Seminars Sorted by Series

For a polynomial $f\in \mathbb Q[x]$, Hilbert's irreducibility theorem asserts that the fiber $f^{-1}(a)$ is irreducible over $\mathbb Q$ for all values $a\in \mathbb Q$ outside a "thin" set of exceptions $R_f$. The problem of describing $R_f$ is...

In this talk, we prove an upper bound on the average number of 2-torsion elements in the class group of monogenised fields of any degree $n \geq 3$ and, conditional on a widely expected tail estimate, compute this average exactly. As an application...

The unbounded denominators conjecture, first raised by Atkin and Swinnerton-Dyer, asserts that a modular form for a finite index subgroup of $SL_2(\mathbb Z)$ whose Fourier coefficients have bounded denominators must be a modular form for some...

In 1986, Hooley applied (what practically amounts to) the general Langlands reciprocity (modularity) conjecture and GRH in a fresh new way, over certain families of cubic 3-folds. This eventually led to conditional near-optimal bounds for the number...

Katz and Oort raised the following question: Given an algebraically closed field $k$, and a positive integer $g > 3$, does there exist an abelian variety over k not isogenous to a Jacobian over $k$? There has been much progress on this question...

We introduce a new orbit parametrization for square roots of the class of the inverse different in rings cut out by binary forms. This parametrization has many applications of interest in arithmetic statistics; for example, we use it to prove that...

A surprising property of the cohomology of locally symmetric spaces is that Hecke operators can act on multiple cohomological degrees with the same eigenvalues. We will discuss this phenomenon for the coherent cohomology of line bundles on modular...

I will present joint work with V. Paškūnas and G. Böckle concerning deformation rings for mod $p$ Galois representations of $p$-adic local fields. After giving a short introduction to the subject, I will explain our main result which says that...

Wiles’s proof of the modularity of (semistable) elliptic curves over the rationals and Fermat’s Last Theorem relied on his invention of a modularity lifting method. There were two strands to the method:

(i) A numerical criterion to for a map of...

In 2001 Croot resolved an old conjecture of Erdos and Graham, proving that in any finite colouring of the positive integers there is a (non-trivial) monochromatic solution to 1/n_1+...+1/n_k = 1 with all n_i distinct. A natural generalisation, also...

I will explain how various results in arithmetic statistics by Bhargava, Gross, Shankar and others on 2-Selmer groups of Jacobians of (hyper)elliptic curves can be organised and reproved using the theory of graded Lie algebras, following earlier...

Let $N$ and $p$ $\geq 5$ be primes such that $p$ divides $N-1$. In his landmark paper on the Eisenstein ideal, Mazur proved the $p$-part of the BSD conjecture for the $p$-Eisenstein quotient $J^{(p)}$ of $J_0(N)$ over $\mathbf{Q}$. Using recent...

Recently, Fargues and Scholze attached a semi-simple L-parameter to any smooth irreducible representation of a p-adic reductive group, realizing the local Langlands correspondence as a geometric Langlands correspondence over the Fargues-Fontaine...

We determine the average size of the 3-torsion in class groups of $G$-extensions of a number field when $G$ is any transitive 2-group containing a transposition, for example, $D4$.  It follows from the Cohen--Lenstra--Martinet heuristics that the...

I will discuss recent work with Maksym Radziwill in which we show that for any fixed tempered cuspidal representation $\pi$ of $\mathrm{GL}(4)$ over the rationals, there exist infinitely many primitive characters $\chi$ such that the twisted $L$...

Of the $(2H+1)^n$ monic integer polynomials $f(x)=x^n+a_1 x^{n-1}+\cdots+a_n$ with $\max\{|a_1|,\ldots,|a_n|\}\leq H$, how many have associated Galois group that is not the full symmetric group $S_n$?  There are clearly $\gg H^{n-1}$ such...

Half-integral weight modular forms are classical objects with many important arithmetic applications.  In terms of automorphic representations, these correspond to objects on the metaplectic double cover of SL(2).  In this talk, I will outline a...

An abstract group is said to have the bounded generation property (BG) if it can be written as a product of finitely many cyclic subgroups. Being a purely combinatorial notion, bounded generation has close relation with many group theoretical...

This lecture will partly survey branching laws for real and p-adic groups which often is related to period integrals of automorphic representations, discuss some of the more recent developments, focusing attention on homological aspects and the...

Over 15 years ago, di Francesco and Itzykson gave an estimate on the growth (as the degree increases) of the number of plane rational curves passing through the appropriate number of points. This provides an example of an upper bound on (primary)...

I will explain how essential information about the structure of symplectic manifolds is captured by algebraic data, and specifically by the non-commutative (mixed) Hodge structure on the cohomology of the Fukaya category. I will discuss how mirror...

I discuss two examples of random symplectic maps in this talk. As the first example consider a stochastic twist map that is defined to be a stationary ergodic twist map on a planar strip. As a natural question, I discuss the fixed point of such maps...

I will first discuss the orientability of the moduli spaces of J-holomorphic maps with Lagrangian boundary conditions. It is known that these spaces are not always orientable and I will explain what the obstruction depends on. Then, in the presence...

I will discuss two of my current projects which have different aims but use similar techniques. The first aims to understand the equivariant K-theory of symplectic orbifolds. The second is about the topology of toric origami manifolds. Neither space...

A bihermitian metric (or generalized Kahler structure) involves a pair of complex structures and a metric compatible with both. The study of these is closely related to that of holomorphic Poisson structures on a manifold. A deformation theorem for...

This is joint work with Peter Albers. We study a Floer gradient equation on a very negative line bundle over a symplectic manifold. If the line bundle is negative enough there are generically no holomorphic spheres. We explain the metamorphosis of...

We discuss a quantum counterpart, in the sense of the Berezin-Toeplitz quantization, of certain constraints on Poisson brackets coming from "hard" symplectic geometry. It turns out that they can be interpreted in terms of the quantum noise of...

We prove homological mirror symmetry for a smooth Calabi-Yau hypersurface in projective space. In the one-dimensional case, this is the elliptic curve, and our result is related to that of Polishchuk-Zaslow; in the two-dimensional case, it is the K3...

Every smooth affine variety has a natural symplectic structure coming from some embedding in complex Euclidean space. This symplectic form is a biholomorphic invariant. An important algebraic invariant of smooth affine varieties is log Kodaira...

The question of what conditions guarantee that a symplectic circle action is Hamiltonian has been studied for many years. In 1998, Sue Tolman and Jonathon Weitsman proved that if the action is semifree and has a non-empty set of isolated fixed...

Welschinger invariants, real analogs of genus 0 Gromov-Witten invariants, provide non-trivial lower bounds in real algebraic geometry. In this talk I will explain how to get some wall-crossing formulas relating Welschinger invariants of the same (up...

We establish a derived equivalence of the Fukaya category of the 2-torus, relative to a basepoint, with the category of perfect complexes on the Tate curve over Z[q]. It specializes to an equivalence, over Z, of the Fukaya category of the punctured...

This talk is part of a circle of ideas that one could call ``categorical dynamics''. We look at how objects of the Fukaya category move under deformations prescribed by fixing an odd degree quantum cohomology class. This is an analogue of moving...

I'll discuss a bound on the length of the boundary of a J-holomorphic curve with Lagrangian boundary conditions by a constant times its area. The constant depends on the symplectic form, the almost complex structure, the Lagrangian boundary...