Student Algebra-Number Theory Archives for Academic Year 2020


Introduction to p-adic Hodge theory

When: Fri, September 11, 2020 - 4:00pm
Where: Online
Speaker: Shin Eui Song (University of Maryland, College Park) - https://www.math.umd.edu/~sesong
Abstract: We introduce p-adic Hodge theory by describing two different aspects of the theory. The arithmetic side of the theory concerns p-adic representations of the absolute Galois group of a p-adic field. We discuss a toy example on how a p-adic representation arising from an elliptic curve comes from a (semi-)linear algebraic object. On the other hands, the geometric side of the theory concerns on the comparison theory of various cohomology theories of proper smooth varieties. If time permits, we discuss Fontaine's formalism to connect to two perspectives.

Dirac operators in representation theory

When: Fri, September 18, 2020 - 4:00pm
Where: Online
Speaker: Arghya Sadhukhan (UMD) -
Abstract: Dirac operator was introduced into representation theory in the 1970s by Parthasarathy and Atiyah-Schidmt in the context of discrete series representations of real reductive Lie groups. Since then, they have played a crucial role in the enormous project of computing unitary dual of such groups. By work of Harish-Chandra, the study of irreducible unitary representations of G can be reduced to analysis of their associated (g, K) modules, thus converting the problem of understanding typically infinite dimensional analytical objects into a purely algebraic one. To further study these (g, K) modules via their infinitesimal characters, one is then led to consider the action of Dirac operator on them. By a conjecture of Vogan settled in the early 2000s, we can identify such infinitesimal characters for (g, K) modules having non-vanishing Dirac cohomology. I’ll survey these developments and time permitting, discuss some of its ramification in classical topics, such as the generalized Weyl character formula and Borel-Weil-Bott theorem.

p-divisible groups, Dieudonne modules and deformations

When: Fri, September 25, 2020 - 4:00pm
Where: Online
Speaker: Weimin Jiang (UMD) -
Abstract: In this talk, we will first motivate the definition of p-divisible group by looking at the Tate modules of geometric objects in characteristic p. Then we try to use Dieudonne modules, some (semi)-linear algebraic objects to classify p-divisible groups. Finally, we talk about the deformation space of 1-dimensional p-divisible groups and its importance in local Langlands (if time permits).

The equivalence of the constructible topology and the ultrafilter topology

When: Fri, October 2, 2020 - 4:00pm
Where: Online
Speaker: Do Hoon Kim (UMD) -
Abstract: Let A be a commutative ring. It is well-known that the Zariski topology on Spec(A) is generally not Hausdorff. The constructible topology is a refinement of the Zariski topology that is always compact and Hausdorff. In this paper, we define a new topology on Spec(A) using the notion of ultrafilters and show that this topology is equivalent to the constructible topology. In particular, the definition of the ultrafilter topology gives a full description of all the closed sets in the constructible topology.

$P_{L^+G} (\mathrm{Gr}_G)$: The Left-hand side of the geometric Satake correspondence

When: Fri, October 9, 2020 - 4:00pm
Where: Online
Speaker: Jackson Hopper (UMD) -
Abstract: Given a reductive group $G$, the geometric Satake correspondence is an isomorphism between two $\overline{\mathbb{Q}}_{\ell}$-linear tensor categories associated to $G$. This isomorphism builds a bridge between two interpretations of an important category, allowing us to apply methods of study naturally suited to the either description, and has even been used to prove a form of the Langlands correspondence for function fields. On the right-hand side, we have the familiar category of finite-dimensional $\ell$-adic representations of the Langlands dual group $\hat{G}$, equipped with the usual tensor product $\otimes$. On the left-hand side, we have a more mysterious category whose definition and basic properties I will establish in this talk. This will build on previous talks defining both the affine Grassmannian $\mathrm{Gr}_G$ and perverse sheaves. I will extend the definition of perverse sheaves to apply to ind-(separated finite-type schemes), define group equivariance of perverse sheaves, and classify the simple objects of the category $P_{L^+G} (\mathrm{Gr}_G)$. Then I will then prove the category is semisimple, and define the convolution product $\star$ on the simple objects. At this point we can in full awareness state the theorem of geometric Satake correspondence.

Perfect submonoids of dominant weights+something for classification of reductive monoids

When: Fri, October 16, 2020 - 4:00pm
Where: Online
Speaker: Chengze Duan (UMD) -
Abstract: Vinberg introduced the notion of perfect submonoids of dominant weights of G in the study of Vinberg monoids. These perfect submonoids are closely related to tensor product decomposition, which is important in representation theory of algebraic groups. I’ll talk about an explicit description of such submonoids of dominant weights and its relation to classfication of reductive monoids and Vinberg monoids (if time permits).

Local Langlands correspondence and Fargues-Fontaine curve

When: Fri, October 23, 2020 - 4:00pm
Where: Online
Speaker: Shin Eui Song (UMD) - https://www.math.umd.edu/~sesong/index.html
Abstract: An important recent development in local Langlands correspondence for an arbitrary reductive group $G$ over the $p$-adic numbers $\mathbb{Q}_p$ is the construction of a map that sends a smooth irreducible representation of $G(\mathbb{Q}_p)$ to a semi-simple Langlands parameter by Fargues and Scholze. The main strategy is to associate a smooth irreducible representation to a sheaf on $\mathrm{Bun}_G$, the moduli space of $G$-bundles on the Fargues-Fontaine curve. In this survey, we briefly review the general strategy in constructing the Fargues-Scholze map. Then we introduce two constructions of a Fargues-Fontaine curve over a perfectoid field: schematic and adic. In both versions, we uniformize the $p$-adic (resp. perfectoid) punctured unit disk to construct a Fargues-Fontaine curve which then is related via a GAGA type theorem.

Of polynomials, representations et al.

When: Fri, October 30, 2020 - 4:00pm
Where: Online
Speaker: Arghya Sadhukhan (UMD) -
Abstract: Introduced in [KL79] in a study of Springer representations of Weyl group , Kazhdan-Lusztig polynomials have since been ubiquitous in representation theory. Indexed by pairs of elements in Coxeter groups, they have non-negative integer coefficients – a deep fact first proved employing perverse sheaves & BBD theorem (for Weyl groups), and later by Elias-Williamson via Soergel bimodules. Among other things, they describe multiplicities of simple \( \mathfrak{g} \) into Verma modules (& therefore gives character formulas for such simple modules), encodes information about intersection homology of Schubert varieties, and can be used to construct interesting representations of Hecke algebras using Lusztig’s idea of cells in Weyl group. I’ll discuss different facets of this beautiful story, and if time permits, we will see some recent developments in Elias-Williamson’s work.

Deformations of Galois Representations

When: Fri, November 6, 2020 - 4:00pm
Where: Online
Speaker: Steven Jin (UMD) -
Abstract: Let $\Pi$ be a profinite group and $h:A_1\to A_0$ a continuous homomorphism of local rings. Then this induces a map $\tilde{h}:GL_N(A_1)\to GL_N(A_0)$. If $\rho_0:\Pi\to GL_N(A_0)$ is a continuous representation, a deformation of $\rho_0$ to $A_1$ is a certain equivalence class of liftings $\rho_1:\Pi\to GL_N(A_1)$. When $\Pi$ is taken to be the Galois group of a number field (or of the maximal algebraic extension of a number field unramified outside of a finite set of primes), it makes sense to speak about Galois deformations. In this talk, we will survey the theory of Galois deformations. Littered throughout, we will also touch on a number of arithmetic and geometric applications to motivate the study.

Vinberg monoids and its application to affine Springer fibers

When: Fri, November 13, 2020 - 4:00pm
Where: Online
Speaker: Chengze Duan (UMD) -
Abstract: Vinberg monoids have many applications and one of them is in the theory of affine Springer fibers. Many important features of affine Springer fibers, such as nonemptiness, dimension formula (and maybe irreducible components) can be extended to generalized affine Springer fibers using Vinberg monoids. In this talk I will talk about the theory and construction of Vinberg monoids and the theory of usual affine Springer fibers first. Then I will talk about the above application of Vinberg monoids.

Non-archimedean geometry, through Tate, Berkovich and Huber.

When: Fri, December 4, 2020 - 4:00pm
Where: Online
Speaker: Weimin Jiang (UMD) -
Abstract: This will be an introductory talk on non-archimedean geometry by Tate, Berkovich and Huber. We can do “geometry” over any field(even any base scheme) by the tools of algebraic geometry, but when the ground field is equipped with a complete absolute value, we hope to get something more. Classical examples and methods in real and complex geometry are not adapted well in non-archimedean case, which leads to the discovery of various constructions. We will mostly focus on Tate’s rigid analytic space and Huber’s adic space constructions.

The simplest properties for any moduli involving primes

When: Tue, February 9, 2021 - 4:00pm
Where: Online
Speaker: Shin Eui Song (UMD) -
Abstract: Ramanujan discovered three strikingly simple congruences involving prime numbers and partition numbers. After the death of Ramanujan, Hardy published a manuscript containing proofs of all three congruences that employ the Eisenstein series under Ramanujan's name. In this talk, we will first briefly review the modular forms and define the Eisenstein series. Next, we will define the Serre derivative which can be thought of as a derivative for modular forms, and study its properties. Finally, we will prove Ramanujan's congruences using the theory of modular forms.

Cells in (finite and affine) Weyl groups

When: Tue, February 16, 2021 - 4:00pm
Where: Online
Speaker: Arghya Sadhukhan (UMD) -
Abstract: Hecke algebras arise naturally in the representation theory of reductive algebraic groups over finite or p-adic fields, and as such they occur as certain deformation of finite or affine Weyl groups respectively. A major breakthrough in this direction was achieved in the 1979 paper of Kazhdan and Lusztig, where they introduced the notion of cells of an arbitrary Coxeter group by establishing a new `canonical’ basis of the corresponding Hecke algebra. In subsequent work, Lusztig formulated 15 conjectural properties which capture essential features of cells for all choices of parameters and developed the representation theory of Hecke algebras associated with finite Weyl groups on the basis of P1–P15. These conjectures are known to hold for finite and affine Weyl groups in the equal parameter case, thanks to geometric interpretation of K-L polynomials in terms of Schubert variety. However in the situation of unequal parameter, very little is known. In this continuation of my talk from last semester, I will survey some of these topics.

Root valuation strata in the adjoint quotient of loop algebras

When: Tue, March 2, 2021 - 4:00pm
Where: Online
Speaker: Chengze Duan (UMD) -
Abstract: In the theory of affine Springer fiber, Bezrukavnikov developed a dimension formula for integral regular semisimple elements. Moreover, we can associate to each element a nice discrete invariant such that the dimension only depends on that invariant. This leads to the root valuation strata in the loop algebra and also in its adjoint quotient. We will talk about this and the geometry of it.