Algebra and Algebraic Geometry Seminar Fall 2024: Difference between revisions

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(Created page with "The seminar normally meets 2:30-3:30pm on Fridays, in the room '''Van Vleck''' '''B139'''. ==Algebra and Algebraic Geometry Mailing List== *Please join the AGS mailing list by sending an email to ags+subscribe@g-groups.wisc.edu to hear about upcoming seminars, lunches, and other algebraic geometry events in the department (it is possible you must be on a math department computer to use this link). ==Spring 2024 Schedule== {| cellpadding="8" ! align="left" |date ! align=...")
 
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The seminar normally meets 2:30-3:30pm on Fridays, in the room '''Van Vleck''' '''B139'''.
The seminar normally meets 2:30-3:30pm on Fridays, in the room '''Van Vleck''' '''B131'''.
==Algebra and Algebraic Geometry Mailing List==
==Algebra and Algebraic Geometry Mailing List==
*Please join the AGS mailing list by sending an email to ags+subscribe@g-groups.wisc.edu to hear about upcoming seminars, lunches, and other algebraic geometry events in the department  (it is possible you must be on a math department computer to use this link).
*Please join the AGS mailing list by sending an email to ags+subscribe@g-groups.wisc.edu to hear about upcoming seminars, lunches, and other algebraic geometry events in the department  (it is possible you must be on a math department computer to use this link).
==Spring 2024 Schedule==
==Fall 2024 Schedule==
{| cellpadding="8"
{| cellpadding="8"
! align="left" |date
! align="left" |date
Line 9: Line 9:
! align="left" |host/link to talk
! align="left" |host/link to talk
|-
|-
|February 16
|September 27
|Sean Cotner (Michigan)
|Joshua Mundinger (Madison)
|[[#Sean Cotner|Schemes of homomorphisms]]
|[[#Joshua Mundinger|Hochschild homology and the HKR spectral sequence]]
|Josh
|local
|-
|-
|February 23
|October 4
|[https://sites.google.com/view/ylf/ Lingfei Yi (Minnesota)]
|Dima Arinkin (Madison)
|[[#Lingfei Yi|Slices in the loop spaces of symmetric varieties]]
|[[#Dima Arinkin|Derived category of the stacky compactified Jacobian]]
|Dima/Josh
|local
|-
|March 1
|Shravan Patankar (UIC)
|[[#Shravan Patankar|The absolute integral closure in equicharacteristic zero]]
|Dima/Josh
|-
|-
|March 18 ('''Monday''') 2:30-3:30pm in '''B123'''
|November 14 (2-3pm, Birge 348)
|[https://www.universiteitleiden.nl/en/staffmembers/marton-hablicsek Marton Hablicsek] (Leiden University)
|Yunfeng Jiang (Kansas)
|[[#Marton Hablicsek|A formality result for logarithmic Hochschild (co)homology]]
|Intro pre-talk in GAGS
|Dima
|Andrei/Ruobing
|-
|-
|April 19
|November 15
|Teresa Yu (Michigan)
|Yunfeng Jiang (Kansas)
|[[#Teresa Yu|Standard monomial theory modulo Frobenius in characteristic two]]
|TBA
|Dima/Jose
|Andrei/Ruobing
|-
|-
|May 2 ('''Thursday''') '''3:30-4:30pm''' in '''B321'''
|December 13
|John Cobb (UW)
|Thomas Brazelton (Harvard)
|[[#John Cobb|Multigraded Stillman’s Conjecture]]
|TBA
|local
|Andrei/Josh
|}
|}


==Abstracts==
==Abstracts==


===Sean Cotner===
===Joshua Mundinger===
'''Schemes of homomorphisms'''
'''Hochschild homology and the HKR spectral sequence'''
 
Given two algebraic groups G and H, it is natural to ask whether the set Hom(G, H) of homomorphisms from G to H can be parameterized in a useful way. In general, this is not possible, but there are well-known partial positive results (mainly due to Grothendieck). In this talk I will describe essentially optimal conditions on G and H under which Hom(G, H) is a scheme. There will be many examples, and we will see how a geometric perspective on Hom(G, H) can be useful in studying concrete questions. Time permitting, I will discuss some aspects of the theory of Hom schemes over a base.
 
===Lingfei Yi===
'''Slices in the loop spaces of symmetric varieties'''
 
Let X be a symmetric variety. J. Mars and T. Springer constructed conical transversal slices to the closure of Borel orbits on X and used them to show that the IC-complexes for the orbit closures are pointwise pure. This is an important geometric ingredient in their work providing a more geometric approach to the results of Lusztig-Vogan. In the talk, I will discuss a generalization of Mars-Springer's construction of transversal slices to the setting of the loop space LX of X where we consider closures of spherical orbits on LX. I will also explain its applications to the formality conjecture in the relative Langlands duality. If time permits, I will discuss similar constructions for Iwahori orbits. This is a joint work with Tsao-Hsien Chen.
 
===Shravan Patankar===
'''The absolute integral closure in equicharacteristic zero'''
 
In spite of being large and non noetherian, the absolute integral closure of a domain R, R^{+}, carries great importance in positive characteristic commutative algebra and algebraic geometry. Recent advances due to Bhatt hint at a similar picture in mixed characteristic. In equicharacteristic zero however, this object seems largely unexplored. We answer a series of natural questions which suggest that it might play a similar central role in the study of singularities and algebraic geometry in equicharacteristic zero. More precisely, we show that it is rarely coherent, and facilitates a characterization of regular rings similar to Kunz's theorem. Both of these results, have in turn, applications back to positive characteristics.
 
===Marton Hablicsek===
'''A formality result for logarithmic Hochschild (co)homology'''
 
Hochschild homology is a foundational invariant for associate algebras, schemes, stacks, etc. For smooth and proper varieties X over a field of characteristic 0, Hochschild homology and its variants, like cyclic homology, are closely related to Hodge cohomology and to de Rham cohomology. For affine schemes, the Hochschild invariants are, in general, infinite dimensional. In this talk, we extend Hochschild homology to logarithmic schemes, in particular to compactifications, i.e, to pairs (X,D) where X is a smooth and proper variety and D is a simple normal crossing divisor. Using the formality theorem of Arinkin and Căldăraru, we recover an HKR isomorphism for logarithmic schemes relating logarithmic Hochschild homology to logarithmic differential forms. I will also discuss simple applications of our framework. This is a joint work with Francesca Leonardi and Leo Herr.


===Teresa Yu===
Hochschild homology of an algebraic variety carries the Hochschild-Konstant-Rosenberg (HKR) filtration. In characteristic zero, this filtration is split, yielding the HKR decomposition of Hochschild homology. In characteristic p, this filtration does not split, giving rise to the HKR spectral sequence. We describe the first nonzero differential of this spectral sequence. Our description is related to the Atiyah class.
'''Standard monomial theory modulo Frobenius in characteristic two'''


Over a field of characteristic zero, standard monomial theory and determinantal ideals provide an explicit decomposition of polynomial rings into simple GL_n-representations, which have characters given by Schur polynomials. In this talk, we present work towards developing an analogous theory for polynomial rings over a field of characteristic two modulo a Frobenius power of the maximal ideal generated by all variables. In particular, we obtain a filtration by modular GL_n-representations whose characters are given by certain truncated Schur polynomials, thus proving a conjecture by Gao-Raicu-VandeBogert in the characteristic two case. This is joint work with Laura Casabella.
===Dima Arinkin===
'''Derived category of the stacky compactified Jacobian'''


===John Cobb===
Abstract:
'''Multigraded Stillman’s Conjecture'''
The Jacobian of a smooth projective curve is an abelian variety which is identified with its own dual. This implies that its derived category carries a non-trivial auto-equivalence - the Fourier-Mukai transform. When the curve has planar singularities, the Jacobian is no longer compact (and, in particular, not an abelian variety), but it turns out that the Fourier-Mukai transform still exists, provided we compactify the Jacobian. The transform can be viewed as the `classical limit' of the geometric Langlands correspondence.


In 2000, Mike Stillman conjectured that the projective dimension of a homogeneous ideal in a standard graded polynomial ring can be bounded just in terms of the number and degrees of its generators. I’ll describe the Ananyan-Hochster principle important to its proof, how to package this up using ultraproducts, and use this to characterize the polynomial rings graded by any abelian group that possess a Stillman bound.
In this talk, I will explore what happens when the curve becomes reducible. From the point of view of the geometric Langlands conjecture, it is important to work with the compactified Jacobian viewed as a stack (rather than the corresponding moduli space).
In my talk I will show that this also leads to certain issues, and in fact that the most general version of the statement is inconsistent, while
more conservative versions are true.

Latest revision as of 12:01, 15 October 2024

The seminar normally meets 2:30-3:30pm on Fridays, in the room Van Vleck B131.

Algebra and Algebraic Geometry Mailing List

  • Please join the AGS mailing list by sending an email to ags+subscribe@g-groups.wisc.edu to hear about upcoming seminars, lunches, and other algebraic geometry events in the department (it is possible you must be on a math department computer to use this link).

Fall 2024 Schedule

date speaker title host/link to talk
September 27 Joshua Mundinger (Madison) Hochschild homology and the HKR spectral sequence local
October 4 Dima Arinkin (Madison) Derived category of the stacky compactified Jacobian local
November 14 (2-3pm, Birge 348) Yunfeng Jiang (Kansas) Intro pre-talk in GAGS Andrei/Ruobing
November 15 Yunfeng Jiang (Kansas) TBA Andrei/Ruobing
December 13 Thomas Brazelton (Harvard) TBA Andrei/Josh

Abstracts

Joshua Mundinger

Hochschild homology and the HKR spectral sequence

Hochschild homology of an algebraic variety carries the Hochschild-Konstant-Rosenberg (HKR) filtration. In characteristic zero, this filtration is split, yielding the HKR decomposition of Hochschild homology. In characteristic p, this filtration does not split, giving rise to the HKR spectral sequence. We describe the first nonzero differential of this spectral sequence. Our description is related to the Atiyah class.

Dima Arinkin

Derived category of the stacky compactified Jacobian

Abstract: The Jacobian of a smooth projective curve is an abelian variety which is identified with its own dual. This implies that its derived category carries a non-trivial auto-equivalence - the Fourier-Mukai transform. When the curve has planar singularities, the Jacobian is no longer compact (and, in particular, not an abelian variety), but it turns out that the Fourier-Mukai transform still exists, provided we compactify the Jacobian. The transform can be viewed as the `classical limit' of the geometric Langlands correspondence.

In this talk, I will explore what happens when the curve becomes reducible. From the point of view of the geometric Langlands conjecture, it is important to work with the compactified Jacobian viewed as a stack (rather than the corresponding moduli space). In my talk I will show that this also leads to certain issues, and in fact that the most general version of the statement is inconsistent, while more conservative versions are true.

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