Algebra-Number Theory Archives for Academic Year 2020


Steenrod operations on the de Rham cohomology of algebraic stacks

When: Wed, September 9, 2020 - 2:00pm
Where: Zoom
Speaker: Federico Scavia (UBC) - http://www.math.ubc.ca/~scavia/
Abstract: Let k be a field. Totaro studied the de Rham cohomology of algebraic stacks
over k, and computed it for classifying stacks of linear algebraic k-groups
in many cases. Combining previous work of Drury, May and Epstein, I define
and study Steenrod operations on the de Rham cohomology of smooth algebraic
stacks over a field k of characteristic p>0. These operations share many
properties with their topological analogues, but there are also important
differences. I then determine the Steenrod operations on the de Rham
cohomology of linear algebraic k-groups computed by Totaro.

Projective manifolds whose tangent bundle contains a strictly nef subsheaf

When: Wed, September 16, 2020 - 9:00am
Where: Zoom
Speaker: Wenhao Ou (Chinese Academy of Sciences) - https://sites.google.com/site/wenhaooumath/
Abstract: After a theorem of Andreatta and Wisniewski, if the tangent bundle of a projective manifold X contains an ample subsheaf, then X is isomorphic to projective space. We show that, if the tangent bundle contains a strictly nef subsheaf, then X is a projective bundle over a hyperbolic manifold. Moreover, if the fundamental group of X is virtually abelian, then X is isomorphic to a projective space. This is joint with Jie Liu and Xiaokui Yang.

Some results on Seshadri constants

When: Mon, September 21, 2020 - 2:00pm
Where: Online
Speaker: Krishna Hanumanthu ( Chennai Mathematical Institute) - https://www.cmi.ac.in/~krishna/
Abstract: Seshadri constants of nef line bundles on projective varieties were defined by Demailly in 1990, motivated by an ampleness criterion of Seshadri. They are a measure of local positivity of line bundles, have interesting connections to the geometry of the variety, and their study is now an active area of research. We will give an overview of the current work in this area and discuss some recent results on Grassmann bundles over curves and Bott towers.

Eichler-Shimura relations for Hodge type Shimura varieties

When: Wed, September 23, 2020 - 2:00pm
Where: Zoom
Speaker: Si Ying Lee (Harvard) - https://www.math.harvard.edu/people/leesi-ying/


Abstract: The well-known classical Eichler-Shimura relation for modular curves asserts that the Hecke operator $T_p$ is equal, as an algebraic correspondence over the special fiber, to the sum of Frobenius and Verschebung. Blasius and Rogawski proposed a generalization of this result for general Shimura varieties with good reduction at $p$, and conjectured that the Frobenius satisfies a certain Hecke polynomial. I will talk about recent work on this conjecture for Shimura varieties of Hodge type.

Eisenstein series on G_2 and the Skinner--Urban method for Sym^3

When: Mon, September 28, 2020 - 2:00pm
Where: https://umd.zoom.us/j/96890967721


Speaker: Sam Mundy (Columbia University) -
Abstract: In this talk I'll give an overview of the method of Skinner--Urban method for constructing Selmer classes for certain Galois representations which have an automorphic origin. I'll explain some recent progress in trying to apply this method for the exceptional group G_2 to obtain Selmer classes for the symmetric cube of certain GL_2 Galois representations.

On the irreducible case of Fargues' conjecture for GL_n - I

When: Wed, October 7, 2020 - 2:00pm
Where: Zoom
Speaker: Arthur-Cesar Le Bras (Institut Galilee, Universite Paris 13) - http://lebras.perso.math.cnrs.fr/

Abstract: In 2014, Fargues formulated a striking conjecture, which veryroughly says that geometric Langlands works over the Fargues-Fontaine
curve and provides a geometrization of the classical local Langlands
correspondence. In my first talk, I will recall what the main geometric
players are, and what the conjecture says, with special emphasis on the
case of GL_n. In my second talk, I would like to discuss work in
progress with Johannes Anschutz, regarding the case where the group is
GL_n and where one starts with an irreducible (instead of any
indecomposable) Weil-Deligne representation in the conjecture.

The Hilbert scheme of infinite affine space

When: Mon, October 12, 2020 - 2:00pm
Where: via Zoom, link on seminar page

Speaker: Maria Yakerson (ETH Zurich) -
https://www.muramatik.com/
Abstract: Various invariants have been computed for Hilbert schemes of surfaces, however our knowledge about Hilbert schemes (of points) of higher dimensional schemes is quite limited. For example, Hilbert schemes of n-dimensional affine spaces have very complicated geometry for high n. In this talk we will present the surprising observation, that the Hilbert scheme of infinite dimensional affine space has homotopy type of a Grassmannian, and so its invariants of homotopical nature have a simple description. We will explain then how this observation allows us to obtain new properties of algebraic and hermitian K-theories as generalized cohomology theories. This is joint work with Marc Hoyois, Joachim Jelisiejew, Denis Nardin, and Burt Totaro.

On the irreducible case of Fargues' conjecture for GL_n - II

When: Wed, October 14, 2020 - 2:00pm
Where: Zoom
Speaker: Arthur-Cesar Le Bras (Institut Galilee, Universite Paris 13) - http://lebras.perso.math.cnrs.fr/

Abstract: In 2014, Fargues formulated a striking conjecture, which veryroughly says that geometric Langlands works over the Fargues-Fontaine
curve and provides a geometrization of the classical local Langlands
correspondence. In my first talk, I will recall what the main geometric
players are, and what the conjecture says, with special emphasis on the
case of GL_n. In my second talk, I would like to discuss work in
progress with Johannes Anschutz, regarding the case where the group is
GL_n and where one starts with an irreducible (instead of any
indecomposable) Weil-Deligne representation in the conjecture.

Hopf-theoretic approach to motives of twisted flag varieties

When: Mon, October 19, 2020 - 2:00pm
Where: Zoom
Speaker: Nikita Semenov (LMU (Munich)) - http://www.mathematik.uni-muenchen.de/~semenov/
Abstract: Let G be a split semisimple algebraic group over a field and
let A be an oriented cohomology theory in the sense of Levine--Morel. We
provide a uniform approach to the A-motives of geometrically cellular
smooth projective G-varieties based on the Hopf algebra structure of
A(G). Using this approach we provide various applications to the
structure of motives of twisted flag varieties. This is a joint work
with Victor Petrov.

Vertex algebras of CohFT-type

When: Wed, October 28, 2020 - 2:00pm
Where: Zoom

Speaker: Nicola Tarasca (Virginia Commonwealth University) - http://people.vcu.edu/~tarascan/
Abstract: This talk will focus on geometric realizations of non-commutative algebras. I will discuss how representations of conformal vertex algebras encode information about the geometry of algebraic curves. The starting point is the Virasoro uniformization, which provides an incarnation of the Virasoro algebra in the tangent space of a tautological line bundle on the moduli space of coordinatized curves. After briefly reviewing vertex algebras, I will discuss how their representations yield new vector bundles of conformal blocks on moduli spaces of curves and new cohomological field theories. This is joint work with Chiara Damiolini and Angela Gibney.

The arithmetic fundamental lemma over a general p-adic field (Note time change!)

When: Wed, November 4, 2020 - 2:45pm
Where: Zoom
Speaker: Wei Zhang (MIT) - http://math.mit.edu/~wz2113/

Abstract: The arithmetic fundamental lemma (AFL) is an identity relating the arithmetic intersection numbers on a Rapoport-Zink space for unitary groups to the first derivative of relative orbital integral on the general linear groups over a p-adic field F. In this talk I will report a work in progress joint with A. Mihatsch to prove the AFL for a general p-adic field. We also establish a partial analog (over totally real fields) of a theorem of Bruinier--Howard--Kudla--Rapoport--Yang on the modularity of generating series of arithmetic special divisors.

Lichtenbaum's conjecture for an arithmetic surface

When: Mon, November 9, 2020 - 2:00pm
Where: Online
Speaker: Niranjan Ramachandran (UMD) - https://www-math.umd.edu/people/faculty/item/442-atma.html

Abstract: A recent conjecture of S. Lichtenbaum provides Euler characteristic-type formulas for the special values of zeta functions of proper regular schemes over Z.
The talk will discuss the case of the special value at s=1 of an arithmetic surface; we shall indicate the relations with the BSD conjecture and the Bloch-Kato conjecture. This is joint work with Lichtenbaum.

An A_{crys}-type specialization of prismatic cohomology

When: Wed, December 2, 2020 - 2:00pm
Where: Online
Speaker: Shizhang Li (University of Michigan, Ann Arbor) - http://shizhang.li/

Abstract: In this talk I will explain an upcoming joint work with Tong Liu establishing a comparison between prismatic and certain A_{crys}-type crystalline cohomology. I shall first introduce some reasons why one expects such a comparison. Then I'll explain the statement of this comparison and give some applications (specialize in the Breuil--Kisin prism setup).

Strictly nef divisors

When: Wed, December 9, 2020 - 2:00pm
Where: Online
Speaker: Priyankur Chaudhuri (University of Maryland) -
Abstract: TBA

TBD

When: Mon, January 25, 2021 - 2:00pm
Where: Online
Speaker: Lukas Brantner (Oxford University) -


(Banerjee) TBA

When: Wed, January 27, 2021 - 2:00pm
Where: Online
Speaker: Oishee Banerjee (HCM Bonn) - https://www.math.uni-bonn.de/people/oishee/
Abstract: TBA

TBA

When: Mon, March 29, 2021 - 2:00pm
Where: Online
Speaker: Kai-Wen Lan (U. Minnesota) -