Algebra-Number Theory Archives for Fall 2023 to Spring 2024
Torelli theorems--manifest and occult
When: Wed, September 14, 2022 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Michael Rapoport (UMD) -
Abstract: I will first survey the few cases in which a hypersurface in projective space is determined by its periods (manifest Torellis). Then I will survey the few cases in which hidden periods accomplish the same.
Derived equivalences of genus one curves
When: Mon, September 19, 2022 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Jonathan Rosenberg (UMd) - http://math.umd.edu/~jmr
Abstract: This is joint work with Niranjan Ramachandran. We show that for a smooth projective genus one curve X over a field k of characteristic zero, the derived category D(X) is equivalent to a twisted derived category D(J, α) on the Jacobian J of X. This replaces Mukai duality over a non-algebraically closed field. We were led to this by a development in physics.
S=T for Shimura varieties
When: Mon, October 24, 2022 - 9:45amWhere: Online
Speaker: Zhiyou Wu (Beijing International Center for Mathematical Research (BICMR)) - https://zhiyou-wu.com/
Abstract: I will explain how the new p-adic geometry developed by Scholze can help prove the Eichler-Shimura relation for Shimura varieties of Hodge type, which has nothing to do with p-adic geometry a priori. I will explain the motivation in the first talk, and more technical details in the second talk.
S=T for Shimura varieties
When: Wed, October 26, 2022 - 9:45amWhere: Online
Speaker: Zhiyou Wu (Beijing International Center for Mathematical Research (BICMR)) - https://zhiyou-wu.com/
Abstract: I will explain how the new p-adic geometry developed by Scholze can help prove the Eichler-Shimura relation for Shimura varieties of Hodge type, which has nothing to do with p-adic geometry a priori. I will explain the motivation in the first talk, and more technical details in the second talk.
Hilbert's 13th Problem for algebraic groups
When: Mon, October 31, 2022 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Zinovy Reichstein (University of British Columbia) - https://personal.math.ubc.ca/~reichst/
Abstract: The algebraic form of Hilbert's 13th Problem asks for the
resolvent degree rd(n) of the general polynomial f(x) = x^n + a_1
x^{n-1} + ... + a_n of degree n, where a_1, ..., a_n are independent
variables. Here rd(n) is the minimal integer d such that every root of
f(x) can be obtained in a finite number of steps, starting with C(a_1,
..., a_n) and adjoining an algebraic function in <= d variables at each
step. It is known that rd(n) = 1 for every n <= 5. It is not known
whether or not rd(n) is bounded as n tends to infinity; it is not even
known whether or not rd(n) > 1 for any n. Recently Farb and Wolfson
defined the resolvent degree rd_k(G), where G is a finite group and k is
a field of characteristic 0. In this setting rd(n) = rd_C(S_n), where
S_n is the symmetric group on n letters and C is the field of complex
numbers. In this talk I will define rd_k(G) for any field k and any
algebraic group G over k. Surprisingly, Hilbert's 13th Problem
simplifies when G is connected. My main result is that rd_k(G) <= 5 for
an arbitrary connected algebraic group G defined over an arbitrary field k.
Dirac geometry
When: Mon, November 7, 2022 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Piotr Pstrągowski (IAS and Harvard University) - https://people.math.harvard.edu/~piotr/
Abstract: The rings which appear naturally in derived contexts, for example as homology of differential graded algebras, are graded-commutative in the Koszul sense; that is, odd degree elements anticommute. In this talk, I will describe joint work with Lars Hesselholt where we develop algebraic geometry built out of such rings, which we call Dirac geometry. The latter can be considered as a natural extension of G_m-equivariant algebraic geometry where the Serre twist has a square-root. I will also discuss how these objects naturally occur in a mysterious extension of the theory of motives arising in algebraic topology.
Frobenius conjugacy classes attached to abelian varieties (Distinguished Lectures in Algebra/Number Theory)
When: Thu, December 8, 2022 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Mark Kisin (Harvard University) - https://people.math.harvard.edu/~kisin/
Abstract: The Mumford-Tate group of an abelian variety A over the complex numbers is an algebraic group G, defined in terms of the complex geometry of A, more specifically its Hodge structure. If A is defined over a number field K, then a remarkable result of Deligne asserts that the ℓ -adic cohomology of A gives rise to a G -valued Galois representation ρℓ:Gal(¯K/K)→G(Qℓ). We will show that for a place of good reduction v∤ℓ of A, the conjugacy class of Frobenius ρℓ(Frobv) does not depend on ℓ. This is joint work with Rong Zhou.
Heights in the isogeny class of an abelian variety (Distinguished Lectures in Algebra/Number Theory)
When: Fri, December 9, 2022 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Mark Kisin (Harvard University) - https://people.math.harvard.edu/~kisin/
Abstract: Let A be an abelian variety over ¯Q. In this talk I will consider the following conjecture of Mocz. Conjecture: Let c>0. In the isogeny class of A, there are only finitely many isomorphism classes of abelian varieties of height c. I will sketch a proof of the conjecture when the Mumford-Tate conjecture - which is known in many cases - holds for A. This result should be compared with Faltings' famous theorem, which is about finiteness for abelian varieties defined over a fixed number field. This is joint work with Lucia Mocz.
Compact moduli and degenerations
When: Mon, January 30, 2023 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Dori Bejleri (Harvard University) - https://people.math.harvard.edu/~bejleri/
Abstract: It has been said that working with non-compact spaces is like trying to hold change in your pocket with a hole in it. One of the central examples of non-compact spaces in algebraic geometry are moduli spaces. Broadly speaking, the points of a moduli space represent equivalence classes of algebraic varieties of a given type, and its geometry reflects the ways these varieties deform in algebraic families. The classification of algebraic varieties of a given type is tantamount to understanding the geometry of the corresponding moduli space. The goal of this talk is to discuss recent progress on compactifying moduli spaces of higher dimensional varieties, focusing on the interplay between compactifications of moduli spaces and singular degenerations of the objects they classify.
Derived categories of torsors over abelian varieties
When: Wed, February 1, 2023 - 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 Jonathan Rosenberg. The main result is that the derived category of a torsor over an abelian variety A is equivalent to the twisted derived category over A.
Mod-p Poincare Duality in p-adic Analytic Geometry
When: Mon, February 6, 2023 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Bogdan Zavyalov (IAS Princeton) - https://bogdanzavyalov.com/
Abstract: Etale cohomology of F_p-local systems does not behave nicely on general
smooth p-adic rigid-analytic spaces; e.g., the F_p-cohomology of the 1-dimensional
closed unit ball is infinite.
However, it turns out that the situation is much better if one considers only proper rigid-analytic spaces. These spaces have finite F_p cohomology groups and these groups satisfy Poincare Duality if X is smooth and proper.
I will explain how one can prove such results using the concept of almost coherent sheaves that allows to "localize" such questions in an appropriate sense and actually reduce to some local computations.
A Quillen-Lichtenbaum Conjecture for Dirichlet L-functions
When: Mon, March 6, 2023 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Ningchuan Zhang (UPenn) - https://sites.google.com/view/ningchuan-zhang
Abstract: The original version of the Quillen-Lichtenbaum Conjecture, proved by Voevodsky and Rost, connects special values of the Dedekind zeta function of a number field with its algebraic $K$-groups. In this talk, I will discuss a generalization of this conjecture to Dirichlet $L$-functions. The key idea is to twist algebraic $K$-theory spectra of number fields with equivariant Moore spectra associated to Dirichlet characters. Rationally, we obtain a Quillen-Borel type theorem for Artin $L$-functions. This is joint work in progress with Elden Elmanto.
Equivariant enumerative geometry
When: Mon, March 27, 2023 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Thomas Brazelton (University of Pennsylvania) - https://www2.math.upenn.edu/~tbraz/
Abstract: Classical enumerative geometry asks geometric questions of the form "how many?" and expects an integral answer. For example, how many circles can we draw tangent to a given three? How many lines lie on a cubic surface? The fact that these answers are well-defined integers, independent upon the initial parameters of the problem, is Schubert’s principle of conservation of number. In this talk we will outline a program of "equivariant enumerative geometry", which wields equivariant homotopy theory to explore enumerative questions in the presence of symmetry. Our main result is equivariant conservation of number, which states roughly that the orbits of solutions to an equivariant enumerative problem are conserved. We leverage this to compute the S4 orbits of the 27 lines on any symmetric cubic surface.
The stacky concentration theorem
When: Wed, April 12, 2023 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Adeel Khan (Academia Sinica, Taiwan) - https://www.preschema.com/
Abstract: Given a torus action on a manifold M, fundamental results of Borel and Atiyah-Bott assert that (i) the equivariant cohomology of M is concentrated in the fixed locus M^T, up to inverting Euler classes of enough line bundles; and moreover (ii) the fundamental class [M] can be computed in terms of [M^T] and the inverse of the Euler class of the normal bundle. In algebraic geometry, Edidin and Graham proved analogues of these facts in Chow theory. In this talk, I'll discuss a stacky generalization of these results, as well as a categorification at the level of equivariant derived categories. This is based on joint work with D. Aranha, A. Latyntsev, H. Park, and C. Ravi, as well as another work in progress with C. Ravi.
Weights and local Langlands parametrization
When: Mon, May 1, 2023 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Michael Harris (Columbia University) -
Abstract: This is a report on several recent results on the Langlands parametrization of irreducible representations of groups over local fields of positive characteristic. The parametrization is defined algebraically in the work of V. Lafforgue, Genestier-Lafforgue, and Fargues-Scholze, using variants of the theory of moduli of vector bundles on curves. I will specifically report on results on ramification of local parameters (joint with Gan and Sawin) and on the generalized Ramanujan conjecture (joint with Ciubotaru). Deligne's theory of weights plays a crucial role in both projects. If time permits, I will also discuss joint work in progress with Beuzart-Plessis and Thorne on a strategy to invert the parametrization.
The period-index problem over the complex numbers
When: Wed, May 3, 2023 - 2:00pmWhere: Kirwan Hall 3206
Speaker: James Hotchkiss (University of Michigan) -
Abstract: The period-index problem is a longstanding question about the complexity of Brauer classes over a field. I will discuss some Hodge-theoretic aspects of the problem for complex function fields, and give some applications to Brauer groups and the integral Hodge conjecture.
Local enumerative invariants for generalized del Pezzo surfaces
When: Mon, May 8, 2023 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Sungwoo Nam (University of Illinois at Urbana-Champaign) - https://scream27.github.io/sungwoo/
Abstract: In this talk, I will discuss a curve-counting theory of generalized del Pezzo surfaces. I will begin by discussing the case of smooth del Pezzo surfaces, which was studied under the name of local mirror symmetry. Then I will discuss how (with motivation from physics) the theory generalizes to some singular surfaces, specifically those with simple normal crossing singularities. Part of this talk is joint work with Sheldon Katz.