(joint w/ B.Bakker) For an elliptic curve E over a field k, the p-torsion E[p] gives a 2-dimensional representation of the Galois group G_k over F_p. For k=Q and p>13, the Frey-Mazur conjecture famously states that one can recover the isogeny class of E from the representaiton E[p]. We state and prove a direct analogue of this question over function fields of complex algebraic curves. Specifically, for any complex algebraic curve C, let k(C) be its field of rational functions. Then there exists a constant A(C), such that for all primes p>A(C), isogeny classes of elliptic curves E over k(C) can be recovered from E[p]. Moreover, we show that A(C) can be made to depend only on the gonality of C, which can be thought of as the analogous notion of degree for number fields. The study of this question will lead us into the realm of moduli spaces and hyperbolic geometry. The use of the latter means that, unfortunately, these methods don't apply in finite characteristic.

In his recent paper,\Beyond Endoscopy", Langlands proposed an approach to (ultimately) attack the general functoriality conjectures by means of the trace formula. For a (reductive algebraic) group G over a global field F and a representation of its L-group, the strategy, among other things, aims at detecting those automorphic representations of G for which the L-function, L(s;\pi ;\rho ), has a pole at s = 1. The method suggested using the the trace formula together with an averaging process to capture these poles. In this talk we will start by recalling the functoriality conjectures and brie y describe the method suggested by Langlands. Then, specializing on the group GL(2) we will discuss some recent work on Beyond Endoscopy. More precisely, we will discuss the elliptic part of the trace formula and the analytic problems caused by the volumes of tori, singularities of orbital integrals and the non-tempered terms. We will then describe how one can use an approximate functional equation in the trace formula to rewrite the elliptic part which resolves these issues. Finally, we will talk about applications of the resulting formula.

I will build a commutative ring that satisfies "standard conjectures", starting from a finite field. What is this ring? What does it say about the finite field? This talk will be elementary: No background beyond the first year graduate algebra will be necessary. Joint with Mats Boij, Bill Huang, and Greg Smith.

The prime numbers are the multiplicative building blocks of the integers, and much thought has been given towards understanding their behavior. In this talk, we will examine prime numbers from two points of view. We will first consider questions on the distribution of the primes. In particular, we will illustrate how the breakthrough work of Maynard and Tao on bounded gaps between primes settles an old problem of Erdos and Turan. Secondly, we will explore the relationship between prime numbers and zeros of the Riemann zeta-function, as a way to motivate the study of the moments of the Riemann zeta function and more general L-functions. In particular, we consider arbitrary products of L-functions attached to irreducible cuspidal automorphic representations of GL(m) over the rationals. The Langlands program suggests essentially all L-functions are of this form. Assuming some standard conjectures, I will discuss how to estimate two types of moments: the continuous moment of an arbitrary product of primitive automorphic L-functions and the discrete moment (taken over fundamental discriminants) of an arbitrary product of primitive automorphic L-functions twisted by quadratic Dirichlet characters.

If $K$ is the rational function field $K=\mathbb{Q}(t)$, then a polynomial $f$ in $K[x]$ can be regarded as a one-parameter family of polynomials over $\mathbb{Q}$. If $f$ is irreducible, then a basic form of Hilbert's irreducibility theorem states that there are infinitely many $t$ in $\mathbb{Q}$ for which the specialized polynomial $f_t$ is irreducible over $\mathbb{Q}$. In this talk we will discuss analogous theorems for an abelian variety $A/K$ regarded as a one-parameter family of abelian varieties over $K$. For example, we will exhibit $A$ which are simple over $K$ and for which there are only finitely many $t$ in $\mathbb{Q}$ such that the abelian variety $A_t$ is not simple over $\mathbb{Q}$.

Piper Harron presents the delightfully mathematical one woman show that answers questions her audience may have never asked itself before now! Such as: What is the shape of a number field? And: How do we show shapes are equidistributed? She will sketch the proof, providing references to old stuff and details to new stuff. Come one, come all (people, especially graduate students, interested in number theory)!

There are well-known explicit families of K3 surfaces equipped with a low degree polarization, e.g., quartic surfaces in P^3. What if one specifies multiple line bundles instead of a single one? We will discuss representation-theoretic constructions of such families, i.e., moduli spaces for K3 surfaces whose Neron-Severi groups contain specified lattices. These constructions, inspired by arithmetic considerations, also involve some fun geometry and combinatorics. This is joint work with Manjul Bhargava and Abhinav Kumar.

In the previous talk, we discussed the algebraic de Rham theory for unipotent fundamental groups of elliptic curves. In this talk, we generalize it to a Q-de Rham theory for the relative completion of the modular group, the (orbifold) fundamental group of the modular curve. Using Chen's method of power series connections, we construct a connection on the modular curve that generalizes the elliptic KZB connection on an elliptic curve. By Tannaka duality, it can be viewed as a universal relative unipotent connection with a regular singularity at the cusp. This connection enables us to construct iterated integrals of modular forms, possibly 'of the second kind', that provide periods called 'multiple modular values' by Brown. These periods include multiple zeta values and periods of modular forms.

Beatty sequences are generalized arithmetic progressions which have been studied intensively in recent years. Thanks to the work of Vinogradov, it is known that every Beatty sequence S contains "appropriately many" prime numbers. For a given pair of Beatty sequences S and T, it is natural to wonder whether there are "appropriately many" primes in S for which the next larger prime lies in T. In this talk, I will show that this is indeed the case if one assumes a certain strong form of the Hardy-Littlewood conjectures. This is recent joint work with Victor Guo.

For a non-degenerate integral quadratic form F(x1,...,xd) in 5 (or more) variables, we prove an optimal strong approximation theorem. Fix any compact subspace Ω⊂Rd of the affine quadric F(x1,...,xd)=1. Suppose that we are given a small ball B of radius 0 < r < 1 inside Ω, and an integer m. Further assume that N is a given integer which satisfies N ≫ (r−1m)4+ε for any ε > 0. Finally assume that we are given an integral vector (λ1, . . . , λd) mod m. Then we show that there exists an integral solution x = (x1, . . . , xd) x of F(x)=N such that xi ≡λi mod m and √N ∈B, provided that all the local conditions are satisfied. We also show that 4 is the best possible exponent. Moreover, for a non-degenerate integral quadratic form F (x1 , . . . , x4 ) in 4 variables we prove the same result if N ≥ (r−1m)6+ε and N is not divisible by 2k where 2k ≫ Nε for any ε. Based on some numerical experiments on the diameter of LPS Ramanujan graphs, we conjecture that the optimal strong approximation theorem holds for any quadratic form F(X) in 4 variables with the optimal exponent 4.

In the early 2000s Gan, Gross and Prasad made remarkable conjectures relating the non-vanishing of central values of certain Rankin-Selberg L-functions to the non-vanishing of certain explicit integrals of automorphic forms, called 'automorphic periods', on classical groups. They have been subsequently refined by Ichino-Ikeda and Neal Harris into precise conjectural identities relating these two invariants thus generalizing a famous result of Waldspurger for toric periods on GL(2). In the case of unitary groups, those have been established by Wei Zhang under some local restrictions. I will review the current state of the art on this and in particular how certain results in local harmonic analysis allow to remove almost all the local restrictions made by Zhang.

Random Matrix Theory has successfully modeled the behavior of zeros of elliptic curve L-functions in the limit of large conductors. We explore the behavior of zeros near the central point for one-parameter families of elliptic curves with rank over Q(T) and small conductors. Zeros of L-functions are conjectured to be simple except possibly at the central point for deep arithmetic reasons; these families provide a fascinating laboratory to explore the effect of multiple zeros on nearby zeros. Though theory suggests the family zeros (those we believe exist due to the Birch and Swinnerton-Dyer Conjecture) should not interact with the remaining zeros, numerical calculations show this is not the case; however, the discrepency is likely due to small conductors, and unlike excess rank is observed to noticeably decrease as we increase the conductors. We shall mix theory and experiment and see some surprisingly results, which leads us to conjecture that a discretized Jacobi ensemble correctly models the small conductor behavior.

The remarkable Cheeger-Muller theorem, of differential geometric origin, provides an analytic means of studying torsion in the cohomology of Riemannian manifolds. We describe how this theorem can be applied to prove a numerical form of Langlands' base change functoriality for torsion in cohomology.

A celebrated result of Burgess proves nontrivial bounds for short multiplicative character sums. In general, bounds for short character sums have utility in a wide range of problems in number theory, and it would be highly desirable to extend Burgess’s method to apply to more general character sums. This talk presents new work in this direction, joint with Roger Heath-Brown, that proves nontrivial bounds for short mixed character sums in which the additive character is evaluated at a real-valued polynomial. Our approach, via a version of the Burgess method, includes a novel application of the recent results of Wooley on the Vinogradov mean value theorem.

Joint work with Jacob Tsimerman. Let B(g,p) denote the number of isomorphism classes of g-dimensional abelian varieties over the finite field of size p. Let A(g,p) denote the number of isomorphism classes of principally polarized g dimensional abelian varieties over the finite field of size p. We derive upper bounds for B(g,p) and lower bounds for A(g,p) for p fixed and g increasing. The extremely large gap between the lower bound for A(g,p) and the upper bound B(g,p) implies some statistically counterintuitive behavior for abelian varieties of large dimension over a fixed finite field.

The Malle-Bhargava heuristics give asymptotic predictions for the density of number fields of bounded discriminant with a given Galois group G, in terms of the number of G-extensions of p-adic fields Q_p. These heuristics can also be applied when the discriminant is replaced by any of a wide variety of other “counting functions”. I’ll discuss how some of these alternate counting functions are built, the idea of global mass formulas, and some cases in which the heuristic predictions can be compared to known results.

For a K3 surface X over a number field with potentially good reduction everywhere, we prove that there are infinitely many primes modulo which the reduction of X has larger geometric Picard rank than that of the generic fiber X. A similar statement still holds true for ordinary K3 surfaces over global function fields. In this talk, I will present the proofs via the intersection theory on GSpin Shimura varieties and also discuss various applications. These results are joint work with Ananth Shankar, Arul Shankar, and Salim Tayou and with Davesh Maulik and Ananth Shankar.

By studying universal motivic unipotent representations of fundamental group of varieties and comparing their different realizations, we combine Kim's recent method in Diophantine geometry with Deligne-Goncharov's theory of motivic fundamental groups to develop a machinery for approaching Diophantine problems concerning integral points.

In this talk, I will introduce the famous result by Gangl-Kaneko-Zagier about a family of period polynomial relations among double zeta value of even weight. Then I will generalize their result in various ways, from which we can see the appearance of modular forms in low levels. At the end, I will give a generalization of the Eichler-Shimura-Manin correspondence to the case of the space of newforms of level 2 and 3 and a certain period polynomial space.

I will begin with an introduction to Galois deformation theory and its role in modularity lifting. This will motivate the study of local deformation rings and more specifically flat deformation rings. I will then discuss Kisin's work on flat deformations and explain how to generalize to Galois representations valued in an arbitrary reductive group. Kisin's techniques led to the successful determination of the connected components of the flat deformation ring in the 2-dimensional case. If time permits, I will touch on difficulties of going beyond GL_2.

Waves in free-space diffractively spread, while waves in a spatially non-homogeneous medium undergo a combination of scattering and localization.
In many applications, e.g. photonic and quantum systems, one is interested in controlled localization of wave energy. Edge states are a type of localization along a line-defect, the interface between different media.
Topologically protected edge states are a class of edge states which are robust to strong local distortions of the edge. They are therefore potential vehicles for robust energy-transfer in the presence of defects and random imperfections. These states arise, for example, in graphene and its photonic analogues.
We first review the mathematics of dispersive waves in periodic media and discuss examples of wave localization by a defect.
We then specialize to the case of honeycomb structures (such as grapheme) and discuss their novel properties. Finally we introduce and discuss a rich family of continuum partial differential equation (Schroedinger) models, admitting edge states which are topologically protected and those which are not.

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 my recent proof of the Gross-Stark conjecture, a p-adic version of Stark's Conjecture that relates the leading term of the Deligne-Ribet p-adic L-function to a determinant of p-adic logarithms of p-units in abelian extensions. Next I will state my refinement of the Gross-Stark conjecture that gives an exact formula for Gross-Stark units. I will conclude with a description of work in progress that aims to prove this conjecture and thereby give a p-adic solution to Hilbert's 12th problem.