## William Sokurski : Fourier operators on GL(2) for odd Adjoint powers

- Number Theory ( 229 Views )Recently A. Braverman, D. Kazhdan, and L. Lafforgue have interpreted Langlands' functoriality in terms of a generalized harmonic analysis on reductive groups that requires the existence of new spaces of functions and an associated, generally non-linear, involutive Fourier transform. This talk will discuss some of these objects involved in the local p-adic situation, after introducing some ideas and basic constructions involved. Specifically, the local Fourier transforms have a nice interpretation in terms of their spectral decomposition giving the gamma factors that appear in functional equations of L functions, which, in the standard case allows one to write down the epsilon factors attached to supercuspidal representations as non-abelian Gauss sums. For G=GL(2), we use the local Langlands correspondence to provide L and epsilon factors for odd adjoint power transfers and use this to interpret the Adjoint power Fourier-transform such that its spectral decomposition on supercuspidal representations is given explicitly by certain non-abelian Kloosterman sums, which we use to give a form of the Fourier operator.

## Chung Pang Mok : Introduction to Mochizukis works on inter-universal Teichmuller theory

- Number Theory ( 183 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.

## Edna Jones : The Kloosterman circle method and weighted representation numbers of positive definite quadratic forms

- Number Theory ( 177 Views )We develop a version of the Kloosterman circle method with a bump function that is used to provide asymptotics for weighted representation numbers of positive definite integral quadratic forms. Unlike many applications of the Kloosterman circle method, we explicitly state some constants in the error terms that depend on the quadratic form. This version of the Kloosterman circle method uses Gauss sums, Kloosterman sums, SaliÃ© sums, and a principle of nonstationary phase. If time permits, we may discuss a potential application of this version of the Kloosterman circle method to a proof of a strong asymptotic local-global principle for certain Kleinian sphere packings.

## Ayla Gafni : Extremal primes for elliptic curves without complex multiplication

- Number Theory ( 174 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.

## Yuanqing Cai : Fourier coefficients of theta functions on metaplectic groups

- Number Theory ( 167 Views )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.

## Dan Goldston : Small Gaps between Zeros of the Riemann Zeta-Function

- Number Theory ( 150 Views )We consider the complex zeros of the Riemann zeta-function &rho = &beta + i &gamma, &gamma > 0. The Riemann Hypothesis (RH) is that &beta = 1/2. If we consider the vertical distribution of these zeros, then the average vertical spacing between zeros at height T is 2&pi / log T. We expect theoretically and find numerically that the distribution of the lengths of these gaps follows a certain continuous GUE distribution where both very small and very large multiples of the average spacing occur. In contrast to this, the existence of a Landau Siegel-zero would force all the gaps in a certain large range to never be closer than half the average spacing, and also have even more bizarre and unlikely properties. There are three methods that have been developed to prove something about small gaps. First, Selberg in the 1940's using moments for the number of zeros in short intervals, was able to prove unconditionally that there are some gaps larger than the average spacing and others smaller than the average spacing. Next assuming RH Montgomery in 1972 introduced a pair correlation method for zeros and produced small gaps less than 0.67 times the average spacing. Finally, in 1981 Montgomery-Odlyzko assuming RH introduced a Dirichlet polynomial weighted method which found small gaps less then 0.5179 times the average spacing. (This method was further developed by Conrey, Ghosh, and Gonek.) These methods all exhibit the presumed barrier at 1/2 times the average spacing for small gaps. I will talk about two projects that are work in progress. The first is joint with Hugh Montgomery and is motivated by the observations that all the results mentioned above do not exclude the possibility that the small gaps found are all coming from multiple zeros and thus gaps of length zero, and at present we do not know if there are any non-zero gaps that are shorter then the average spacing. While we have not yet be able to prove there are any smaller than average non-zero gaps, we can quantify the relationship between non-zero gaps and multiple zeros and show there is a positive proportion of one or the other. The second project is joint work with Caroline Turnage-Butterbaugh where we have developed a Dirichlet Polynomial Weighted Pair Correlation Method which potentially can be applied to a number of questions on zeros.

## Ding Ma : Multiple Zeta Values and Modular Forms in Low Levels

- Number Theory ( 132 Views )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.

## Pam Gu : A family of period integrals related to triple product $L$-functions

- Number Theory ( 131 Views )Let $F$ be a number field with ring of adeles $\mathbb{A}_F$. Let $r_1,r_2,r_3$ be a triple of positive integers and let $\pi:=\otimes_{i=1}^3\pi_i$ where the $\pi_i$ are all cuspidal automorphic representations of $\mathrm{GL}_{r_i}(\mathbb{A}_F)$. We denote by $L(s,\pi, \otimes^3)=L(s, \pi_1\times \pi_2 \times \pi_3)$ the corresponding triple product $L$-function. It is the Langlands $L$-function defined by the tensor product representation $\otimes^3:{}^L(\mathrm{GL}_{r_1} \times \mathrm{GL}_{r_2} \times \mathrm{GL}_{r_3}) \to \mathrm{GL}_{r_1r_2r_3}(\mathbb{C})$. In this talk I will present a family of Eulerian period integrals, which are holomorphic multiples of the triple product -function in a domain that nontrivially intersects the critical strip. We expect that they satisfy a local multiplicity one statement and a local functional equation. This is joint work with Jayce Getz, Chun-Hsien Hsu and Spencer Leslie.

## Chen Wan : Multiplicity one theorem for the Ginzburg-Rallis model

- Number Theory ( 122 Views )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.

## Jeff Streets : A parabolic flow of Hermitian metrics

- Number Theory ( 121 Views )I will introduce a parabolic flow of Hermitian metrics which is a generalization of Kahler-Ricci flow. This flow preserves the pluriclosed condition, and its existence and convergence properties are closely related to the underlying topology of the given complex manifold. I will classify static solutions to the flow on various classes of complex surfaces, and show that no static solutions exist on Class VII surfaces, an important first step in using this flow to classify these surfaces. Joint with G. Tian.

## Samit Dasgupta : Starks Conjectures and Hilberts 12th Problem

- Number Theory ( 118 Views )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.

## Dante Bonolis : 2-torsion in class groups of number fields

- Number Theory ( 110 Views )In 2020, Bhargava, Shankar, Taniguchi, Thorne, Tsimerman, and Zhao established that, for a given number field $K$ with a degree $n\geq 5$, the size of the $2$-torsion is bounded by $h_{2}(K) \ll D^{\frac{1}{2}-\frac{1}{2n}}$, where $D_{K}$ is the discriminant of $K$ over $\mathbb{Q}$. In this presentation, we will introduce new bounds that take into account the geometry of the lattice underlying the ring of integers of $K$. This research is a joint project with Pierre Le Boudec.

## Romyar Sharifi : Modular symbols and arithmetic

- Number Theory ( 110 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.

## Naser Tabeli Zadeh : Optimal strong approximation for quadratic forms

- Number Theory ( 109 Views )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.

## Wei Ho : Families of lattice-polarized K3 surfaces

- Number Theory ( 109 Views )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.

## Efrat Bank : Primes in short intervals on curves over finite fields.

- Number Theory ( 107 Views )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.

## Ben Howard : Periods of CM abelian varieties

- Number Theory ( 107 Views )Colmez conjectured a formula relating periods of abelian varieties with complex multiplication to derivatives of Artin L-functions. IÂ?ll explain how to prove an averaged version of ColmezÂ?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.

## Kate Stange : Visualising the arithmetic of imaginary quadratic fields

- Number Theory ( 105 Views )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.

## Valentin Blomer : Spectral summation formulae and their applications

- Number Theory ( 105 Views )Starting from the Poisson summation formula, I discuss spectral summation formulae on GL(2) and GL(3) and present a variety of applications to automorphic forms, analytic number theory, and arithmetic.

## Piper Harron : The Equidistribution of Lattice Shapes of Rings of Integers in Cubic, Quartic, and Quintic Number Fields

- Number Theory ( 96 Views )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)!

## Robin Zhang : Harris-Venkatesh plus Stark

- Number Theory ( 64 Views )The class number formula describes the behavior of the Dedekind zeta function at s = 0. The Stark conjecture extends the class number formula, describing the behavior of Artin L-functions at s = 0 in terms of units. The Harrisâ??Venkatesh conjecture, originally motivated by the conjectures of Venkatesh and Prasannaâ??Venkatesh on derived Hecke algebras, can be viewed as an analogue to the Stark conjecture modulo p. In this talk, I will draw an introductory picture, formulate a unified conjecture combining Harrisâ??Venkatesh and Stark for modular forms of weight 1, and describe the proof of this in the imaginary dihedral case. Time permitting, I will also describe some new questions and in-progress work modulo pn.