Welcome to the Maryland Distinguished Lectures in Algebra and Number Theory! This is an annual lecture series concerning recent developments in Algebra and Number Theory, delivered by the world's foremost experts. It is organized by Thomas Haines.
2025 Davesh Maulik
All talks will take place at Kirwan Hall Room 3206
Wednesday, March 26 at 3:15PM
The P=W Conjecture
Given a compact Riemann surface, nonabelian Hodge theory relates topological and algebro-geometric objects associated to it. Namely, complex representations of the fundamental group are in correspondence with algebraic vector bundles, equipped with an extra structure called a Higgs field. This gives a transcendental matching between two very different moduli spaces associated with the Riemann surface: the character variety (parameterizing representations of the fundamental group) and the Hitchin moduli space (parameterizing Higgs bundles). In 2010, de Cataldo, Hausel, and Migliorini proposed the P=W conjecture, which predicted that the Hodge theory of the character variety is determined by the topology of the Hitchin space, imposing surprising constraints on each side. In this talk, I will introduce the conjecture and review its recent proofs; time permitting, I will try to explain how this phenomenon relates to other geometric questions.
Thursday, March 27 at 2:00PM
Algebraic Cycles and Hitchin Fibration
In the first lecture, given a proper map $f: X \rightarrow Y$, I introduced the perverse filtration on the cohomology of $X$, which measures the singularities of the fibers of $f$. In this talk, when $X$ is an abelian fibration over $Y$, I will explain a technique for studying this filtration via the Fourier-Mukai transform on DCoh(X), the derived category of coherent sheaves of X. This approach is a natural extension of ideas of Beauville and Deninger-Murre for studying Chow groups of abelian schemes. As an application, we get a proof of the P=W conjecture, introduced in the last lecture, but also other conjectures lifting these filtrations to Chow groups. Joint work with Junliang Shen and Qizheng Yin.
Friday, March 28 at 2:00PM
D-Equivalence Conjecture for Hyperkahler Varieties of K3[n] type.
The D-equivalence conjecture of Bondal and Orlov predicts that birational Calabi-Yau varieties have equivalent derived categories of coherent sheaves. I will explain how to prove this conjecture for hyperkahler varieties of K3^[n] type (i.e. those that are deformation equivalent to Hilbert schemes of K3 surfaces). This is joint work with Junliang Shen, Qizheng Yin, and Ruxuan Zhang.
2024 Zhiwei Yun
Theme: Higher theta series
Abstract: Theta series play an important role in the classical theory of modular forms. In the modern language of automorphic representations, they are constructed from a pair of groups $G$ and $H$ (one orthogonal and one symplectic, or both unitary groups) and the remarkable Weil representation of $G\times H$. Kudla introduced an analogue of theta series in arithmetic geometry, by forming a generating series of algebraic cycles on Shimura varieties. The arithmetic theta series has since become a very active program.
In joint work with Tony Feng and Wei Zhang, we consider analogues of arithmetic theta series over function fields, and try to go further than what was done over number fields. Our work concentrated on unitary groups. We defined a generating series of algebraic cycles on the moduli stack of unitary Drinfeld Shtukas (called the higher theta series). We made the Modularity Conjecture: the higher theta series is an automorphic form valued in a certain Chow group. This is a function field analogue of the special cycles generating series defined by Kudla and Rapoport, but with an extra degree of freedom namely the number of legs of the Shtukas.
One concrete formula we proved was a higher derivative version of the Siegel-Weil formula. It is an equality between degrees of 0-dimensional special cycles on the moduli of unitary Shtukas and higher derivatives of the Siegel-Eisenstein series of another unitary group. More recently, we have obtained a proof of a weaker version of the Modularity Conjecture, confirming that the cycle class of the higher theta series (valued in the cohomology of the generic fiber) is automorphic.
The series of talks will feature a colloquium-style introduction to some representation-theoretic and geometric background (the second talk), the other two being more technical talks in which I will explain some ingredients in the proofs of the higher Siegel-Weil formula and the weak Modularity Conjecture.
Higher Theta series, Part I
Tuesday, February 27 at 3:30 pm
University of Maryland - Kirwan Hall Rm 3206
Higher Theta series, Part II (Colloquium)
Wednesday, February 28 at 3:15 pm
University of Maryland - Kirwan Hall Rm 3206
Higher Theta series, Part III
Thursday, February 29 at 3:30 pm
University of Maryland - Kirwan Hall Rm 3206
2022 Mark Kisin
Theme: Arithmetic of abelian varieties and their moduli
Essential dimension and prismatic cohomology (Colloquium)
Wednesday, December 7 at 3:15pm
University of Maryland - Kirwan Hall Rm 3206
Abstract: The smallest number of parameters needed to define an algebraic covering space is called its essential dimension. Questions about this invariant go back to Klein, Kronecker and Hilbert and are related to Hilbert's 13th problem. In this talk, I will give a little history, and then explain a new approach which relies on recent developments in $p$-adic Hodge theory. This is joint work with Benson Farb and Jesse Wolfson.
Frobenius conjugacy classes attached to abelian varieties
Thursday, December 8 at 2:00pm
University of Maryland - Kirwan Hall Rm 3206
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($\bar K/K$)→$G(\mathbb Q_\ell)$. 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
Friday, December 9 at 2:00pm
University of Maryland - Kirwan Hall Rm 3206
Abstract: Let $A$ be an abelian variety over $\bar{\mathbb Q}.$ 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.
2021 Wei Zhang
2019 Ngô Bảo Châu