The Geometric Manin's conjectures - talk discussing multiple potential formulations of a geometric analogue of Manin's conjecture, including one for which my work with Tim Browning on the cohomology of moduli spaces of rational curves provides evidence.
The lucky logarithmic derivative - talk on my work on sums of the divisor function in short intervals over function fields and its application to moments of L-functions, using the stable cohomology method
Applications of Exponential Sums - talk on my work with Emmanuel Kowalski and Philippe Michel proving exponential sum estimates with applications to moments of L-functions and level of distribution estimates for modular forms
Abstract: There are many problems about counting special types of numbers (primes or other numbers with special factorizations) in arithmetic progressions, or summing arithmetic functions in arithmetic progressions. These all have analogues polynomials over a finite field. Recently I proved, by a geometric method, strong bounds for these analogues (approaching level of distribution 1 and square-root cancellation as the size of the finite field goes to infinity). I will explain how these bounds relate to those obtained from a simpler approach using the Riemann hypothesis (i.e. by using Fourier analysis on the multiplicative group) and how we can deduce, using a classical probability-theoretic method, a result that applies to every factorization type at once.
Abstract: Faltings proved the statement, previously conjectured by Shafarevich, that there are finitely many abelian varieties of dimension n, defined over a fixed number field, with good reduction outside a fixed finite set of primes, up to isomorphism. In joint work with Brian Lawrence, we prove an analogous finiteness statement for hypersurfaces in a fixed abelian variety with good reduction outside a finite set of primes. I will give a broad introduction to some of the ideas in the proof, which builds on p-adic Hodge theory techniques from work of Lawrence and Venkatesh as well as sheaf convolution in algebraic geometry.
Abstract: Manin's conjecture on rational points suggests a number of different conjectures on moduli spaces of curves on algebraic varieties, of varying levels of generality and strength, that can all be considered its geometric analogue. I will survey a few of these conjectures and then discuss my work with Tim Browning making progress towards one of them.
Abstract: The Bateman-Horn conjecture predicts the fraction of integers n such that n2+1 is prime, and makes similar predictions for polynomials of higher degree. In joint work in progress with Mark Shusterman, we prove an analogue of the n2+1 case, replacing natural numbers n with polynomials in Fq[u], which for instance counts the fraction of polynomials f such that f2+u is an irreducible polynomial. The proof combines geometric methods, unusual algebraic properties of polynomials, and some (very) classical number theory.
Abstract: The cohomology of the space of degree d holomorphic maps from the complex projective line to a sufficiently nice algebraic variety is expected to stabilize as d goes to infinity. The limit is expected to be the cohomology of the double loop space, i.e. the space of degree d continuous maps from the sphere to that variety. This was shown for projective space by Segal, and there has been further subsequent work. In joint work with Tim Browning, we give a new approach to the problem for smooth affine hypersurfaces of low degree (which should also work for projective hypersurfaces, complete intersections, and/or higher genus curves), based on methods from analytic number theory. We take an argument of Birch that solves the number-theoretic analogue of this problem and translate it, step by step, into the language of ell-adic sheaf theory using the sheaf-function dictionary. This produces a spectral sequence that computes the cohomology, whose degeneration would imply that the rational compactly-supported cohomology matches that of the double loop space.
Abstract: Poonen and Voloch have conjectured that almost every degree d Fano hypersurface in Pn defined over the field of rational numbers satisfies the Hasse principle. In joint work with T. Browning and P. Le Boudec, we establish this conjecture under the mild assumption that n ≥ d + 1. This talk is the first of two talks. Our goal in this first talk will be to introduce our main results and to present the strategy of the proof.
Abstract: We study the function field analogue of a classical problem in analytic number theory on the sums of the generalized divisor function in short intervals, in the limit as the degrees of the polynomials go to infinity. As a corollary, we calculate arbitrarily many moments of a certain family of L-functions, in the limit as the conductor goes to infinity. This is done by showing a cohomology vanishing result using a general bound due to Katz and some elementary calculations with polynomials. This method is based on work of Hast and Matei, except that thanks to a trick involving the logarithmic derivative, we are able to achieve a much smaller error term than is possible by this method for a "typical" problem of this type.
Abstract: Let Fq be a finite field and Fq[T] the
ring of polynomials in one variable over Fq. There is a
theory of modular forms invariant under congruence subgroups of GL2(Fq[T]) that is analogous to the classical theory of modular
forms invariant under congruence subgroups of GL2(Z). We
study an analogue of the classical sup-norm problem, which asks for
bounds on the largest value of a cusp form, in this setting. We
obtain, for forms of squarefree level with trivial central character,
a bound stronger than the analogous bounds in the classical setting,
as long as q>134. This uses a new geometric method which should also
apply to automorphic forms on more general groups over function
fields. I will explain the background material and some of the key
ideas that go into the proof.
This talk is based on work in progress. Thanks to Columbia University for recording this video.
Abstract: The configuration spaces of n-tuples of points in the affine line which are unordered but grouped into k different "colors" have a multitude of one-dimensional local systems. The intersection cohomology complexes of these local systems are a natural extension of them to a larger space where these points may collide. These complexes have a deep relation with the "multiple Dirichlet series" studied in analytic number theory. I will describe some of their nice geometric properties, inspired by ideas from analytic number theory, and express the hope that further useful geometric properties could be established.
Abstract: Sums of Kloosterman sums against other functions often arise in analytic number theory problems, but it is difficult to prove bounds when the length of the sum is below the square root of the modulus. In joint work with Emmanuel Kowalski and Philippe Michel, we proved a bound on bilinear forms in Kloosterman sums where the length is just below this range. This has applications to moments of L-values of modular forms and to the level of distribution of Eisenstein series. I will explain this result and discuss possible generalizations.
Abstract: The Riemann hypothesis over function fields is a theorem of Weil, with a more general form due to Deligne. This raises the hope of answering more precise questions on the zeroes of L- functions, such as the distribution of the zeroes in a suitable family of L-functions. Powerful results on this have been proved, starting with the book of Katz and Sarnak, but the theory is not yet complete, and it is an area of active research. I will survey the history, explain recent developments, and discuss what is in reach for the future.
Abstract: The breakthrough work of Marcus, Spielman, and Srivastava showed that every bipartite Ramanujan graph has a bipartite Ramanujan double cover. Chris Hall, Doron Puder, and I generalized this to covers of arbitrary degree. I will explain the proof, with emphasis on how group theory and representation theory are useful for this problem.
Abstract: I will explain the analogy between trace functions over finite fields defined by exponential sums and certain classical functions on the complex numbers defined by integrals of exponentials. There are close analogies, largely due to Katz, that sometimes allow one to guess results in one domain from results in the other. For instance, many important properties of Kloosterman sums are related to facts about Bessel functions. I will explain some of these correspondences, and how to use them to understand exponential sums.
Abstract: In joint work with Emmanuel Kowalski and Philippe Michel, we prove two different estimates on sums of coefficients of modular forms---one related to L-functions and another to the level of distribution. A key step in the argument is a careful analysis of vanishing cycles, a tool originally developed by Lefschetz to study the topology of algebraic varieties. We will explain why this is helpful for these problems.