We prove an analogue of the Prime Number Theorem for short intervals on a smooth proper curve of arbitrary genus over a finite field. Our main result gives a uniform asymptotic count of those rational functions, inside short intervals defined by a very ample effective divisor E, whose principal divisors are prime away from E. In this talk, I will discuss the setting and definitions we use in order to make sense of such a count, and will give a rough sketch of the proof. This is a joint work with Tyler Foster.

Following the method developed by Waldspurger and Beuzart-Plessis in their proof of the local Gan-Gross-Prasad conjecture, we were able to prove the multiplicity one theorem on Vogan L-packet for the Ginzburg-Rallis model. In some cases, we also proved the epsilon dichotomy conjecture which gives a relation between the multiplicity and the value of the exterior cube epsilon factor.

We provide a construction of free factorization algebras in algebraic geometry and link factorization homology of a scheme with coefficients in a free factorization algebra to the homology of its (unordered) configuration spaces. As an application, we obtain a purely algebro-geometric proof of homological stability of configuration spaces.

The speaker will discuss recent work on Manin's theory of zeta polynomials for modular forms. He will describe recent results which confirm Manin's speculation that there is such a theory which arises from periods of newforms. More precisely, for each even weight k>2 newform f, the speaker will describe a canonical polynomial Zf(s) which satisfies a functional equation of the form Zf(s)=Zf(1−s), and also satisfies the Riemann Hypothesis: if Zf(ρ)=0, then Re(ρ)=1/2. This zeta function is arithmetic in nature in that it encodes the moments of the critical values of L(f,s). This work builds on earlier results of many people on period polynomials of modular forms. This is joint work with Seokho Jin, Wenjun Ma, Larry Rolen, Kannan Soundararajan, and Florian Sprung.

I will first introduce some generalities about exponential sums, in particular that square-root cancellation is expected for many algebraic character sums over the rational points of an algebraic variety over a finite field. I will then set the stage for my work: we consider a family of exponential sums, which in our case is parameterized by the rational points of a variety (the parameter space). Our task is to obtain a good bound on the number of exceptional ("bad") parameters for which square-root cancellations fail. Following an idea of Michael Larsen, we consider even moments of the family of exponential sums. If the summands are of certain product form, a transformation can be applied to produce another family of exponential sums (of the same type). If the summands are products of multiplicative characters composed with certain polynomial functions, a weak bound can then be applied to the character sums in this family (with few bad parameters), yielding bounds for the moments. We know from the theory of l-adic sheaves that the parameter space for the original family have a stratification by smooth varieties, which is uniform in some sense as long as the degrees of the characters and polynomials are bounded. Moreover, on each stratum the character sum behave in certain uniform way, so that we can talk about good and bad strata. The bounds on moments yield bounds on dimensions of bad strata, which in turn yield bounds on the number of bad parameters (in any box) of the original family. Though not optimal, the bounds already imply nontrivial Burgess bounds for forms, in joint work with Lillian Pierce.

The theory of congruences between automorphic forms traces back to Ramanujan, who observed various congruence properties between coefficients of generating functions related to the partition function. Since then, the subject has evolved to become a central piece of contemporary number theory, lying at the heart of spectacular achievements such as the proof of Fermat's Last Theorem and the Sato-Tate conjecture. In my talk I will explain how the modern theory gives satisfactory explanations of some concrete phenomena for modular forms (the GL_2 case), and discuss recent progress concerning automorphic forms for higher rank groups.

A result of Gelfand and Graev shows that the supercuspidal representations of SL_2 are neatly parameterized by characters of elliptic tori, and that the stable character data for all such representations may be collected into a single function by means of a Fourier Transform. Using recent advances in the computation of characters of supercuspidal representations, we prove analogous results for the group SL_n.

TBD

Abstract: We will describe the multizeta for function fields and the relevant results, conjectures and structures.

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.

Calculating multiple zeta values at arguments of mixed signs in a way that is compatible with both the quasi-shuffle product and the meromorphic continuation, is commonly referred to as the renormalization problem for multiple zeta values. In this talk, we consider the set of all solutions to this problem and provide a framework for comparing its elements in terms of a free and transitive action of a particular subgroup of the group of characters of the quasi-shuffle Hopf algebra. This provides a transparent way of relating different solutions at non-positive values, which answers an open question in the recent literature. This is a joint work with Ebrahimi-Fard, Manchon and Singer.

A number of computational problems on graphs can be solved using algorithms based on the spectrum of a matrix associated with the graph. On very sparse graphs the traditionally-considered matrices develop spurious large eigenvalues associated with localized eigenvectors that harm the algorithmic performance. Inspired by the theory of spin glasses, we introduce the non-backtracking operator that is able to mitigate this problem. We discuss properties of this operator, as well as its applications to several algorithmic problems such as clustering of networks, percolation, matrix completion or inference from pairwise comparisons.

Over the last decades, the study of "spin glasses" in physics has stimulated a large amount of theoretical activity in physics, and led to several breakthroughs. While the original puzzle of spin glass materials is still not fully solved, their theoretical analysis has created powerful techniques as well as a rich conceptual framework, to study emergent properties of strongly disordered and interacting systems. In this talk, I will use these tools and discuss how apparently unrelated complex problems such as: how to pack many objects in a given volume, how to color a graph with a given number of colors, why a liquid is turning into a glass when the temperature is lowered, and why some computational (classical and quantum) problems are hard while other are easy — actually (and surprisingly) do share many characteristics when looking at them through the (spin) glass.

In this talk, I will introduce a complete proof of a standard conjecture on the local converse theorem for generic representations of GLn(F), where F is a non-archimedean local field. This is a joint work with Prof. Herve Jacquet.

Our starting point will be the relationship between hydrodynamics from the macroscopic continuum point of view and its atomistic interpretation in statistical mechanical terms. Then we extend stationary state (equilibrium and nonequilibrium) Molecular Dynamics to time dependent situations, including response and relaxation. We call the procedure Dynamical Non-Equilibrium Molecular Dynamics (D-NEMD), to distinguish it from standard NEMD, rigorously useful only to simulate stationary nonequilibrium states. It is, in essence, a generalization of linear response theory. The idea, formulated by Onsager in the thirties in metaphysical language, has received a solid foundation in the fifties by the work of Kubo (in the linear and nonlinear regimes). Adapted to MD simulations by G.Jacucci, I.R.Mac Donald and myself in the seventies, it has been baptized as the (nonlinear) Kubo-Onsager relation, connecting dynamical nonequilibrium averages or dynamical relaxations to initial distribution which can be sampled in MD by stationary processes. Finally we go back to hydrodynamics, to illustrate the method, by studying the hydrodynamic relaxation of an interface between two immiscible liquids.

Colmez conjectured a formula relating periods of abelian varieties with complex multiplication to derivatives of Artin L-functions. I’ll explain how to prove an averaged version of Colmez’s conjectural formula, using the arithmetic of integral models of orthogonal Shimura varieties. This is joint work with F. Andreatta, E. Goren, and K. Madapusi Pera.

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.

Kazhdan and Patterson constructed generalized theta representations on covers of general linear groups as multi-residues of the Borel Eisenstein series. These representations and their unique models were used by Bump and Ginzburg in the Rankin-Selberg constructions of the symmetric square L-functions for GL(r). In this talk, we will discuss the two other types of models that the theta representations may support. We first talk about semi-Whittaker models, which generalize the models used in the work of Bump and Ginzburg. Secondly, we determine the unipotent orbits attached to theta functions, in the sense of Ginzburg. We also determine the covers when these models are unique. Time permitting, we will discuss some applications in Rankin-Selberg constructions.

I will discuss recent results towards the weight part of Serre's conjecture for GL_n as formulated by Herzig. The conjecture predicts the set of weights where an odd n-dimensional mod p Galois representation will appear in cohomology (modular weights) in terms of the restriction of the representation to the decomposition group at p. We show that the set of modular weights is always contained in the predicted set in generic situations. This is joint work with Daniel Le and Bao V. Le Hung.

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.

In this talk we describe a partially successful attempt to describe a characteristic p > 0 analog of the locally symmetric spaces for GL(n,R), by interpreting this as a "moduli space" for abelian varieties with real structure.

I will explain how to attach ideal classes of cyclotomic fields to geodesics in the complex upper half-plane. A conjecture of mine states this construction is inverse to another arising from the Galois action on cohomology of modular curves modulo an Eisenstein ideal. I hope to use this to motivate a broader philosophy, developed jointly with Takako Fukaya and Kazuya Kato, that certain arithmetic objects attached to Galois representations of global fields can be described using higher-dimensional modular symbols.

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)!

Let $K$ be an imaginary quadratic field with ring of integers $\mathcal{O}_K$. The Schmidt arrangement of $K$ is the orbit of the extended real line in the extended complex plane under the Mobius transformation action of the Bianchi group $\operatorname{PSL}(2,\mathcal{O}_K)$. The arrangement takes the form of a dense collection of intricately nested circles. Aspects of the number theory of $\mathcal{O}_K$ can be characterised by properties of this picture: for example, the arrangement is connected if and only if $\mathcal{O}_K$ is Euclidean. I'll explore this structure and its connection to Apollonian circle packings. Specifically, the Schmidt arrangement for the Gaussian integers is a disjoint union of all primitive integral Apollonian circle packings. Generalizing this relationship to all imaginary quadratic $K$, the geometry naturally defines some new circle packings and thin groups of arithmetic interest.

I will discuss the relation between singularities in formal arc spaces of certain reductive monoids and local factors of automorphic L-functions.

Given an elliptic curve $E/\mathbb{Q}$, let $E[2^k]$ denote the set of points on $E$ that have order dividing $2^k$. The coordinates of these points are algebraic numbers and using them, one can build a Galois representation $\rho : G_{\mathbb{Q}} \to \GL_{2}(\mathbb{Z}_{2})$. We give a classification of all possible images of this Galois representation. To this end, we compute the 'arithmetically maximal' tower of 2-power level modular curves, develop techniques to compute their equations, and classify the rational points on these curves.

The zeta function of a curve C over a finite field can be expressed in terms of the characteristic polynomial of its Frobenius endomorphism. We will see how one can compute the trace of high powers of this endomorphism in various moduli spaces. Finally, we will discuss how one can use this information to compute the one-level density -- which concerns low-lying zeros of the zeta function -- in the case of cubic covers of the projective line.

We give an overview of a general trend of results that say that the distribution of the number of F_q-points of certain families of curves of genus g is asymptotically given by a sum of q+1 independent, identically distributed random variables as g goes to infinity. In particular, we discuss the distribution of the number of F_q-points for cyclic l-covers of genus g. (This is joint work with Bucur, David, Feigon, Kaplan, Ozman, Wood.) This work generalizes previous results in which only connected components of the moduli space were considered.

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.