Student Algebra-Number Theory Archives for Fall 2022 to Spring 2023
Introduction to Deformation Theory
When: Mon, October 4, 2021 - 5:00pmWhere: Kirwan Hall 3206
Speaker: Shin Eui Song (UMD) - https://www.math.umd.edu/~sesong/
Moduli Space of Elliptic Curves
When: Mon, October 11, 2021 - 5:00pmWhere: Kirwan Hall 3206
Speaker: Samuel Bachhuber (UMD) -
Global moduli problems and representability
When: Mon, October 18, 2021 - 5:00pmWhere: Kirwan Hall 3206
Speaker: Jackson Hopper (UMD) -
Local deformation and Algebraization
When: Mon, October 25, 2021 - 5:00pmWhere: Kirwan Hall 3206
Speaker: Shin Eui Song (UMD) - https://www.math.umd.edu/~sesong
Abstract: We first review local deformation functors. Studying local deformations is useful as there may not be any global moduli space. We will review the definition of versal deformations and effective formal deformation. One of the fascinating results is that having an effective formal versal deformation implies that an algebraization exists. This follows from a deep approximation theorem of Artin.
The cotangent complex and applications to deformation theory
When: Mon, November 1, 2021 - 5:00pmWhere: Kirwan Hall 3206
Speaker: Steven Jin (UMD) -
Abstract: Suppose $X$ is a scheme (or Artin stack) over a base scheme $S$. The cotangent complex of $X/S$ is a certain complex $L_{X/S}$ of locally free $\mathcal{O}_X$-modules in the nonpositive degrees that in some sense generalizes the sheaf of relative Kahler differentials. In this talk, we will motivate why such an object might be desirable, and we will sketch some of its most important properties. We will conclude with a discussion of how the cotangent complex controls the deformation theory of $X/S$.
Twining Character Formula for Reductive Groups
When: Fri, February 25, 2022 - 2:00pmWhere: Kirwan Hall 1313
Speaker: Jackson Hopper (UMD)
Abstract: TBA
How shtukas were developed
When: Fri, March 4, 2022 - 2:00pmWhere: Kirwan Hall 1313
Speaker: Shin Eui Song (UMD) - https://www.math.umd.edu/~sesong/
Abstract: To Be Announced
On the dimension of certain affine Deligne-Lusztig varieties
When: Wed, March 9, 2022 - 3:00pmWhere: Kirwan Hall 1313
Speaker: Arghya Sadhukhan (UMD) - https://www-math.umd.edu/people/all-directory/item/1339-arghyas0.html
Abstract: The study of affine Deligne-Lusztig varieties (ADLVs) $X_w(b)$ and their certain union $X(\mu,b)$ in the affine flag varieties arose from the study of Shimura varieties with Iwahori level structure. As such, understanding their geometric properties have been crucial in studying reductions of Shimura varieties in the context of arithmetic geometry; on the other hand, work of Chan et al have explored representation theoretic significance of certain variations of ADLVs in the framework of local Langlands conjectures. In this talk, I will outline some of these developments and discuss certain dimension formula for ADLVs.
Minimal reduction type of affine Springer fibers
When: Wed, March 16, 2022 - 3:00pmWhere: Kirwan Hall 1308
Speaker: Chengze Duan (UMD) -
Abstract: In Kazhdan and Lusztig's original paper on affine Springer fibers, they defined the Kazhdan-Lusztig map from the set of nilpotent orbits to the conjugacy classes in the Weyl group using the loop group. In 2011, Lusztig defined a map in the inverse direction using only the geometry of the group itself and not the loop group. It is conjectured that KL map is a section of the later one. In 2020, Yun introduced a new map, called the minimal reduction type, using the affine Springer fibers, and showed that this is the same as the map defined by Lusztig and is a section of KL map. We will introduce Yun's theory and also talk about the strata he used to prove the results.
Using blowups to solve Steiner's conic problem
When: Wed, March 30, 2022 - 3:00pmWhere: Kirwan Hall 1308
Speaker: Dohoon Kim (UMD) - https://www-math.umd.edu/people/all-directory/item/1506-dohoonk.html
Abstract: In 1848, Jakob Steiner asked for--and answered incorrectly--the number of conics that are tangent to five fixed conics. The problem in his solution was that the intersection of the five corresponding hypersurfaces is not transverse. We can resolve this by blowing up the moduli space of conics along the extraneous intersection. The proper transforms of the five hypersurfaces will then intersect transversally, and we can utilize the Chow ring of the blowup to calculate their intersection class.
Twisted Weyl Character Formula II
When: Wed, April 6, 2022 - 3:00pmWhere: Kirwan Hall 1308
Speaker: Jackson Hopper (UMD) - jacksondhopper.com
Abstract: In my first talk, I presented the statement of the twisted Weyl character formula and gave an overview of its geometric proof, as well as a thorough introduction to the classification of reductive groups as it relates to this theorem. In my second talk I will focus on more of the details of the proof, particularly the combinatorial indexing of varieties used to prove the bijection at the heart of the proof.
A perfectoid approach to the Hodge-Tate decomposition
When: Wed, April 13, 2022 - 3:00pmWhere: Kirwan Hall 1308
Speaker: Steven Jin (UMD) -
Abstract: Suppose $C$ is a complete and algebraically closed extension of $\mathbb{Q}_p$. Let $X/C$ be a smooth proper variety (or more generally a smooth proper rigid space). There is an $E_2$-spectral sequence $E_{2}^{i,j}: H^i(X, \Omega_{X/C}^j)(-j) \implies H^{i+j}(X_{et}, \mathbb{Z}_p) \otimes_{\mathbb{Z}_p} C$. When $X$ is defined over a finite extension $K$ of $\mathbb{Q}_p$, this spectral sequence degenerates (canonically) to yield the Galois-equivariant decomposition
$H^n(X_{\overline{K}, et}, \mathbb{Q}_p) \otimes_{\mathbb{Q}_p} C \cong \bigoplus_{i+j=n} H^i(X, \Omega_{X/K}^j) \otimes_K C(-j).$
In this talk, we will explain how modern perfectoid methods can be used to prove this result. Key ingredients include (i) a study of the pro-étale site of a rigid space and (ii) various properties of the cotangent complex as discussed in my talk last semester. This talk will be aimed at a general audience familiar with algebraic geometry, and I will assume no background in rigid geometry or perfectoid theory.
Affine Deligne Lusztig Varieties and Quantum Bruhat Graphs
When: Wed, April 20, 2022 - 3:00pmWhere: Kirwan Hall 1308
Speaker: Arghya Sadhukhan (UMD) - https://www-math.umd.edu/people/all-directory/item/1339-arghyas0.html
Abstract: First introduced in the context of enumerative geometry to describe the multiplicative structure of the quantum cohomology ring of the complex flag variety, quantum Bruhat graphs have proved to be useful in recent years in Lie theoretic and arithmetic geometric problems. For instance, they encode covering relation in affine Weyl group and give rise to useful description of the admissible set and the Demazure product; these in turn allows us to understand the generic Newton point in Iwahori double cosets of loop groups, and hence find applications in the problems of dimension and irreducible components of certain affine Deligne-Lusztig varieties in the affine flag variety. I’ll discuss these developments and if time permits, report an ongoing work around this theme.
Using equivariant cohomology to solve Steiner's problem
When: Wed, April 27, 2022 - 3:00pmWhere: Kirwan Hall 1308
Speaker: Dohoon Kim (UMD) - https://www-math.umd.edu/people/all-directory/item/1506-dohoonk.html
Abstract: In the previous talk, we solved Steiner’s problem—finding the number conics tangent to five fixed conics—by using the Chow ring of a blowup. This time, we will solve the same problem by using techniques from equivariant cohomology. In particular, we will be using the Atiyah–Bott localization formula, which allows us to calculate—under certain constraints—the integral of an equivariant class was a certain sum over the fixed points. To that end, we will be utilizing a natural torus action on the blowup to calculate the various ingredients of the localization formula, namely the restriction of our class to the fixed points and the (top) Chern classes of the tangent spaces at the fixed points.
Overview of the proof of global Langlands conjecture for GL2
When: Wed, May 4, 2022 - 3:00pmWhere: Kirwan Hall 1308
Speaker: Shin Eui Song (UMD) - https://www.math.umd.edu/~sesong/