Algebra-Number Theory Archives for Fall 2022 to Spring 2023
Nef and abundant divisors, semiampleness and canonical bundle formula
When: Mon, September 13, 2021 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Priyankur Chaudhuri (UMD) - https://sites.google.com/view/priyankurc/home
Abstract: In the 1980's, Kawamata showed that if the canonical divisor of a normal projective variety with possibly mild singularities is nef and abundant, then it is semiample (i.e. some multiple of it moves without base points). From the point of view of the abundance conjecture, this is considered to be an important result. In this talk, using canonical bundle formulas, I will discuss some generalizations of this well known theorem. The question we consider is this: if L in Pic X is a line bundle such that both L and L + K_X are nef and L + K_X abundant, then is L + K_X semiample? We will answer some special cases of this question, in particular the cases when either K_X is effective or when the stable base locus of L has codimension at
least 2. In particular, this extends Kawamata's theorem to the setting of generalized abundance.
2-dimensional stable pairs on 4-folds
When: Wed, September 15, 2021 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Amin Gholampour (University of Maryland) - www.math.umd.edu/~amingh
Abstract: I will discuss a 2-dimensional stable pair theory of nonsingular complex 4-folds that is parallel to Pandharipande-Thomas' 1-dimensional stable pair theory of 3-folds. These stable pairs are related to 2-dimensional subschemes of the 4-fold via wall-crossings in the space of polynomial stability conditions. In Calabi-Yau case, we apply Oh-Thomas theory to define invariants counting these stable pairs under some restrains. If time allows, I will talk about some examples and applications. This is a joint work with Yunfeng Jiang and Jason Lo.
Higher arithmetic theta series
When: Mon, September 20, 2021 - 2:00pmWhere: Online
Speaker: Tony Feng (MIT) - https://www.mit.edu/~fengt/ -
Abstract: Arithmetic theta series are incarnations of theta functions in arithmetic algebraic geometry. The first examples were constructed by Kudla as generating series of special cycles on Shimura varieties. Their conjectural key features are (1) modularity of the generating series, and (2) the arithmetic Siegel-Weil formula, relating their enumerative geometry to the first derivative of Eisenstein series at special values. In joint work with Zhiwei Yun and Wei Zhang, we construct "higher" arithmetic theta series on moduli spaces of shtukas, which we conjecture to also enjoy (1) modularity and (2) a higher arithmetic Siegel-Weil formula relating their enumerative geometry to all derivatives of Eisenstein series at special values. We prove several results towards these conjectures, drawing upon ideas from Ngo's proof of the Fundamental Lemma in addition to new ingredients from Springer theory and derived algebraic geometry.
Stable trace formula for Shimura varieties of abelian type, I
When: Mon, October 4, 2021 - 2:00pmWhere: Kirwan Hall 3206 AND on zoom: https://umd.zoom.us/j/96890967721
Speaker: Yihang Zhu (UMD) - http://math.umd.edu/~yhzhu/
Abstract:
In this series of three talks, we report on the joint work with M. Kisin and S. W. Shin. (preprint, http://math.umd.edu/~yhzhu/KSZ.pdf) (The third talk will be given by Shin.)
We consider the alternating trace of a Hecke operator away from p and a Frobenius power at p acting on the compact support cohomology of a Shimura variety of abelian type with hyperspecial level at p. We show that this is equal to the sum of the elliptic parts of the stable trace formulas for the endoscopic groups with respect to well-chosen test functions, proving a conjecture of Kottwitz.
In the first talk, we give some historical background, state the problem, and discuss our strategy of reducing the result to a form of the Langlands-Rapoport Conjecture where certain "controlled twists" are allowed. This form of Langlands-Rapoport strengthens what was proved in earlier work of Kisin.
Stable trace formula for Shimura varieties of abelian type, II
When: Wed, October 6, 2021 - 2:00pmWhere: Kirwan Hall 3206 AND on zoom: https://umd.zoom.us/j/96890967721
Speaker: Yihang Zhu (UMD) - http://math.umd.edu/~yhzhu/
Abstract:
In this series of three talks, we report on the joint work with M. Kisin and S. W. Shin. (preprint, http://math.umd.edu/~yhzhu/KSZ.pdf) (The third talk will be given by Shin.)
We consider the alternating trace of a Hecke operator away from p and a Frobenius power at p acting on the compact support cohomology of a Shimura variety of abelian type with hyperspecial level at p. We show that this is equal to the sum of the elliptic parts of the stable trace formulas for the endoscopic groups with respect to well-chosen test functions, proving a conjecture of Kottwitz.
In the second talk, we discuss some ingredients in the proof of the Langlands-Rapoport-tau Conjecture introduced in the first talk. In the case of Hodge type, we use Breuil-Kisin modules and a recent purity result of Ansch\"utz to construct certain "non-standard" lattices in the rational Dieudonn\'e modules of the reductions of special points. These lattices serve as "marking points" that play a key role in the proof.
Prismatic interpretation of torsion and pathology in p-adic geometry
When: Mon, October 18, 2021 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Shizhang Li (Michigan) -
Abstract: I will report an ongoing joint work with Tong Liu, concerning the structure of a certain submodule inside prismatic cohomology of a smooth proper scheme over a p-adic ring of integers. I will explain how this part of prismatic cohomology causes various pathologies, then say a few corresponding consequences of our structural result. If time permits, I shall also mention an interesting example, which negatively answers a question of Breuil.
Canonical heights on Shimura varieties and the Andre-Oort conjecture
When: Tue, October 26, 2021 - 2:00pmWhere: Online
Speaker: Ananth Shankar (University of Wisconsin, Madison) - https://sites.google.com/view/ashankar/home
Abstract: Let S be a Shimura variety. The Andre-Oort conjecture posits that the Zariski closure of special points must be a sub Shimura subvariety of S. The Andre-Oort conjecture for A_g (the moduli space of principally polarized Abelian varieties) — and therefore its sub Shimura varieties — was proved by Jacob Tsimerman. However, this conjecture was unknown for Shimura varieties without a moduli interpretation. I will describe joint work with Jonathan Pila and Jacob Tsimerman (with an appendix by Esnault-Groechenig) where we prove the Andre Oort conjecture in full generality.
Relating the Artin-Tate and Birch-Swinnerton-Dyer conjectures
When: Mon, November 1, 2021 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Niranjan Ramachandran (UMD) - https://www.math.umd.edu/~atma/
Abstract: This will be a report on recent work with S. Lichtenbaum and T. Suzuki.
Geometry of Hilbert modular eigenvarieties
When: Fri, November 5, 2021 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Lynnelle Ye (Stanford) - https://lynnelle.github.io/
Abstract: It is a longstanding question, first asked by Coleman and Mazur in 1998, whether eigenvarieties satisfy the valuative criterion for properness over weight space. We will show that eigenvarieties parametrizing $p$-adic overconvergent cuspidal Hilbert modular eigenforms for a totally real field $F$ are proper over integer weights in this sense, generalizing a result of Hattori for the case when the residue degrees of $p$ are at most $2$ (itself a generalization of Buzzard-Calegari's original work for the Coleman-Mazur eigencurve). This requires extending overconvergent Hilbert eigenforms farther than they have been constructed in the literature (by Andreatta-Iovita-Pilloni). We will also discuss conditional results for non-integer weights when $p$ is totally split.
On the Kottwitz conjecture for local shtuka spaces
When: Mon, November 8, 2021 - 2:00pmWhere: Online
Speaker: David Hansen (Max-Planck Institut für Mathematik, Bonn) -
Abstract: The cohomology of local Shimura varieties, and of more general spaces of local shtukas, is of fundamental interest in the Langlands program. On the one hand, it is supposed to realize instances of the local Langlands correspondence. On the other hand, there is a tight relationship with the cohomology of global Shimura varieties. In recent joint work with Kaletha and Weinstein, we proved the first general results towards the Kottwitz conjecture, which predicts how supercuspidal L-packets contribute to the cohomology of local shtuka spaces. I will review this story, and give some overview of the ideas which enter into our proof. The key idea in our argument - namely, that the Kottwitz conjecture should follow from some form of the Lefschetz-Verdier fixed point formula - was already formulated by Michael Harris in the '90s. However, executing this idea brings substantial technical challenges. I will try to emphasize the new ingredients which allow us to implement this idea in full generality.
Recent developments in étale cohomology
When: Wed, November 10, 2021 - 2:00pmWhere: Online
Speaker: David Hansen (Max-Planck Institut für Mathematik, Bonn) -
Abstract: I'll talk about some recent foundational developments in etale cohomology:
i) A flexible six-functor formalism for "Zariski-constructible" sheaves on rigid spaces (joint work with Bhargav Bhatt).
ii) A new "relative" variant of perverse sheaves (joint work with Peter Scholze).
If time permits, I'll mention some tantalizing open problems in these directions.
Galois action on the pro-algebraic fundamental group
When: Mon, November 15, 2021 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Alexander Petrov (Harvard) - https://people.math.harvard.edu/~apetrov/
Abstract: Given a smooth variety over a number field, it turns out that the action of the Galois group on the pro-algebraic completion of the etale fundamental group is almost everywhere unramified and is de Rham (that is, any finite-dimensional subrepresentation of the ring of functions on this pro-algebraic completion is de Rham in the sense of Fontaine). The main ingredient for this is a statement in relative p-adic Hodge theory that, roughly speaking, says that the geometric monodromy of all local systems is captured by the geometric monodromy of de Rham local systems: for any Q_p-local system L on a smooth variety X over a p-adic field K there exists a de Rham local system M such that the restriction of L to X_{\bar{K}} embeds into the restriction of M.
Therefore, the space of functions on the pro-algebraic completion of the fundamental group serves as a source of Galois representations satisfying the natural conditions satisfied by Galois representations coming from etale cohomology of algebraic varieties. Converesely, it turns out that any semi-simple representation coming from etale cohomology of an algebraic variety is a subquotient of the space of functions on the pro-algebraic completion of the pi_1 of the projective line with 3 punctures. The proof is a purely algebro-geometric argument relying on the Belyi theorem.
The references are https://arxiv.org/abs/2012.13372 and https://arxiv.org/abs/2109.09301
Decomposition of the de Rham complex and Steenrod operations
When: Wed, November 17, 2021 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Alexander Petrov (Harvard) - https://people.math.harvard.edu/~apetrov/
Abstract: A classical result of Deligne and Illusie states that for a smooth variety X over F_p that admits a lift to Z/p^2 the de Rham complex decomposes as a direct sum of its cohomology sheaves, if dimension of X is less than p. It is expected that this result is false without the bound on the dimension, but, surprisingly, no counterexample has been found so far.
I'll explain how one can describe the first potentially non-trivial extension in the de Rham complex using other invariants of the variety X. The argument is motivated by the definition of Steenrod power operations in topology and relies crucially on the existence of prismatic cohomology. Along the way, it gives a way to compute a part of the Sen operator on Hodge-Tate cohomology, recently defined by Bhatt and Lurie, and allows to produce an example where this Sen endomorphism is not semisimple.
Stable trace formula for Shimura varieties of abelian type, III
When: Mon, November 29, 2021 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Sug Woo Shin (UC Berkeley) -
Abstract: In a recent paper with Mark Kisin and Yihang Zhu, we proved the stable trace formula for Shimura varieties of abelian type. (This was the subject of Zhu’s talks in early October.) We will discuss applications of this formula. After a broad introduction to such applications, we will specialize to the problem of describing the cohomology of Shimura varieties (joint work with Kisin and Zhu).
Robba cohomology for dagger spaces in positive characteristic
When: Mon, December 6, 2021 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Koji Shimizu (UC Berkeley) -
Abstract: We will discuss a p-adic cohomology theory for rigid analytic varieties with overconvergent structure (dagger spaces) over a local field of characteristic p. After explaining the motivation, we will define a site (Robba site) and discuss its basic properties.
Completed prismatic F-crystals in the relative case
When: Wed, December 8, 2021 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Koji Shimizu (UC Berkeley) -
Abstract: Bhatt and Scholze introduced the absolute prismatic site of a p-adic ring and proved the equivalence between prismatic F-crystals and lattices in crystalline representations in the CDVR case with perfect residue field. We will define a wider category of completed prismatic F-crystals in the relative case and explain its relation to the category of crystalline local systems. (Joint work with Heng Du, Tong Liu, and Yong Suk Moon)
Compatibility of the Gan-Takeda and Fargues-Scholze local Langlands, I
When: Mon, January 31, 2022 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Linus Hamann (Princeton University) -
Abstract: Given a prime p, a finite extension L/Qp, a connected p-adic
reductive group G/L,
and a smooth irreducible representation \pi of G(L), Fargues-Scholze
recently attached a
semisimple Weil parameter to such \pi, giving a general candidate for
the local Langlands correspondence. It is natural to ask whether this
construction is compatible with known instances of
the correspondence after semisimplification. For G = GL_n and its
inner forms, Fargues-Scholze
and Hansen-Kaletha-Weinstein show that the correspondence is
compatible with the
correspondence of Harris-Taylor/Henniart. We verify a similar
compatibility for
G = GSp_4 and its unique non-split inner form G = GU_2(D), where D is
the quaternion division
algebra over L, assuming that L/Q_p is unramified and p > 2. In this
case, the local Langlands
correspondence has been constructed by Gan-Takeda and Gan-Tantono. Analogous
to the case of GL_n and its inner forms, this compatibility is proven
by describing the Weil group
action on the cohomology of a local Shimura variety associated to
GSp_4, using basic uniformization of abelian type Shimura varieties
due to Shen, combined with various global results
of Kret-Shin and Sorensen on Galois representations in the cohomology
of global
Shimura varieties associated to inner forms of GSp_4 over a totally
real field.
Compatibility of the Gan-Takeda and Fargues-Scholze local Langlands, II
When: Wed, February 2, 2022 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Linus Hamann (Princeton University) -
Abstract: In this talk, we will explain some applications of the
compatibility result proven in the previous talk. In particular, using
recent results of Hansen and Fargues and Scholze on the spectral action
one can use compatibility to verify a strong form of the Kottwitz
conjecture that works even beyond the minuscule case, verifying some
form of Fargues' conjecture on eigensheaves attached to supercuspidal
parameters in the process. Time permitting, we will explain some joint
work in progress with Alexander Bertoloni-Meli and Kieh-Hieu Nguyen
extending compatibility to odd unramified unitary groups, as well as
possible future applications to torsion in the cohomology of Shimura
varieties and generalizations of an averaging formula of Shin.
Jets spaces and SL_n(Z)
When: Wed, February 23, 2022 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Nero Budur (KU Leuven) - https://perswww.kuleuven.be/%7Eu0089821/
Abstract: Using jet spaces we show that the space of representations of the fundamental group of a compact Riemann surface of genus >1 has rational singularities. We apply this to show that the number of irreducible complex representations of SL_n(Z) of dimension at most m grows at most as the square of m, for fixed n>2.
Reduction of Brauer classes on K3 surfaces
When: Wed, March 9, 2022 - 2:00pmWhere: Online
Speaker: Sarah Frei (Rice University) - https://math.rice.edu/~sf31/
Abstract: For a very general polarized K3 surface over the rational
numbers, it is a consequence of the Tate conjecture that the Picard
rank jumps upon reduction modulo any prime. This jumping in the Picard
rank is countered by a drop in the size of the Brauer group. In this
talk, I will report on joint work with Brendan Hassett and Anthony
Várilly-Alvarado, in which we consider the following: Given a
non-trivial Brauer class on a very general polarized K3 surface over
Q, how often does this class become trivial upon reduction modulo
various primes? This has implications for the rationality of
reductions of cubic fourfolds as well as reductions of twisted derived
equivalent K3 surfaces.
Intersection K-theory of quotient singularities
When: Mon, March 14, 2022 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Tudor Padurariu (Columbia University ) - https://www.math.columbia.edu/~tpad
Abstract: For a class of quotient stacks V/G for G a reductive group and V a symmetric G-representation, we propose a definition of the intersection K-theory of the quotient V//G, which is a Q-vector space denoted IK(V//G). There is a Chern character from IK(V//G) to intersection cohomology IH(V//G) and IK(V//G) satisfies Kirwan surjectivity. We construct IK(V//G) as a direct summand of the Grothendieck group (tensor with Q) of a noncommutative resolution of singularities of V//G constructed by Spenko-Van den Bergh. We explain extensions of some of these results for stacks with good moduli spaces. Time permitting, we discuss an application of this construction to a PBW-type theorem for (Kontsevich-Soibelman) K-theoretic Hall algebras.
Derived microlocalization
When: Wed, March 16, 2022 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Adeel Khan (Academia Sinica (Taiwan)) - https://www.preschema.com/
Abstract: I will discuss a certain categorification of Kontsevich's virtual fundamental class, which I call derived microlocalization, and some applications to topics such as singular support of étale sheaves and categorified Donaldson-Thomas theory.
Gopakumar-Vafa invariants and support theorems
When: Mon, March 28, 2022 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Lutian Zhao (UMD) - https://www-math.umd.edu/people/postdocs-and-visitors/item/1649-lutianzhao.html
Abstract: In this talk I will define and and describe the curve counting invariant called the Gopakumar-Vafa invariant. This is a conjectural counting invariant that involves many support theorems for perverse sheaves. I will describe the support theorem of Ngo and how this theorem predicts one of the counting invariant correspondence: the Gopakumar-Vafa/Pandharipande-Thomas correspondence. This will be a vast generalization of the Macdonald formula for family of algebraic curves.
Values of zeta functions at s=1/2
When: Wed, March 30, 2022 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Niranjan Ramachandran (UMD) - https://www.math.umd.edu/~atma/
Abstract: We will discuss some results about the special values at s=1/2 of zeta functions of varieties over finite fields.
Gopakumar-Vafa invariants and support theorems, part 2
When: Mon, April 4, 2022 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Lutian Zhao (University of Maryland) -
Abstract: In this talk I will define and and describe the curve counting invariant called the Gopakumar-Vafa invariant. This is a conjectural counting invariant that involves many support theorems for perverse sheaves. I will describe the support theorem of Ngo and how this theorem predicts one of the counting invariant correspondence: the Gopakumar-Vafa/Pandharipande-Thomas correspondence. This will be a vast generalization of the Macdonald formula for family of algebraic curves.
Tableau formulas for skew Schubert polynomials
When: Wed, April 6, 2022 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Harry Tamvakis (UMD) - https://www.math.umd.edu/~harryt/
Abstract: Schubert polynomials are canonical representatives for the
(equivariant) Schubert classes on complete flag manifolds for the classical
Lie groups. In general, they are given by explicit but complicated formulas
in terms of combinatorial data coming from the Weyl group. We will define
the skew elements of the Weyl group and show how their Schubert polynomials
satisfy tableau formulas which are the natural generalization in this setting of
Littlewood's tableau formula for Schur polynomials.
Surfaces on cubic fourfolds via stability conditions
When: Mon, April 11, 2022 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Alexander Perry (University of Michigan) - http://www-personal.umich.edu/~arper/
Abstract: Any cubic fourfold has an associated K3 category, defined by Kuznetsov as a semiorthogonal component of the derived category. I will explain some geometric applications of stability conditions on this K3 category, in particular the construction of special surfaces on cubic fourfolds and relations to the rationality problem. This is based on joint work in progress with Arend Bayer, Aaron Bertram, and Emanuele Macri.
Obstructions to rationality of conic bundles threefolds
When: Wed, April 13, 2022 - 2:00pmWhere: Online
Speaker: Isabel Vogt (Brown) -
Abstract: n this talk I'll discuss joint work with Sarah Frei, Lena Ji, Soumya Sankar and Bianca Viray on the problem of determining when a geometrically rational variety is birational to projective space over its field of definition. Benoist--Wittenberg recently refined the classical intermediate Jacobian obstruction of Clemens--Griffiths by considering torsors under the intermediate Jacobian of a geometrically rational threefold. By work of Benoist--Wittenberg and Kuznetsov--Prokhorov, this obstruction is strong enough to characterize rationality of geometrically rational Fano threefolds of geometric Picard rank 1. Moving into higher Picard rank, we compute this obstruction for conic bundles over P^2. As a consequence of our work, when the ground field is the real numbers, we show that neither the topological obstruction nor the refined intermediate Jacobian obstruction is sufficient to determine rationality.
On canonical integral models of Shimura varieties
When: Fri, April 15, 2022 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Patrick Daniels (University of Michigan) - http://www-personal.umich.edu/~pjdaniel/
Abstract: Pappas and Rapoport have recently conjectured the existence of "canonical integral models" for Shimura varieties with parahoric level structure, which are characterized using Scholze's theory of p-adic shtukas. We will discuss the characterization of Pappas and Rapoport, and illustrate the conjecture using the example of zero-dimensional Shimura varieties, where (surprisingly) the theory is already nontrivial.
Canonical forms of neural ideals
When: Wed, April 20, 2022 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Rebecca R.G. () - https://sites.google.com/site/rebeccargmath/
Abstract: A neural ideal captures the firing pattern of a set of neurons (called a neural code), turning problems in coding theory into algebraic questions. In Curto, Itskov, et al. 2013, the authors gave an algorithm for computing the canonical form of a neural ideal, a unique set of pseudomonomial generators (products of $x_i$ and $1-x_j$) corresponding to the neural code. In Gunturkun, Jeffries, and Sun 2020, the authors gave a technique for polarizing neural ideals, to turn them into monomial ideals while retaining the structure of the canonical form. In joint work with Hugh Geller (Sewanee, The University of the South), we give a simple criterion for determining whether a polarized neural ideal is in canonical form. In order to do this, we examine in detail how the generators of an ideal change throughout the algorithm for computing the canonical form. We also describe the canonical forms of some classes of polarized neural ideals.
Enumerative geometry of small resolutions of a singular octic double solid
When: Mon, April 25, 2022 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Sheldon Katz (University of Illinois) - https://faculty.math.illinois.edu/~katz/
Abstract: Forty years ago, Herb Clemens studied the geometry and topology of the octic double solid X, a double cover of P^3 branched along a degree 8 hypersurface S. He also studied the situation where S has nodes, in which case X correspondingly has nodes; and then he studied small resolutions X’ of X. In this talk, I focus on the case where S is the determinant of a generic 8x8 matrix of linear forms. This X’ is a compact complex manifold of dimension 3 that is not even symplectic, yet string theory suggests that it supports a rich theory of enumerative geometry. Furthermore, considerations of mirror symmetry suggest that H_2(X’,Z) has non-trivial torsion subgroup Z_2, partitioning the space of B-fields into two connected components. This observation leads to an assignment of a Z_2 “charge” to curve classes on X’. The result is a Z_2 refinement of the usual formula for the (all genus) Gromov-Witten free energy in terms of Gopakumar-Vafa invariants, expressed as a Z_2 refinement of the usual Gopakumar-Vafa invariants. The validity of many of these numbers can be checked directly in low degree by familiar methods of algebraic geometry. A derived equivalence between a noncommutative resolution of X and the complete intersection of four quadrics in P^7 plays a key role in obtaining the prediction for the Gromov-Witten free energies using ideas and techniques of string theory. This is the simplest of many examples of non-symplectic small resolutions of nodal Calabi-Yau threefolds which are amenable to techniques of physics and appear to have similar enumerative properties.
This talk is based on joint work in progress with Albrecht Klemm, Thorsten Schimannek, and Eric Sharpe.
Connected components of affine Deligne-Lusztig varieties
When: Wed, April 27, 2022 - 8:30pmWhere: Online
Speaker: Sian Nie (Chinese Academy of Sciences) -
Abstract: Affine Deligne-Lusztig varieties, first introduced by Rapoport, are affine analogues of Deligne-Lusztig varieties. They play an important role in understanding geometric and arithmetic aspects of Shimura varieties. In this talk, I will talk about recent progresses on the classification of their connected components and relevant applications in the theory of Shimura varieties.