Algebra-Number Theory Archives for Fall 2021 to Spring 2022
Steenrod operations on the de Rham cohomology of algebraic stacks
When: Wed, September 9, 2020 - 2:00pmWhere: 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 - 9:00amWhere: 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 - 2:00pmWhere: 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 - 2:00pmWhere: 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 - 2:00pmWhere: 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 - 2:00pmWhere: 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 - 2:00pmWhere: 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 - 2:00pmWhere: 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 - 2:00pmWhere: 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 - 2:00pmWhere: 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 - 2:45pmWhere: 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 - 2:00pmWhere: 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 - 2:00pmWhere: 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 - 2:00pmWhere: Online
Speaker: Priyankur Chaudhuri (University of Maryland) -
Abstract: This talk will be mainly focussed on a few important conjectures surrounding strictly nef divisors. Serrano conjectured that if $L$ is a strictly nef divisor on a smooth projective variety $X$, then $K_X+tL$ is ample for all $t> dim X+1$. I will talk about a few special cases of this conjecture and some generalizations. Time permitting, I will also talk about a simple criterion for global generation of homogeneous vector bundles on flag varieties.
Purely inseparable Galois theory
When: Mon, January 25, 2021 - 2:00pmWhere: Online
Speaker: Lukas Brantner (University of Oxford) -
Abstract: An algebraic extension of fields F/K of characteristic p is purely inseparable if for each x in F, some power x^{p^n} belongs to K. Using homotopical methods, we construct a Galois correspondence for finite purely inseparable field extensions F/K, generalising a classical result of Jacobson for extensions of exponent one (where x^p belongs to K for all x in F). This is joint work with Waldron.
Volumes of definable sets in o-minimal expansions and affine GAGA theorems
When: Mon, February 22, 2021 - 2:00pmWhere: Online
Speaker: Patrick Brosnan (UMCP) - http://www2.math.umd.edu/~pbrosnan/
Abstract: The affine GAGA theorem of Peterzil and Starchenko says that a closed analytic subset of complex n-space which is definable in an o-minimal expansion of the ordered field R is actually algebraic. It is a one of the main tools in recent work on Hodge theory by Bakker, Brunebarbe and Tsimerman. In my talk I'll show how to give a very short proof of Peterzil--Starchenko using a much older affine GAGA theorem of Stoll along with a volume estimate for definable sets in an o-minimal structure (due to Kurdyka and Raby in a slightly different form and to and Nguyen and Valette in the o-minimal setting). The main point of my talk is really to advertise the o-minimal methods. So I'll give the (relatively easy) proof the volume estimate. But I'll also try to say something about the way Peterzil--Starchenko has been used in Hodge theory.
Zeta functions of surfaces over finite fields
When: Mon, March 1, 2021 - 2:00pmWhere: Online
Speaker: Niranjan Ramachandran (UMD) - https://www-math.umd.edu/people/faculty/item/442-atma.html
Abstract: This talk will report on the links between the Artin-Tate conjecture and the Birch-Swinnerton-Dyer conjecture over function fields. This is joint with S. Lichtenbaum.
Computing cohomology via cohomological descent over the symmetric simplex category
When: Wed, March 3, 2021 - 2:00pmWhere: Online
Speaker: Oishee Banerjee (HCM Bonn) - https://www.math.uni-bonn.de/people/oishee/
Abstract: In 1979, Graeme Segal proved homological stability for the moduli space of degree d self-maps of the complex projective space CP1 ; it was later generalised to morphisms on smooth projective curves of higher genus as well. Ever since then, despite various attempts, the answer to the question of (co)homological stability in the case when the domain of the morphisms is complex projective m-space, CPm, for m greater than 1, stayed elusive. In this talk, I will report on how, by a combination of the theory of symmetric simplicial objects (developed independently by Fiederowicz-Loday and Krasauskas) and Deligne's theory of cohomological descent, we can answer (among other things) the question raised above.
Incoherent definite orthogonal spaces and Shimura varieties
When: Mon, March 8, 2021 - 2:00pmWhere: Online
Speaker: Benedict Gross (UCSD/Harvard) -
Abstract: I will review the classification of orthogonal spaces over local and global fields, then introduce the notion of an incoherent definite orthogonal space and its neighbors. I will show how the neighbors can be used to define a Shimura variety of orthogonal type, and to give information on its local points.
Wall-crossing for compact moduli of higher dimensional varieties
When: Mon, March 22, 2021 - 2:00pmWhere: Online
Speaker: Dori Bejleri (Harvard University) - http://people.math.harvard.edu/~bejleri/
The Deligne-Mumford space of pointed stable curves is a central object in algebraic geometry with deep connections to many other fields. The higher dimensional analogue is the moduli space of stable log varieties or stable pairs (X,D) consisting of a variety X and a divisor D satisfying certain conditions. The existence of compact moduli spaces of such stable pairs in all dimensions is one of the crowning achievements of the last several decades of progress on the minimal model program. For a given class of varieties, there may be many different stable pair compactifications of the moduli space depending on the choice of divisor D. In this talk, I will given an overview of the theory of stable log varieties and discuss recent wall-crossing results which explain how these compactifications change as D varies. This is based on joint work with Ascher, Inchiostro and Patakfalvi.
Cohomology of Shimura varieties and Hodge-Tate weights
When: Mon, March 29, 2021 - 2:00pmWhere: Online
Speaker: Kai-Wen Lan (U. Minnesota) -
Abstract: I will report on some results on the cohomology of a (general) Shimura variety with coefficients in automorphic etale local systems. I will start with some background materials, and explain that such cohomology is de Rham with Hodge-Tate weights computable using relative Lie algebra cohomology. I will also explain that the same is true for the cohomology with compact support and for the interior cohomology, and hence also for the intersection cohomology when the automorphic local systems are attached to algebraic representations of regular highest weights. If time permits, I will briefly mention what can be done for the boundary cohomology. (These are based on joint work with Hansheng Diao, Ruochuan Liu, and Xinwen Zhu, and on joint work in progress with David Sherman.)
Local-global principles over semi-global fields
When: Mon, April 12, 2021 - 2:00pmWhere: Online
Speaker: David Harbater (UPenn) - https://www2.math.upenn.edu/~harbater/
Abstract: Local-global principles have classically been studied in the
context of global fields; i.e., number fields or function fields of
curves over finite fields. In recent years, they have also been studied
over what have come to be known as semi-global fields, a class that
includes function fields of p-adic curves. Classical results such as
the Hasse-Minkowski theorem have been carried over to this context,
though with very different proofs. The talk will present results in
this direction, including ongoing work of the speaker with J-L.
Colliot-Thélène, J. Hartmann, D. Krashen, R. Parimala, and V. Suresh.
Stark's Conjectures and Hilbert's 12th Problem
When: Wed, April 14, 2021 - 2:00pmWhere: Online
Speaker: Samit Dasgupta (Duke University) - https://services.math.duke.edu/~dasgupta/#home
Abstract: In this talk we will discuss two central problems in algebraic number theory and their interconnections: explicit class field theory (also known as Hilbert's 12th Problem), and the special values of L-functions. The goal of explicit class field theory is to describe the abelian extensions of a ground number field via analytic means intrinsic to the ground field. Meanwhile, there is an abundance of conjectures on the special values of L-functions at certain integer points. Of these, Stark's Conjecture has special relevance toward explicit class field theory. I will describe two recent joint results with Mahesh Kakde on these topics. The first is a proof of the Brumer-Stark conjecture away from p=2. This conjecture states the existence of certain canonical elements in CM abelian extensions of totally real fields. The second is a proof of an exact formula for Brumer-Stark units that has been developed over the last 15 years. We show that these units together with other easily written explicit elements generate the maximal abelian extension of a totally real field, thereby giving a p-adic solution to Hilbert's 12th problem.
Scholze Shtukas and Shimura Varieties
When: Mon, April 19, 2021 - 10:00amWhere: Online
Speaker: George Pappas (Michigan State) -
Abstract:
I will present some joint work with M. Rapoport: We use Scholze’s theory of p-adic shtukas to relate local and global Shimura varieties and obtain results about their integral models.
Naive A^1-homotopy equivalences and theorems of Whitehead and Zariski
When: Wed, April 21, 2021 - 2:00pmWhere: Online
Speaker: Eion Mackall (UMD) - https://www.eoinmackall.com/
Abstract: A naive A^1-homotopy between morphisms f,g from a variety X to a variety Y is a
cycle on (XxA^1)xY whose support is finite and surjective over XxA^1 and whose fibers over
0 and 1 are the graphs of f and g respectively. Using this notion of naive A^1-homotopy, one
can define naive A^1-homotopy equivalences of varieties. In this talk, we'll discuss how an
analog of a theorem of Whitehead can be used to show that there are no nontrivial
A^1-homotopy equivalences between smooth projective varieties.
Mathematics in the Computer
When: Mon, April 26, 2021 - 2:00pmWhere: Online
Speaker: Mario Carneiro (Carnegie Mellon, Philosophy ) - https://www.cmu.edu/dietrich/philosophy/people/phd/mario-carneiro.html
Abstract: The idea of using computers for checking mathematics has been around almost as long
as computers themselves, but they have gradually started to see more mainstream
recognition, and increasingly large and ambitious projects are being attempted in these systems.
My own history with theorem provers started with the Metamath system,
and then the Lean theorem prover, and it has since expanded into other theorem provers.
Maintaining a formal mathematical library is very different
from the mathematics they teach you in school,
requiring a strange mix of programming and mathematics skills.
But the experience is fun and rewarding,
and the field itself is growing in relevance and popularity.
Peering into the foundations of the enterprise reveals even more questions -
why should we trust computers any more than humans?
What can we do to make computer-powered mathematical arguments as air-tight as possible?
Is it possible to build a formally verified theorem prover?
How will all this affect the practice of mathematics in the future?
p-adic cohomology of p-adic period domains
When: Mon, May 3, 2021 - 2:00pmWhere: Online
Speaker: Gabriel Dospinescu (ENS-Lyon) - http://perso.ens-lyon.fr/gabriel.dospinescu/
Abstract: We will explain how to adapt Orlik's beautiful computation of the l-adic (l different from p) compactly supported cohomology of p-adic period domains (attached to basic isocrystals and quasi-split reductive groups) to the p-adic setting. The key input turns out to be a computation of Ext^1 between two generalized Steinberg representations with mod p coefficients. We will also mention some subtleties related to the passage from torsion to p-adic coefficients, where the situation is quite different with respect to the l-adic setting. This is joint work with Colmez, Hauseux and Niziol.
The Zariski topology, linear systems, and algebraic varieties
When: Wed, May 12, 2021 - 2:00pmWhere: Online
Speaker: Max Lieblich (U Washington Seattle) - https://max.lieblich.us/
Abstract: This is joint work with with János Kollár, Martin Olsson, and Will Sawin. I will discuss an extension of the classical Veblen-Young theorem on axiomatic projective geometry to arbitrary algebraic varieties. In particular, I will explain how most proper, normal varieties are uniquely determined (up to isomorphisms of the base field) from their underlying Zariski topological spaces.