Geometry and Topology Seminar 2019-2020: Difference between revisions

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== Fall 2011 ==
The [[Geometry and Topology]] seminar meets in room '''901 of Van Vleck Hall''' on '''Fridays''' from '''1:20pm - 2:10pm'''.
<br>
For more information, contact Shaosai Huang.


The seminar will be held  in room 901 of Van Vleck Hall on Fridays from 1:20pm - 2:10pm
[[Image:Hawk.jpg|thumb|300px]]
 
 
== Spring 2020 ==


{| cellpadding="8"
{| cellpadding="8"
Line 9: Line 14:
!align="left" | host(s)
!align="left" | host(s)
|-
|-
|September 9
|Feb. 7
|[http://www.math.wisc.edu/~maribeff/ Gloria Mari Beffa] (UW Madison)
|Xiangdong Xie  (Bowling Green University)
|[[#Gloria Mari Beffa (UW Madison)|
| Minicourse 1: Quasiconformal structure on the ideal boundary of Gromov hyperbolic spaces
''The pentagram map and generalizations: discretizations of AGD flows'']]
|(Dymarz)
|[local]
|-
|-
|September 16
|Feb. 14
|[http://www.math.umn.edu/~zhux0086/ Ke Zhu] (University of Minnesota)
|Xiangdong Xie  (Bowling Green University)
|[[#Ke Zhu (University of Minnesota)|
| Minicourse 2: Quasiconformal structure on the ideal boundary of Gromov hyperbolic spaces
''Thin instantons in G2-manifolds and
|(Dymarz)
Seiberg-Witten invariants'']]
|[http://www.math.wisc.edu/~oh/ Yong-Geun]
|-
|-
|September 23
|Feb. 21
|[http://www.math.wisc.edu/~ache/ Antonio Ache] (UW Madison)
|Xiangdong Xie (Bowling Green University)
|[[#Antonio Ache (UW Madison)|
| Minicourse 3: Quasiconformal structure on the ideal boundary of Gromov hyperbolic spaces
''Obstruction-Flat Asymptotically Locally Euclidean Metrics'']]
|(Dymarz)
|[local]
|-
|-
|September 30
|Feb. 28
|[http://people.maths.ox.ac.uk/mackayj/ John Mackay] (Oxford University)
|Kuang-Ru Wu (Purdue University)
|[[#John Mackay (Oxford University)|
|Griffiths extremality, interpolation of norms, and Kahler quantization
''What does a random group look like?'']]
|(Huang)
|[http://www.math.wisc.edu/~dymarz/ Tullia]
|-
|-
|October 7
|Mar. 6
|[http://mypage.iu.edu/~fisherdm/ David Fisher] (Indiana University)
|Yuanqi Wang (University of Kansas)
|[[#David Fisher (Indiana University)|
|Moduli space of G2−instantons on 7−dimensional product manifolds
''Hodge-de Rham theory for infinite dimensional bundles and local rigidity'']]
|(Huang)
|[http://www.math.wisc.edu/~rkent/ Richard and Tullia]
|-
|-
|October 14
|Mar. 13 <b>CANCELED</b>
|[http://www.cpt.univ-mrs.fr/~lanneau/ Erwan Lanneau] (University of Marseille, CPT)
|Karin Melnick (University of Maryland)
|[[#Erwan Lanneau (University of Marseille, CPT)|
|A D'Ambra Theorem in conformal Lorentzian geometry
''Dilatations of pseudo-Anosov homeomorphisms and Rauzy-Veech induction'']]
|(Dymarz)
|[http://www.math.wisc.edu/~jeanluc/ Jean Luc]
|-
|-
|October 21
|<b>Mar. 25</b> <b>CANCELED</b>
|[http://www.math.wisc.edu/~rsong/ Ruifang Song] (UW Madison)
|Joerg Schuermann (University of Muenster, Germany)
|[[#Ruifang Song (UW Madison)|
|An introduction to Poincare-Hopf theorems for singular spaces via characteristic cycles
''The Picard-Fuchs equations of Calabi-Yau hypersurfaces in partial flag varieties'']]
|(Maxim)
|[local]
|-
|-
|October 24 ( with Geom. analysis seminar)
|Mar. 27 <b>CANCELED</b>
|[http://math.univ-lyon1.fr/~ovsienko/ Valentin Ovsienko] (University of Lyon)
|David Massey (Northeastern University)
|[[#Valentin Ovsienko (University of Lyon)|
|Extracting easily calculable algebraic data from the vanishing cycle complex
''The pentagram map and generalized friezes of Coxeter'']]
|(Maxim)
|[http://www.math.wisc.edu/~maribeff/ Gloria]
|-
|<b>Apr. 10</b> <b>CANCELED</b>
|Antoine Song (Berkeley)
|TBA
|(Chen)
|}
 
== Fall 2019 ==
 
{| cellpadding="8"
!align="left" | date
!align="left" | speaker
!align="left" | title
!align="left" | host(s)
|-
|-
|November 4
|Oct. 4
| Steven Simon (NYU)
|Ruobing Zhang (Stony Brook University)
|[[#Steven Simon (NYU))|
| Geometric analysis of collapsing Calabi-Yau spaces
''Equivariant Analogues of the Ham Sandwich Theorem'']]
|(Chen)
|[http://www.math.wisc.edu/~maxim/ Max]
|-
|-
|November 18
|[http://www.math.tamu.edu/~zelenko/ Igor Zelenko] (Texas A&M University)
|[[#Igor Zelenko (Texas A&M University)|
''On geometry of curves of flags of constant type'']]
|[http://www.math.wisc.edu/~maribeff/ Gloria]
|-
|-
|December 1 at 2 PM in Ingraham 114
|Oct. 25
| Bing Wang (Simons Center for Geometry and Physics)
|Emily Stark (Utah)
|[[#Bing Wang (Simons Center for Geometry and Physics)|
| Action rigidity for free products of hyperbolic manifold groups
''Uniformization of algebraic varieties.''
|(Dymarz)
NOTE SPECIAL PLACE AND TIME: Thursday, December 1 at 2 PM in Ingraham 114.]]
|[Jeff]
|-
|-
|December 2
|Nov. 8
|[http://www.math.uic.edu/~ddumas/ David Dumas] (University of Illinois at Chicago)
|Max Forester (University of Oklahoma)
|[[#David Dumas (University of Illinois at Chicago)|
|Spectral gaps for stable commutator length in some cubulated groups
''Real and complex boundaries in the character variety'']]
|(Dymarz)
|[http://www.math.wisc.edu/~rkent/ Richard]
|-
|-
|December 9
|Nov. 22
|[http://math.stanford.edu/~bfclarke/home/Home.html Brian Clarke]  (Stanford)
|Yu Li (Stony Brook University)
|[[#Brian Clarke  (Stanford)|
|On the structure of Ricci shrinkers
''TBA'']]
|(Huang)
|[http://www.math.wisc.edu/~jeffv/ Jeff]
|-
|-
|}
|}


== Abstracts ==
==Spring Abstracts==
 
===Gloria Mari Beffa (UW Madison)===
''The pentagram map and generalizations: discretizations of AGD flows''
 
GIven an n-gon one can join every other vertex with a segment and find the intersection
of two consecutive segments. We can form a new n-gon with these intersections, and the
map taking the original n-gon to the newly found one is called the pentagram map. The map's
properties when defined on pentagons are simple to describe (it takes its name from this fact),
but the map turns out to have a unusual  number of other properties and applications.
 
In this talk I will give a quick review of recent results by Ovsienko, Schwartz and Tabachnikov on the
integrability of the pentagram map and I will describe on-going efforts to generalize the pentagram
map to higher dimensions using possible connections to Adler-Gelfand-Dikii flows. The talk will
NOT be for experts and will have plenty of drawings, so come and join us.
 
===Ke Zhu (University of Minnesota)===
''Thin instantons in G2-manifolds and
Seiberg-Witten invariants''
 
For two nearby disjoint coassociative submanifolds $C$ and $C'$ in a $G_2$-manifold, we construct thin instantons with boundaries lying on $C$
and $C'$ from regular $J$-holomorphic curves in $C$. It is a high dimensional analogue of holomorphic stripes with boundaries on two nearby Lagrangian submanifolds $L$ and $L'$. We explain its relationship with the Seiberg-Witten invariants for $C$. This is a joint work with Conan Leung and Xiaowei Wang.
 
===Antonio Ache  (UW Madison)===
Obstruction-Flat Asymptotically Locally Euclidean Metrics
 
Given an even dimensional Riemannian manifold <math>(M^{n},g)</math> with <math>n\ge 4</math>, it was shown in the work of Charles Fefferman and Robin Graham on conformal invariants the existence of a non-trivial 2-tensor which involves <math>n</math> derivatives of the metric, arises as the first variation of a conformally invariant and vanishes for metrics that are conformally Einstein. This tensor is called the Ambient Obstruction tensor and is a higher dimensional generalization of the Bach tensor in dimension 4. We show that any asymptotically locally Euclidean (ALE) metric which is obstruction flat and scalar-flat must be ALE of a certain optimal order using a technique developed by Cheeger and Tian for Ricci-flat metrics. We also show a singularity removal theorem for obstruction-flat metrics with isolated <math>C^{0}</math>-orbifold singularities. In addition, we show that our methods apply to more general systems. This is joint work with Jeff Viaclovsky.
 
===John Mackay (Oxford University)===
''What does a random group look like?''
 
Twenty years ago, Gromov introduced his density model for random groups, and showed when the density parameter is less than one half a random group is, with overwhelming probability, (Gromov) hyperbolic. Just as the classical hyperbolic plane has a circle as its boundary at infinity, hyperbolic groups have a boundary at infinity which carries a
canonical conformal structure.


In this talk, I will survey some of what is known about random groups, and how the geometry of a hyperbolic group corresponds to the structure of its boundary at infinity.  I will outline recent work showing how Pansu's conformal dimension, a variation on Hausdorff dimension, can be
===Xiangdong Xie===
used to give a more refined geometric picture of random groups at small densities.


===David Fisher (Indiana University)===
The quasiconformal structure on the ideal boundary of Gromov hyperbolic spaces has played
''Hodge-de Rham theory for infinite dimensional bundles and local rigidity''
an important role in various  rigidity questions in geometry and group theory.
In these talks I  shall give an introduction to this topic.  In the first talk I will introduce Gromov hyperbolic spaces, define their ideal boundary,  and  discuss their basic properties.  In the second and third talks I will define the visual metrics on the ideal boundary, explain the connection between quasiisometries of  Gromov hyperbolic space and quasiconformal maps on  their ideal boundary, and indicate  how the quasiconformal structure on the ideal boundary can be used to deduce rigidity.


It is well known that every cohomology class on a manifold
===Kuang-Ru Wu===
can be represented by a harmonic form.  While this fact continues to hold
for cohomology with coefficients in finite dimensional vector bundles, it
is also fairly well known that it fails for infinite dimensional bundles.  In
this talk, I will formulate a notion of a harmonic cochain in group cohomology
and explain what piece of the cohomology can be represented by
harmonic cochains.
I will use these ideas to prove a vanishing theorem that motivates a family of
generalizations of property (T) of Kazhdan.  If time permits, I will
discuss applications
to local rigidity of group actions.


===Erwan Lanneau (University of Marseille, CPT)===
Following Kobayashi, we consider Griffiths negative complex Finsler bundles, naturally leading us to introduce Griffiths extremal Finsler metrics. As we point out, this notion is closely related to the theory of interpolation of norms, and is characterized by an equation of complex Monge– Ampere type, whose corresponding Dirichlet problem we solve. As applications, we prove that Griffiths extremal Finsler metrics quantize solutions to a natural PDE in Kahler geometry, related to the construction of flat maps for the Mabuchi metric. This is joint work with Tamas Darvas.
''Dilatations of pseudo-Anosov homeomorphisms and Rauzy-Veech induction''


In this talk I will explain the link between pseudo-Anosov homeomorphisms and Rauzy-Veech induction. We will see how to derive properties on the dilatations of these homeomorphisms (I will recall the definitions) and as an application, we will use the Rauzy-Veech-Yoccoz induction to give lower bound on dilatations.
===Yuanqi Wang===
This is a common work with Corentin Boissy (Marseille).
$G_{2}-$instantons are 7-dimensional analogues of flat connections in dimension 3. It is part of Donaldson-Thomas’ program to generalize the fruitful gauge theory in dimensions 2,3,4 to dimensions 6,7,8. The moduli space of $G_{2}-$instantons, with virtual dimension $0$, is  expected to have interesting  geometric structure and yield enumerative invariant for the underlying $7-$dimensional manifold.  


In this talk, in some reasonable special cases and a fairly complete manner, we will describe the relation between the moduli space of $G_{2}-$instantons and an algebraic geometry moduli on a Calabi-Yau 3-fold.


===Ruifang Song (UW Madison)===
===Karin Melnick===
''The Picard-Fuchs equations of Calabi-Yau hypersurfaces in partial flag varieties''


We introduce a system of differential equations associated to a smooth algebraic variety X acted by a complex Lie group G and a G-linearlized line bundle L on X. We show that this system is holonomic and thus its solution space is finite dimensional assuming G acts on X with finitely many orbits. When X is a partial flag variety, we show that this system gives the Picard-Fuchs system of Calabi-Yau hypersurfaces in X. When X is a toric variety, our construction recovers GKZ systems and extended GKZ systems, which play important roles in studying periods of Calabi-Yau hypersurfaces in toric varieties. This is based on joint work with Bong Lian and Shing-Tung Yau.
D'Ambra proved in 1988 that the isometry group of a compact, simply connected, real-analytic Lorentzian manifold must be compact. I will discuss my recent theorem that the conformal group of such a manifold must also be compact, and how it relates to the Lorentzian Lichnerowicz Conjecture.


===Valentin Ovsienko (University of Lyon)===
===Joerg Schuermann===
''The pentagram map and generalized friezes of Coxeter''


The pentagram map is a discrete integrable system on the moduli space of n-gons in the projective plane (which is a close relative of the moduli space of genus 0 curves with n marked points). The most interesting properties of the pentagram map is its relations to the theory of cluster algebras and to the classical integrable systems (such as the Boussinesq equation). I will talk of the recent results proving the integrability as well as of the algebraic and arithmetic properties of the pentagram map.
We give an introduction to Poincare-Hopf theorems for singular spaces via characteristic cycles, based on stratified Morse theory for constructible functions. The corresponding local index of an isolated critical point (in a stratified sense) of a one-form depends on the constructible function, specializing for different choices to well known indices like the radial, GSV or Euler obstruction index.
In particular, I will introduce the space of 2-frieze patterns generalizing that of the classical Coxeter friezes and define the structure of cluster manifold on this space. The talk is based on joint works with Sophie Morier-Genoud, Richard Schwartz and Serge Tabachnikov.


===Steven Simon (NYU)===
===David Massey===
''Equivariant Analogues of the Ham Sandwich Theorem''


The Ham Sandwich Theorem, one of the earliest applications of algebraic topology to geometric combinatorics, states that under generic conditions any n finite Borel measures on R^n can be bisected by a single hyperplane. Viewing this theorem as a Z_2-symmetry statement for measures, we generalize the theorem to other finite groups, notably the finite subgroups of the spheres S^1 and S^3, in each case realizing group symmetry on Euclidian space as group symmetries of its Borel measures by studying the topology of associated spherical space forms. Direct equipartition statements for measures are given as special cases. We shall also discuss the contributions of the tangent bundles of these manifolds in answering similar questions.  
Given a complex analytic function on an open subset U  of C<sup>n+1</sup>, one may consider the complex of sheaves of vanishing cycles along f of the constant sheaf Z<sub>U</sub>. This complex encodes on the cohomological level the reduced cohomology of the Milnor fibers of f at each of f<sup>-1</sup>(0). The question is: how does one calculate (ideally, by hand)  any useful numbers about this vanishing cycle complex? One answer is to look at the Lê numbers of f. We will discuss the precise relationship between these objects/numbers.


===Igor Zelenko (Texas A&M University)===
===Antoine Song===
''On geometry of curves of flags of constant type''


The talk is devoted to the (extrinsic) geometry of curves of flags of a vector space $W$ with respect to the action of a subgroup $G$ of the $GL(W)$. We develop an algebraic version of Cartan method of equivalence or an analog of Tanaka prolongation for such problem. Under some natural assumptions on the subgroup $G$ and on the flags, one can pass from the filtered objects to the corresponding graded objects and describe the construction of canonical bundles of moving frames for these curves in the language of pure Linear Algebra. The scope of applicability of the theory includes geometry of natural classes of curves of flags with respect to reductive linear groups or their parabolic subgroups. As simplest examples, this includes the projective and affine geometry of curves.
TBA


Our motivation to study such equivalence problems comes from the new approach to the geometry of structures of nonholonomic nature on manifolds such as vector distributions, sub-Riemannian structure etc. This approach is based on the Optimal Control Theory and it consists of the reduction of the equivalence problem for such nonholonomic geometric structures to the (extrinsic) differential geometry of curves in Lagrangian Grassmannians and, more generally, of curves of flags of isotropic and coisotropic subspaces in a linear symplectic space with respect to the action of the Linear Symplectic Group. The application of the general theory to the geometry of such curves case will be discussed in more detail.
==Fall Abstracts==


===Bing Wang (Simons Center for Geometry and Physics)===
===Ruobing Zhang===
''Uniformization of algebraic varieties''


For algebraic varieties of general type with
This talk centers on the degenerations of Calabi-Yau metrics. We will focus on the interactions between algebraic degenerations and metric convergence with highly singular behaviors in the collapsing case. As the complex structures degenerate, the collapsing Calabi-Yau metrics may exhibit various wild geometric properties with highly non-algebraic features.
mild singularities, we show the Bogmolov-Yau inequality
holds. If equality is attained, then this variety is a
global quotient of complex hyperbolic space away from
a subvariety. This is will be a technical version of
the speaker's colloquium talk.


First, as motivating examples, we will describe our recent results on the new collapsing mechanisms of K3 surfaces. Next, we will switch to higher dimensions and we will exhibit some entirely new constructions of degenerating Calabi-Yau metrics which are expected to work in broader contexts. Complex structures degeneration will be accurately characterized by the bubbling and singularity analysis in a geometric manner.


===David Dumas (University of Illinois at Chicago)===
===Emily Stark===
''Real and complex boundaries in the character variety''


The set of holonomy representations of complex projective structures
The relationship between the large-scale geometry of a group and its algebraic structure can be studied via three notions: a group's quasi-isometry class, a group's abstract commensurability class, and geometric actions on proper geodesic metric spaces. A common model geometry for groups G and G' is a proper geodesic metric space on which G and G' act geometrically. A group G is action rigid if every group G' that has a common model geometry with G is abstractly commensurable to G. For example, a closed hyperbolic n-manifold group is not action rigid for all n at least three. In contrast, we show that free products of closed hyperbolic manifold groups are action rigid. Consequently, we obtain the first examples of Gromov hyperbolic groups that are quasi-isometric but do not virtually have a common model geometry. This is joint work with Daniel Woodhouse.
on a compact Riemann surface is a submanifold of the SL(2,C) character
variety of the fundamental group. We will discuss the real- and
complex-analytic geometry of this manifold and its interaction with
the Morgan-Shalen compactification of the character variety. In
particular we show that the subset consisting of holonomy
representations that extend over a given hyperbolic 3-manifold group
(of which the surface is an incompressible boundary) is discrete.


===Brian Clarke (Stanford)===
===Max Forester===
''Ricci Flow, Analytic Stability, and the Space of Kähler Metrics''


I will consider the space of all Kähler metrics on a fixed, compact, complex manifold as a submanifold of the manifold of all Riemannian metrics.  The geometry induced on it in this way coincides with a Riemannian metric first defined by Calabi in the 1950s. After giving a detailed study of the Riemannian distance function - in particular determining the completion of the space of Kähler metrics - I will give a new analytic stability criterion for the existence of a Kähler--Einstein metric on a Fano manifold in terms of the Ricci flow and the distance function.  Additionally, I will describe a result showing that the Kähler--Ricci flow converges as soon as it converges in the very weak metric sense. This is joint work with Yanir Rubinstein.
I will discuss stable commutator length (scl) in groups, and some gap theorems for the scl spectrum. Such results say that for various groups, scl of an element is always either zero or is larger than some uniform constant. I will discuss the cases of right-angled Artin groups and certain right-angled Coxeter groups. This is joint work with Pallavi Dani, Ignat Soroko, and Jing Tao.


===Yu Li===
We develop a structure theory for non-collapsed Ricci shrinkers without any curvature condition. As an application, we show that any Ricci shrinker whose second eigenvalue of the curvature operator is positive must be a quotient of sphere.


[[Fall-2010-Geometry-Topology]]
== Archive of past Geometry seminars ==
2018-2019  [[Geometry_and_Topology_Seminar_2018-2019]]
<br><br>
2017-2018 [[Geometry_and_Topology_Seminar_2017-2018]]
<br><br>
2016-2017  [[Geometry_and_Topology_Seminar_2016-2017]]
<br><br>
2015-2016: [[Geometry_and_Topology_Seminar_2015-2016]]
<br><br>
2014-2015: [[Geometry_and_Topology_Seminar_2014-2015]]
<br><br>
2013-2014: [[Geometry_and_Topology_Seminar_2013-2014]]
<br><br>
2012-2013: [[Geometry_and_Topology_Seminar_2012-2013]]
<br><br>
2011-2012: [[Geometry_and_Topology_Seminar_2011-2012]]
<br><br>
2010: [[Fall-2010-Geometry-Topology]]

Latest revision as of 18:56, 3 September 2020

The Geometry and Topology seminar meets in room 901 of Van Vleck Hall on Fridays from 1:20pm - 2:10pm.
For more information, contact Shaosai Huang.

Hawk.jpg


Spring 2020

date speaker title host(s)
Feb. 7 Xiangdong Xie (Bowling Green University) Minicourse 1: Quasiconformal structure on the ideal boundary of Gromov hyperbolic spaces (Dymarz)
Feb. 14 Xiangdong Xie (Bowling Green University) Minicourse 2: Quasiconformal structure on the ideal boundary of Gromov hyperbolic spaces (Dymarz)
Feb. 21 Xiangdong Xie (Bowling Green University) Minicourse 3: Quasiconformal structure on the ideal boundary of Gromov hyperbolic spaces (Dymarz)
Feb. 28 Kuang-Ru Wu (Purdue University) Griffiths extremality, interpolation of norms, and Kahler quantization (Huang)
Mar. 6 Yuanqi Wang (University of Kansas) Moduli space of G2−instantons on 7−dimensional product manifolds (Huang)
Mar. 13 CANCELED Karin Melnick (University of Maryland) A D'Ambra Theorem in conformal Lorentzian geometry (Dymarz)
Mar. 25 CANCELED Joerg Schuermann (University of Muenster, Germany) An introduction to Poincare-Hopf theorems for singular spaces via characteristic cycles (Maxim)
Mar. 27 CANCELED David Massey (Northeastern University) Extracting easily calculable algebraic data from the vanishing cycle complex (Maxim)
Apr. 10 CANCELED Antoine Song (Berkeley) TBA (Chen)

Fall 2019

date speaker title host(s)
Oct. 4 Ruobing Zhang (Stony Brook University) Geometric analysis of collapsing Calabi-Yau spaces (Chen)
Oct. 25 Emily Stark (Utah) Action rigidity for free products of hyperbolic manifold groups (Dymarz)
Nov. 8 Max Forester (University of Oklahoma) Spectral gaps for stable commutator length in some cubulated groups (Dymarz)
Nov. 22 Yu Li (Stony Brook University) On the structure of Ricci shrinkers (Huang)

Spring Abstracts

Xiangdong Xie

The quasiconformal structure on the ideal boundary of Gromov hyperbolic spaces has played an important role in various rigidity questions in geometry and group theory. In these talks I shall give an introduction to this topic. In the first talk I will introduce Gromov hyperbolic spaces, define their ideal boundary, and discuss their basic properties. In the second and third talks I will define the visual metrics on the ideal boundary, explain the connection between quasiisometries of Gromov hyperbolic space and quasiconformal maps on their ideal boundary, and indicate how the quasiconformal structure on the ideal boundary can be used to deduce rigidity.

Kuang-Ru Wu

Following Kobayashi, we consider Griffiths negative complex Finsler bundles, naturally leading us to introduce Griffiths extremal Finsler metrics. As we point out, this notion is closely related to the theory of interpolation of norms, and is characterized by an equation of complex Monge– Ampere type, whose corresponding Dirichlet problem we solve. As applications, we prove that Griffiths extremal Finsler metrics quantize solutions to a natural PDE in Kahler geometry, related to the construction of flat maps for the Mabuchi metric. This is joint work with Tamas Darvas.

Yuanqi Wang

$G_{2}-$instantons are 7-dimensional analogues of flat connections in dimension 3. It is part of Donaldson-Thomas’ program to generalize the fruitful gauge theory in dimensions 2,3,4 to dimensions 6,7,8. The moduli space of $G_{2}-$instantons, with virtual dimension $0$, is expected to have interesting geometric structure and yield enumerative invariant for the underlying $7-$dimensional manifold.

In this talk, in some reasonable special cases and a fairly complete manner, we will describe the relation between the moduli space of $G_{2}-$instantons and an algebraic geometry moduli on a Calabi-Yau 3-fold.

Karin Melnick

D'Ambra proved in 1988 that the isometry group of a compact, simply connected, real-analytic Lorentzian manifold must be compact. I will discuss my recent theorem that the conformal group of such a manifold must also be compact, and how it relates to the Lorentzian Lichnerowicz Conjecture.

Joerg Schuermann

We give an introduction to Poincare-Hopf theorems for singular spaces via characteristic cycles, based on stratified Morse theory for constructible functions. The corresponding local index of an isolated critical point (in a stratified sense) of a one-form depends on the constructible function, specializing for different choices to well known indices like the radial, GSV or Euler obstruction index.

David Massey

Given a complex analytic function on an open subset U of Cn+1, one may consider the complex of sheaves of vanishing cycles along f of the constant sheaf ZU. This complex encodes on the cohomological level the reduced cohomology of the Milnor fibers of f at each of f-1(0). The question is: how does one calculate (ideally, by hand) any useful numbers about this vanishing cycle complex? One answer is to look at the Lê numbers of f. We will discuss the precise relationship between these objects/numbers.

Antoine Song

TBA

Fall Abstracts

Ruobing Zhang

This talk centers on the degenerations of Calabi-Yau metrics. We will focus on the interactions between algebraic degenerations and metric convergence with highly singular behaviors in the collapsing case. As the complex structures degenerate, the collapsing Calabi-Yau metrics may exhibit various wild geometric properties with highly non-algebraic features.

First, as motivating examples, we will describe our recent results on the new collapsing mechanisms of K3 surfaces. Next, we will switch to higher dimensions and we will exhibit some entirely new constructions of degenerating Calabi-Yau metrics which are expected to work in broader contexts. Complex structures degeneration will be accurately characterized by the bubbling and singularity analysis in a geometric manner.

Emily Stark

The relationship between the large-scale geometry of a group and its algebraic structure can be studied via three notions: a group's quasi-isometry class, a group's abstract commensurability class, and geometric actions on proper geodesic metric spaces. A common model geometry for groups G and G' is a proper geodesic metric space on which G and G' act geometrically. A group G is action rigid if every group G' that has a common model geometry with G is abstractly commensurable to G. For example, a closed hyperbolic n-manifold group is not action rigid for all n at least three. In contrast, we show that free products of closed hyperbolic manifold groups are action rigid. Consequently, we obtain the first examples of Gromov hyperbolic groups that are quasi-isometric but do not virtually have a common model geometry. This is joint work with Daniel Woodhouse.

Max Forester

I will discuss stable commutator length (scl) in groups, and some gap theorems for the scl spectrum. Such results say that for various groups, scl of an element is always either zero or is larger than some uniform constant. I will discuss the cases of right-angled Artin groups and certain right-angled Coxeter groups. This is joint work with Pallavi Dani, Ignat Soroko, and Jing Tao.

Yu Li

We develop a structure theory for non-collapsed Ricci shrinkers without any curvature condition. As an application, we show that any Ricci shrinker whose second eigenvalue of the curvature operator is positive must be a quotient of sphere.

Archive of past Geometry seminars

2018-2019 Geometry_and_Topology_Seminar_2018-2019

2017-2018 Geometry_and_Topology_Seminar_2017-2018

2016-2017 Geometry_and_Topology_Seminar_2016-2017

2015-2016: Geometry_and_Topology_Seminar_2015-2016

2014-2015: Geometry_and_Topology_Seminar_2014-2015

2013-2014: Geometry_and_Topology_Seminar_2013-2014

2012-2013: Geometry_and_Topology_Seminar_2012-2013

2011-2012: Geometry_and_Topology_Seminar_2011-2012

2010: Fall-2010-Geometry-Topology

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