prove that its center is equivalent to sheaves on the derived loop<br />
space LX. The structure of this category of sheaves defines an<br />
extended 2-dimensional topological quantum field theory.<br />
<br />
===Sukhendu Mehrotra===<br />
<br />
''Moduli spaces of sheaves on K3 surfaces and generalized deformations''<br />
<br />
It is a result of Mukai that connected components of the moduli space of stable sheaves on<br />
a K3 surface X are holomorphic symplectic varities. As any such component Y deforms in a<br />
21 dimensional family, while the moduli space of K3 surfaces is 20 dimensional, the general<br />
deformation Y of Y will not be be a moduli space of sheaves on a K3. This talk presents an<br />
attempt to associate to such a Y a "non-commutative K3 surface" X<br />
for which the modular description carries over. This joint work with Eyal Markman.</div>Mehrotrahttps://wiki.math.wisc.edu/index.php?title=Algebraic_Geometry_Seminar_Fall_2011&diff=3083Algebraic Geometry Seminar Fall 20112011-11-15T05:05:20Z<p>Mehrotra: /* Abstracts */</p>
<hr />
<div>The seminar meets on Fridays at 2:25 pm in Van Vleck B215.<br />
<br />
The schedule for the previous semester is [https://www.math.wisc.edu/wiki/index.php/Algebraic_Geometry_Seminar_Spring_2011 here].<br />
<br />
== Fall 2011 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|Sep. 23<br />
|Yifeng Liu (Columbia)<br />
|Enhanced Grothendieck's operations and base change theorem for<br />
sheaves on Artin stacks<br />
|Tonghai Yang<br />
|-<br />
|Sep. 30<br />
|Matthew Ballard (UW-Madison)<br />
|''You got your Hodge Conjecture in my matrix factorizations''<br />
|(local)<br />
|-<br />
|Oct. 7<br />
|[http://math.mit.edu/~zyun Zhiwei Yun] (MIT)<br />
|''Cohomology of Hilbert schemes of singular curves''<br />
|Shamgar Gurevich<br />
|-<br />
|Oct. 14<br />
|[http://www.icmat.es/research/international-grants/bobadilla Javier Fernández de Bobadilla] (Instituto de Ciencias Matematicas, Madrid)<br />
|''Nash problem for surfaces''<br />
| L. Maxim<br />
|-<br />
|Oct. 21<br />
|Andrei Caldararu (UW-Madison)<br />
|''The Hodge theorem as a derived self-intersection''<br />
|(local)<br />
|-<br />
|Nov. 11<br />
|John Francis (Northwestern)<br />
|''Integral transforms and Drinfeld centers in derived algebraic geometry''<br />
|Andrei Caldararu<br />
|-<br />
|Nov. 18<br />
|Sukhendu Mehrotra (Madison)<br />
|''Moduli spaces of sheaves on K3 surfaces and generalized deformations''<br />
|(local)<br />
|-<br />
|Dec. 2<br />
|Shamgar Gurevich (Madison)<br />
|''Canonical Hilbert Space: Why? How? and its Categorification''<br />
|-<br />
|Dec. 9<br />
|Sean Paul (Madison)<br />
|''Semistable pairs and quasi-closed orbits''<br />
|-<br />
<br />
|}<br />
<br />
== Spring 2012 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s) <br />
|-<br />
|March 16<br />
|[http://www.math.columbia.edu/~zheng/ Weizhe Zheng] (Columbia)<br />
|''TBD''<br />
|Tonghai Yang<br />
|-<br />
|March 23<br />
|[http://www.nd.edu/~rgrady/cv.pdf Ryan Grady] (Notre Dame)<br />
|''Twisted differential operators as observables in QFT.''<br />
|Caldararu<br />
|-<br />
|May 4<br />
|[http://www.math.sunysb.edu/~mde/ Mark Andrea de Cataldo] (Stony Brook)<br />
|''TBA''<br />
|Maxim<br />
<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Yifeng Liu=== <br />
''TBA''<br />
<br />
===Matthew Ballard=== <br />
''You got your Hodge Conjecture in my matrix factorizations''<br />
<br />
Abstract: I will describe how to prove some new cases of Hodge conjecture<br />
using the following tools: categories of graded matrix factorizations, <br />
the homotopy category of dg-categories, Orlov's Calabi-Yau/Landau-Ginzburg<br />
correspondence, Kuznetsov's relationship between the derived categories<br />
of a certain K3 surface and the Fermat cubic fourfold, and Hochschild homology.<br />
This is joint work with David Favero (Wien) and Ludmil Katzarkov (Miami/Wien).<br />
<br />
===Zhiwei Yun=== <br />
''Cohomology of Hilbert schemes of singular curves''<br />
<br />
Abstract: For a smooth curve, the Hilbert schemes are just symmetric<br />
powers of the curve, and their cohomology is easily computed by the<br />
H^1 of the curve. This is known as Macdonald's formula. In joint work<br />
with Davesh Maulik, we generalize this formula to curves with planar<br />
singularities (which was conjectured by L.Migliorini). In the singular<br />
case, the compactified Jacobian will play an important role in the<br />
formula, and we make use of Ngo's technique in his celebrated proof of<br />
the fundamental lemma.<br />
<br />
===Javier Fernández de Bobadilla=== <br />
''Nash problem for surfaces''<br />
<br />
The space of arcs through the singular set of an algebraic variety has a infinite dimensional scheme structure. In the late sixties Nash proved that it has finitely many irreducible components. He defined a natural mapping from this set of irreducible components to the set of essential divisors of a resolution of singularities. Roughly speaking the set of essential divisors is the set of irreducible components of the exceptional divisor of a resolution whose birational transform is an irreducible component of the exceptional divisor of any other resolution. <br />
<br />
Nash proved that this mapping is injective and proposed to study its bijectivity. In 2003 S. Ishii and J. Kollar gave a counterexample to the surjectivity in dimension at least 4. Recently, in a joint work with M. Pe Pereira, the speaker has settled the bijectivity for surfaces. In this talk I will explain the proof.<br />
<br />
===Andrei Caldararu===<br />
<br />
''The Hodge theorem as a derived self-intersection''<br />
<br />
The Hodge theorem is one of the most important results in complex geometry. It asserts that for a complex projective variety X the topological invariants H^*(X, C) can be refined to new ones that reflect the complex structure. The traditional statement and proof of the Hodge theorem are analytic. Given the multiple applications of the Hodge theorem in algebraic geometry, for many years it has been a major challenge to eliminate this analytic aspect and to obtain a purely algebraic proof of the Hodge theorem.<br />
An algebraic formulation of the Hodge theorem has been known since Grothendieck's work in the early 1970's. However, the first purely algebraic (and very surprising) proof was obtained only in 1991 by Deligne and Illusie, using methods involving reduction to characteristic p. In my talk I shall try to explain their ideas, and how recent developments in the field of derived algebraic geometry make their proof more geometric.<br />
<br />
===John Francis===<br />
<br />
''Integral transforms and Drinfeld centers in derived algebraic geometry''<br />
<br />
For a finite group G, conjugation invariant vector bundles on G have a<br />
universal property with respect to Rep(G): they form its Drinfeld<br />
center. Joint work with David Ben-Zvi and David Nadler generalizes<br />
this result, extending work of Hinich, in the setting of derived<br />
algebraic geometry. We describe a generalization of the Drinfeld<br />
center for a monoidal stable infinity category as a Hochschild<br />
cohomology category. For quasi-coherent sheaves on a perfect stack X,<br />
we prove that its center is equivalent to sheaves on the derived loop<br />
space LX. The structure of this category of sheaves defines an<br />
extended 2-dimensional topological quantum field theory.<br />
<br />
===Sukhendu Mehrotra===<br />
<br />
''Moduli spaces of sheaves on K3 surfaces and generalized deformations''<br />
<br />
It is a result of Mukai that connected components of the moduli space of stable sheaves on<br />
a K3 surface X are holomorphic symplectic varities. As any such component Y deforms in a<br />
21 dimensional family, while the moduli space of K3 surfaces is 20 dimensional, the general<br />
deformation Y of Y will not be be a moduli space of sheaves on a K3. This talk presents an<br />
attempt to associate to such a Y a ''non-commutative K3 surface" X<br />
for which the modular description carries over. This joint work with Eyal Markman.</div>Mehrotrahttps://wiki.math.wisc.edu/index.php?title=Algebraic_Geometry_Seminar_Fall_2011&diff=3082Algebraic Geometry Seminar Fall 20112011-11-15T05:02:19Z<p>Mehrotra: /* Fall 2011 */</p>
<hr />
<div>The seminar meets on Fridays at 2:25 pm in Van Vleck B215.<br />
<br />
The schedule for the previous semester is [https://www.math.wisc.edu/wiki/index.php/Algebraic_Geometry_Seminar_Spring_2011 here].<br />
<br />
== Fall 2011 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|Sep. 23<br />
|Yifeng Liu (Columbia)<br />
|Enhanced Grothendieck's operations and base change theorem for<br />
sheaves on Artin stacks<br />
|Tonghai Yang<br />
|-<br />
|Sep. 30<br />
|Matthew Ballard (UW-Madison)<br />
|''You got your Hodge Conjecture in my matrix factorizations''<br />
|(local)<br />
|-<br />
|Oct. 7<br />
|[http://math.mit.edu/~zyun Zhiwei Yun] (MIT)<br />
|''Cohomology of Hilbert schemes of singular curves''<br />
|Shamgar Gurevich<br />
|-<br />
|Oct. 14<br />
|[http://www.icmat.es/research/international-grants/bobadilla Javier Fernández de Bobadilla] (Instituto de Ciencias Matematicas, Madrid)<br />
|''Nash problem for surfaces''<br />
| L. Maxim<br />
|-<br />
|Oct. 21<br />
|Andrei Caldararu (UW-Madison)<br />
|''The Hodge theorem as a derived self-intersection''<br />
|(local)<br />
|-<br />
|Nov. 11<br />
|John Francis (Northwestern)<br />
|''Integral transforms and Drinfeld centers in derived algebraic geometry''<br />
|Andrei Caldararu<br />
|-<br />
|Nov. 18<br />
|Sukhendu Mehrotra (Madison)<br />
|''Moduli spaces of sheaves on K3 surfaces and generalized deformations''<br />
|(local)<br />
|-<br />
|Dec. 2<br />
|Shamgar Gurevich (Madison)<br />
|''Canonical Hilbert Space: Why? How? and its Categorification''<br />
|-<br />
|Dec. 9<br />
|Sean Paul (Madison)<br />
|''Semistable pairs and quasi-closed orbits''<br />
|-<br />
<br />
|}<br />
<br />
== Spring 2012 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s) <br />
|-<br />
|March 16<br />
|[http://www.math.columbia.edu/~zheng/ Weizhe Zheng] (Columbia)<br />
|''TBD''<br />
|Tonghai Yang<br />
|-<br />
|March 23<br />
|[http://www.nd.edu/~rgrady/cv.pdf Ryan Grady] (Notre Dame)<br />
|''Twisted differential operators as observables in QFT.''<br />
|Caldararu<br />
|-<br />
|May 4<br />
|[http://www.math.sunysb.edu/~mde/ Mark Andrea de Cataldo] (Stony Brook)<br />
|''TBA''<br />
|Maxim<br />
<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Yifeng Liu=== <br />
''TBA''<br />
<br />
===Matthew Ballard=== <br />
''You got your Hodge Conjecture in my matrix factorizations''<br />
<br />
Abstract: I will describe how to prove some new cases of Hodge conjecture<br />
using the following tools: categories of graded matrix factorizations, <br />
the homotopy category of dg-categories, Orlov's Calabi-Yau/Landau-Ginzburg<br />
correspondence, Kuznetsov's relationship between the derived categories<br />
of a certain K3 surface and the Fermat cubic fourfold, and Hochschild homology.<br />
This is joint work with David Favero (Wien) and Ludmil Katzarkov (Miami/Wien).<br />
<br />
===Zhiwei Yun=== <br />
''Cohomology of Hilbert schemes of singular curves''<br />
<br />
Abstract: For a smooth curve, the Hilbert schemes are just symmetric<br />
powers of the curve, and their cohomology is easily computed by the<br />
H^1 of the curve. This is known as Macdonald's formula. In joint work<br />
with Davesh Maulik, we generalize this formula to curves with planar<br />
singularities (which was conjectured by L.Migliorini). In the singular<br />
case, the compactified Jacobian will play an important role in the<br />
formula, and we make use of Ngo's technique in his celebrated proof of<br />
the fundamental lemma.<br />
<br />
===Javier Fernández de Bobadilla=== <br />
''Nash problem for surfaces''<br />
<br />
The space of arcs through the singular set of an algebraic variety has a infinite dimensional scheme structure. In the late sixties Nash proved that it has finitely many irreducible components. He defined a natural mapping from this set of irreducible components to the set of essential divisors of a resolution of singularities. Roughly speaking the set of essential divisors is the set of irreducible components of the exceptional divisor of a resolution whose birational transform is an irreducible component of the exceptional divisor of any other resolution. <br />
<br />
Nash proved that this mapping is injective and proposed to study its bijectivity. In 2003 S. Ishii and J. Kollar gave a counterexample to the surjectivity in dimension at least 4. Recently, in a joint work with M. Pe Pereira, the speaker has settled the bijectivity for surfaces. In this talk I will explain the proof.<br />
<br />
===Andrei Caldararu===<br />
<br />
''The Hodge theorem as a derived self-intersection''<br />
<br />
The Hodge theorem is one of the most important results in complex geometry. It asserts that for a complex projective variety X the topological invariants H^*(X, C) can be refined to new ones that reflect the complex structure. The traditional statement and proof of the Hodge theorem are analytic. Given the multiple applications of the Hodge theorem in algebraic geometry, for many years it has been a major challenge to eliminate this analytic aspect and to obtain a purely algebraic proof of the Hodge theorem.<br />
An algebraic formulation of the Hodge theorem has been known since Grothendieck's work in the early 1970's. However, the first purely algebraic (and very surprising) proof was obtained only in 1991 by Deligne and Illusie, using methods involving reduction to characteristic p. In my talk I shall try to explain their ideas, and how recent developments in the field of derived algebraic geometry make their proof more geometric.<br />
<br />
===John Francis===<br />
<br />
''Integral transforms and Drinfeld centers in derived algebraic geometry''<br />
<br />
For a finite group G, conjugation invariant vector bundles on G have a<br />
universal property with respect to Rep(G): they form its Drinfeld<br />
center. Joint work with David Ben-Zvi and David Nadler generalizes<br />
this result, extending work of Hinich, in the setting of derived<br />
algebraic geometry. We describe a generalization of the Drinfeld<br />
center for a monoidal stable infinity category as a Hochschild<br />
cohomology category. For quasi-coherent sheaves on a perfect stack X,<br />
we prove that its center is equivalent to sheaves on the derived loop<br />
space LX. The structure of this category of sheaves defines an<br />
extended 2-dimensional topological quantum field theory.</div>Mehrotrahttps://wiki.math.wisc.edu/index.php?title=User_talk:Mehrotra&diff=3081User talk:Mehrotra2011-11-15T04:58:42Z<p>Mehrotra: Removing all content from page</p>
<hr />
<div></div>Mehrotrahttps://wiki.math.wisc.edu/index.php?title=User_talk:Mehrotra&diff=3080User talk:Mehrotra2011-11-15T04:57:24Z<p>Mehrotra: New page: Test</p>
<hr />
<div>Test</div>Mehrotra