Lenka Zdeborova : The spectral redemption comes from no backtracking
- Number Theory ( 102 Views )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.
Ayla Gafni : Extremal primes for elliptic curves without complex multiplication
- Number Theory ( 167 Views )Fix an elliptic curve $E$ over $\mathbb{Q}$. An ''extremal prime'' for $E$ is a prime $p$ of good reduction such that the number of rational points on $E$ modulo $p$ is maximal or minimal in relation to the Hasse bound. In this talk, I will discuss what is known and conjectured about the number of extremal primes $p\le X$, and give the first non-trivial upper bound for the number of such primes when $E$ is a curve without complex multiplication. The result is conditional on the hypothesis that all the symmetric power $L$-functions associated to $E$ are automorphic and satisfy the Generalized Riemann Hypothesis. In order to obtain this bound, we use explicit equidistribution for the Sato-Tate measure as in recent work of Rouse and Thorner, and refine certain intermediate estimates taking advantage of the fact that extremal primes have a very small Sato-Tate measure.
John Voight : Presentations for rings of modular forms
- Number Theory ( 105 Views )We give an explicit presentation for the ring of modular forms for a Fuchsian group with cofinite area, depending on the signature of the group. Our work can be seen as a generalization of the classical theorem of Petri: we give a presentation for the canonical ring of a stacky curve. This is joint work with David Zureick-Brown.
Florent Krzakala : From spin glasses to Packing, Freezing and Computing problems
- Number Theory ( 109 Views )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.
Michal Zydor : Relative trace formula of Jacquet-Rallis, recent progress
- Number Theory ( 178 Views )I will discuss the relative trace formula approach to the global Gan-Gross-Prasad conjectures for unitary groups. The focus will be on the spectral side. I will present the various terms that appear in the spectral development of the relative trace formula and discuss what is still missing. This is a joint work with Pierre-Henri Chaudouard.
Baiying Liu : On the Local Converse Theorem for p-adic GLn.
- Number Theory ( 116 Views )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.
Ma Luo : Algebraic iterated integrals on elliptic curves
- Number Theory ( 132 Views )I will give two talks on algebraic iterated integrals. In this first one, I will focus on the case of once punctured elliptic curves over a field of characteristic zero, and describe an algebraic de Rham theory for their unipotent fundamental groups by using the elliptic KZB connection. This connection is explicitly expressed by algebraic 1-forms, which are used to construct algebraic iterated integrals on elliptic curves. It also gives an explicit version of Tannaka duality for unipotent connections over an elliptic curve with a regular singular point at the identity.
Abhishek Parab : Absolute convergence of the twisted Arthur-Selberg trace formula
- Number Theory ( 103 Views )We show that the distributions occurring in the geometric and spectral side of the twisted Arthur-Selberg trace formula extend to non-compactly supported test functions. The geometric assertion is modulo a hypothesis on root systems proven among other cases, when the group is split. This result extends the work of Finis-Lapid (and Muller, spectral side) in the non-twisted setting. In the end, we will give an application towards residues of Rankin-Selberg L-functions suggested by J. Getz.
Romyar Sharifi : Modular symbols and arithmetic
- Number Theory ( 103 Views )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.
Hunter Brooks : Special Value Formulas for Rankin-Selberg p-adic L-Functions
- Number Theory ( 162 Views )We discuss special value formulas for a p-adic L-function L_p(f, \chi), where f is a fixed newform and \chi varies over the space of Hecke characters of a fixed imaginary quadratic field, as well as some recent applications. These formulas, first found by Bertolini, Darmon, and Prasanna, relate L_p(f, \mathbb{1}), a value which is outside the range of interpolation defining L_p, to arithmetic invariants of cycles on varieties fibered over modular curves.
Ken Ono : Special values of modular shifted convolution Dirichlet series
- Number Theory ( 129 Views )Rankin-Selberg convolution L-functions are important functions in number theory. Their properties play a central role in many of deepest works on the Ramanujan-Petersson Conjecture. In a recent paper, Hoffstein and Hulse defined generalizations of these L-functions, the so-called "shifted-convolution" L-functions. They obtained the meromorphic continuation of the functions in many cases. Here we consider symmetrizations of these L-functions, and we exactly evaluate their special values at diagonal weights for all shifts. This is joint work with Michael Mertens.
Ben Howard : Periods of CM abelian varieties
- Number Theory ( 99 Views )Colmez conjectured a formula relating periods of abelian varieties with complex multiplication to derivatives of Artin L-functions. Ill explain how to prove an averaged version of Colmezs 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.
P. E. Herman : ON PATTERSONS CONJECTURE: SUMS OF EXPONENTIAL SUMS.
- Number Theory ( 118 Views )It is well known that for an exponential sum with a prime modulus the best bound for the sum comes from Weil's famous estimation. In this talk, we discuss when this bound can be improved on average over integral modulus in a number field. Investigations into exponential sums on average, or sums of exponential sums, have many applications including the Riemann hypothesis and the Ramanujan conjecture for automorphic forms. In particular, we will get an asymptotic for sums of quartic exponential sums over the Gaussian integers. Tools we will use to get this asymptotic include automorphic forms and the trace formula.
Jack Buttcane : Kuznetsov, higher weight and exponential sums on GL(3)
- Number Theory ( 117 Views )I will discuss the relationship between the Kuznetsov formula and certain exponential sums that arise naturally on GL(3). This will lead us to consider the structure of GL(3) Maass forms having non-trivial dependence on the SO(3) part of the Iwasawa decomposition.
Chung Pang Mok : Introduction to Mochizukis works on inter-universal Teichmuller theory
- Number Theory ( 165 Views )Inter-universal Teichmuller theory, as developed by Mochizuki in the past decade, is an analogue for number fields of the classical Teichmuller theory, and also of the p-adic Teichmuller theory of Mochizuki. In this theory, the ring structure of a number field is subject to non-ring theoretic deformation. Absolute anabelian geometry, a refinement of anabelian geometry, plays a crucial role in inter-universal Teichmuller theory. In this talk, we will try to give an introduction to these ideas.
Jacek Brodzki : A generalised Julg-Valette complex for CAT(0)-cube complexes.
- Number Theory ( 138 Views )This talk will introduce a very natural and interesting differential complex associated with a CAT(0)-cube complex. The construction builds on ideas first introduced by Pytlik and Szwarc for the free group and extended by Julg and Valette in the case of groups acting on trees. We will extend ideas of Julg-Valette to show how this construction can be used to study K-amenability and K-homology of groups acting on CAT(0)-cube complexes. This talk is based on joint work with Erik Guentner and Nigel Higson.
Junyan Xu : Bounds for certain families of character sums: how to obtain strong bounds with more exceptions from weak bounds with fewer exceptions
- Number Theory ( 213 Views )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.
Ramesh Sreekantan : Cycles on Abelian surfaces
- Number Theory ( 156 Views )In this talk we use generalizations of classical geometric constructions of Kummer and Humbert to construct new higher Chow cycles on Abelian surfaces and K3 surfaces over p-adic local fields, generalising some work of Collino. The existence of these cycles is predicted by the poles of the local L-factor at p of the L-function of the Abelian surface. The techniques involve using some recent work of Bogomolov, Hassett and Tschinkel on the deformations of rational curves on K3 surfaces. As an application we use these cycles to prove an analogue of the Hodge-D-conjecture for Abelian surfaces.
Ila Varma : Counting $D_4$-quartic fields ordered by conductor
- Number Theory ( 154 Views )We consider the family of $D_4$-quartic fields ordered by the Artin conductors of the corresponding 2-dimensional irreducible Galois representations. In this talk, I will describe ways to compute the number of such $D_4$ fields with bounded conductor. Traditionally, there have been two approaches to counting quartic fields, using arithmetic invariant theory in combination of geometry-of-number techniques, and applying Kummer theory together with L-function methods. Both of these strategies fall short in the case of $D_4$ fields since counting quartic fields containing a quadratic subfield of large discriminant is difficult. However, when ordering by conductor, these techniques can be utilized due to additional algebraic structure that the Galois closures of such quartic fields have, arising from the outer automorphism of $D_4$. This result is joint work with Ali Altug, Arul Shankar, and Kevin Wilson.
Alina Bucur : Traces of high powers of Frobenius for cubic covers of the projective line over finite fields
- Number Theory ( 120 Views )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.
Jiuya Wang : Inductive Method in Counting Number Fields
- Number Theory ( 189 Views )We propose general frameworks to inductively count number fields building on previously known counting results and good uniformity estimates in different flavors. By this method, we prove new results in counting number fields with Galois groups ranging from direct product to wreath product. We will also mention interesting applications en route. This involves my thesis and on going project with Melanie Matchett Wood and Robert J. Lemke Oliver.
Ma Luo : Algebraic de Rham theory for relative completion of $\mathrm{SL}_2(\mathbb{Z})$
- Number Theory ( 166 Views )In this talk, I will first review relative (unipotent) completions of discrete groups in general, and $\mathrm{SL}_2(\mathbb{Z})$ in particular. We then develop an explicit $\mathbb{Q}$-de Rham theory for the relative completion of $\mathrm{SL}_2(\mathbb{Z})$, which enables us to construct iterated integrals of modular forms of the second kind that provide its periods. Following Francis Brown, these periods are called `multiple modular values'. They contain periods of modular forms.
Michael Mossinghoff : Oscillation problems in number theory
- Number Theory ( 159 Views )The Liouville function λ(n) is the completely multiplicative arithmetic function defined by λ(p) = −1 for each prime p. Pólya investigated its summatory function L(x) = Σn≤x λ(n), and showed for instance that the Riemann hypothesis would follow if L(x) never changed sign for large x. While it has been known since the work of Haselgrove in 1958 that L(x) changes sign infinitely often, oscillations in L(x) and related functions remain of interest due to their connections to the Riemann hypothesis and other questions in number theory. We describe some connections between the zeta function and a number of oscillation problems, including Pólya's question and some of its weighted relatives, and, in joint work with T. Trudgian, describe a method involving substantial computation that establishes new lower bounds on the size of these oscillations.