| Talk_id | Date | Speaker | Title |
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26960
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Wednesday 1/27
4:00 PM
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Vaidehee Thatte, SUNY Binghamton
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Arbitrary Valuation Rings and Wild Ramification
- Vaidehee Thatte, SUNY Binghamton
- Arbitrary Valuation Rings and Wild Ramification
- 01/27/2021
- 4:00 PM - 5:00 PM
- Online (virtual meeting)
(Virtual Meeting Link)
- Preston Wake (wakepres@msu.edu)
We aim to develop ramification theory for arbitrary valuation fields, extending the classical theory of complete discrete valuation fields with perfect residue fields. By studying wild ramification, we hope to understand the mysterious phenomenon of the $\textit{defect}$ (or ramification deficiency) unique to the positive residue characteristic case and is one of the main obstacles in obtaining resolution of singularities.
Extensions of degree $p$ in residue characteristic $p>0$ are building blocks of the general case. We present a generalization of ramification invariants for such extensions. These results enable us to construct an upper ramification filtration of the absolute Galois group of Henselian valuation fields (joint with K.Kato).
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26988
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Wednesday 2/3
4:00 PM
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Laure Flapan, MSU
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Fano manifolds associated to hyperkähler manifolds
- Laure Flapan, MSU
- Fano manifolds associated to hyperkähler manifolds
- 02/03/2021
- 4:00 PM - 5:00 PM
- Online (virtual meeting)
(Virtual Meeting Link)
- Laure Flapan (flapanla@msu.edu)
Many of the known examples of hyperkähler manifolds arise from geometric constructions that begin with a Fano manifold whose cohomology looks like that of a K3 surface. In this talk, I will focus on a program whose goal is to reverse this process, namely to begin with a hyperkähler manifold and from it produce geometrically a Fano manifold. This is joint work in progress with K. O’Grady, E. Macrì, and G. Saccà.
Passcode: MSUALG
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26954
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Wednesday 2/10
4:00 PM
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Tudor Padurariu, IAS
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Noncommutative resolutions and intersection cohomology for quotient singularities
- Tudor Padurariu, IAS
- Noncommutative resolutions and intersection cohomology for quotient singularities
- 02/10/2021
- 4:00 PM - 5:00 PM
- Online (virtual meeting)
(Virtual Meeting Link)
- Laure Flapan (flapanla@msu.edu)
It is an important problem to define a K-theoretic version of intersection cohomology, with expected applications in representation theory. One step further is to look for a categorification of intersection cohomology. For good moduli spaces $X$ of Artin stack $Y$ (as defined by Alper), we construct some noncommutative resolutions $D(X)$ inside the category $D^b(Y)$. Further, we construct subcategories $I(X)$ of $D(X)$ whose periodic cyclic homology is given by the intersection cohomology of $X$. In particular, the K-theory of $I(X)$ is a natural definition of intersection K-theory for the variety $X$.
Passcode: MSUALG
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26957
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Wednesday 2/17
4:00 PM
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François Greer, IAS
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A tale of two Severi curves
- François Greer, IAS
- A tale of two Severi curves
- 02/17/2021
- 4:00 PM - 5:00 PM
- Online (virtual meeting)
(Virtual Meeting Link)
- Laure Flapan (flapanla@msu.edu)
Let $(S,L)$ be a general polarized K3 surface with $c_1(L)^2=2g-2$. A general member of the linear system $|L|\simeq \mathbb P^g$ is a smooth curve of genus $g$. For $0\leq h\leq g$, define the Severi variety $V_h(S,L)\subset |L|$ to be the locus of curves with geometric genus $\leq h$. As expected, $V_h(S,L)$ has dimension $h$. We consider the case $h=1$, where the Severi variety is a (singular) curve. Our first result is that the geometric genus of $V_1(S,L)$ goes to infinity with $g$; we give a lower bound $\sim e^{c\sqrt{g}}$. Next we consider the analogous question for Severi curves of a rational elliptic surface, and give a polynomial upper bound instead. Modular forms play a central role in both arguments.
Passcode: MSUALG
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26958
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Wednesday 2/24
4:00 PM
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Ignacio Barros, Université Paris-Saclay
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Pencils on surfaces with normal crossings and the Kodaira dimension of $M_{g,n}$
- Ignacio Barros, Université Paris-Saclay
- Pencils on surfaces with normal crossings and the Kodaira dimension of $M_{g,n}$
- 02/24/2021
- 4:00 PM - 5:00 PM
- Online (virtual meeting)
(Virtual Meeting Link)
- Laure Flapan (flapanla@msu.edu)
The first half of the talk will be a colloquium style talk, where I will recall the history of the problem of determining the Kodaira dimension of the moduli space of curves. In the second half I will report on joint work with D. Agostini, where we study smoothing of pencils of curves on surfaces with normal crossings. As a consequence we obtain new instances of $(g,n)$ where $M_{g,n}$ is of negative Kodaira dimension and provide bounds for the Kodaira dimension of $M_{16}$ and $M_{12,8}$.
Passcode: MSUALG
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26956
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Wednesday 3/3
4:00 PM
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Rankeya Datta, University of Illinois at Chicago
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Openness of splinter loci in prime characteristic.
- Rankeya Datta, University of Illinois at Chicago
- Openness of splinter loci in prime characteristic.
- 03/03/2021
- 4:00 PM - 5:00 PM
- Online (virtual meeting)
(Virtual Meeting Link)
- Joe Waldron (waldro51@msu.edu)
A splinter is a notion of singularity that has seen numerous applications of late, especially in connection with the direct summand theorem, the mixed characteristic minimal model program, Cohen-Macaulayness of absolute integral closures and vanishing theorems. However, many basic questions about splinters remain elusive. One such problem is whether the splinter condition spreads from a point to an open neighborhood of a noetherian scheme. In this talk, we will address this question in prime characteristic and show that a locally noetherian scheme whose associated
absolute Frobenius is finite map has an open splinter locus. In particular,
all varieties over perfect fields of positive characteristic have open splinter loci. If time permits, we will show how our methods also give openness of splinter loci for a large class of schemes that do not necessarily have finite Frobenius. This talk is based on joint work in progress with Kevin Tucker.
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26987
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Wednesday 3/10
4:00 PM
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Joe Waldron, MSU
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Minimal model program for threefolds of mixed characteristic
- Joe Waldron, MSU
- Minimal model program for threefolds of mixed characteristic
- 03/10/2021
- 4:00 PM - 5:00 PM
- Online (virtual meeting)
- Laure Flapan (flapanla@msu.edu)
A major obstacle to extending the minimal model program away from characteristic zero is the lack of cohomology vanishing theorems such as Kodaira vanishing. In this talk we describe the minimal model program and then discuss a new way to overcome this difficulty in the arithmetic situation, which has enabled the development of the minimal model program for arithmetic threefolds of residue characteristic greater than 5. This is joint work with Bhatt, Ma, Patakfalvi, Schwede, Tucker and Witaszek.
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26955
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Wednesday 3/17
4:00 PM
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Daniel Bragg, University of California, Berkeley
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Compactifications of supersingular twistor spaces
- Daniel Bragg, University of California, Berkeley
- Compactifications of supersingular twistor spaces
- 03/17/2021
- 4:00 PM - 5:00 PM
- Online (virtual meeting)
(Virtual Meeting Link)
- Rajesh S Kulkarni (kulkar23@msu.edu)
Supersingular twistor spaces are certain families of K3 surfaces over A^1 associated to a supersingular K3 surface. We will describe a geometric construction that produces families of K3 surfaces over P^1 which compactify supersingular twistor spaces. The key input is a construction relating Brauer classes of order p on a scheme of characteristic p to certain sheaves of twisted differential operators. We will give some results on the geometry of compactified supersingular twistor spaces, and some applications.
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26977
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Wednesday 3/24
4:00 PM
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Stefan Patrikis, Ohio State University
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Lifting Galois representations
- Stefan Patrikis, Ohio State University
- Lifting Galois representations
- 03/24/2021
- 4:00 PM - 5:00 PM
- Online (virtual meeting)
(Virtual Meeting Link)
- Igor Rapinchuk (rapinchu@msu.edu)
I will survey joint work with Najmuddin Fakhruddin and Chandrashekhar Khare in which we prove in many cases existence of geometric p-adic lifts of "odd" mod p Galois representations, valued in general reductive groups. Then I will discuss applications to modularity of reducible mod p Galois representations, including most cases of a generalization of Serre's modularity conjecture to reducible (but not necessarily indecomposable) odd two-dimensional representations of the Galois group of Q. Passcode: MSUALG
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26997
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Wednesday 3/31
4:00 PM
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Oscar Rivero, University of Warwick
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Motivic congruences and Sharifi's conjecture
- Oscar Rivero, University of Warwick
- Motivic congruences and Sharifi's conjecture
- 03/31/2021
- 4:00 PM - 5:00 PM
- Online (virtual meeting)
(Virtual Meeting Link)
- Preston Wake (wakepres@msu.edu)
Let f be a cuspidal eigenform of weight two, and let p be a prime at which f is congruent to an Eisenstein series. Beilinson constructed a class arising from the cup-product of two Siegel units and proved a striking relationship with the first derivative L'(f,0) at the near central point s=0 of the L-series of f. In this talk, I will motivate the study of congruences between modular forms at the level of cohomology classes, and will report on a joint work with Victor Rotger where we prove two congruence formulas relating the motivic part of L'(f,0) modulo p and L''(f,0) modulo p with circular units. The proofs make use of delicate Galois properties satisfied by various integral lattices and exploits Perrin-Riou's, Coleman's and Kato's work on the Euler systems of circular units and Beilinson--Kato elements and, most crucially, the work of Fukaya--Kato.
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26952
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Wednesday 4/7
4:00 PM
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Sarah Frei, Rice University
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Rational points on moduli spaces of sheaves on K3 surfaces
- Sarah Frei, Rice University
- Rational points on moduli spaces of sheaves on K3 surfaces
- 04/07/2021
- 4:00 PM - 5:00 PM
- Online (virtual meeting)
(Virtual Meeting Link)
- Rajesh S Kulkarni (kulkar23@msu.edu)
In this talk, I will report on ongoing work with Ryan Takahashi in which we study Brauer-Manin obstructions to the existence of rational points on moduli spaces of sheaves on K3 surfaces. There are Brauer classes naturally arising out of geometric constructions, and we aim to find conditions under which these Brauer classes obstruct the existence of certain kinds of sheaves on a K3 surface over a number field.
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26959
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Wednesday 4/14
4:00 PM
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Yihang Zhu, University of Maryland
|
Irreducible components of affine Deligne-Lusztig varieties
- Yihang Zhu, University of Maryland
- Irreducible components of affine Deligne-Lusztig varieties
- 04/14/2021
- 4:00 PM - 5:00 PM
- Online (virtual meeting)
(Virtual Meeting Link)
- Igor Rapinchuk (rapinchu@msu.edu)
Affine Deligne-Lusztig varieties (ADLV) naturally arise from the study of Shimura varieties. We prove a formula for the number of their irreducible components, which was a conjecture of Miaofen Chen and Xinwen Zhu. Our method is to count the number of F_q points, and to relate it to certain twisted orbital integrals. We then study the growth rate of these integrals using the Base Change Fundamental Lemma of Clozel and Labesse. In an ongoing work we also give the number of irreducible components in the basic Newton stratum of a Shimura variety. This is joint work with Rong Zhou and Xuhua He. Password: MSUALG
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26996
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Wednesday 4/21
4:00 PM
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Jonathan Wang, MIT
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Intersection complexes and unramified L-factors
- Jonathan Wang, MIT
- Intersection complexes and unramified L-factors
- 04/21/2021
- 4:00 PM - 5:00 PM
- Online (virtual meeting)
(Virtual Meeting Link)
- Preston Wake (wakepres@msu.edu)
A series of conjectures of Braverman-Kazhdan, Sakellaridis and Sakellaridis-Venkatesh propose that affine spherical varieties should provide a source for new integral representations of automorphic L-functions. This global problem is conjecturally (and sometimes provably) related to a certain local problem in harmonic analysis. In particular, it is conjectured that unramified local L-factors are related to intersection complexes of formal arc spaces of spherical varietes. I will explain how we establish this connection for a large class of spherical varieties over a local function field, using techniques from geometric representation theory. This is joint work with Yiannis Sakellaridis.
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