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.
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.
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).
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.
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.
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).
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.
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