Recent Work on Singularities of Reductive Monoids
 

Alan Zhao, Columbia University

On shifted convolution sums of GL(3) Fourier coefficients with average over shifts
 

Sampurna Pal, Indian Statistical Institute, Kolkata
 

Let $A(m,n)$ be normalised Fourier coefficients of $F$, a Hecke Maass cusp form for $SL(3,Z)$. In this talk, we will discuss an upper bound for the sum $\sum_{h\sim H} \sum{n\sim N} A(n) A(n+h)$ which is non-trivial in the range $N^{1/6+\varepsilon}<H<N^{1/2}$. This improves upon the best known range of $H$ ($H>N^{1/4}$ by Dasgupta-Leung-Young) for which cancellation can be shown in the above shifted convolution sum. This is an ongoing project with Ritwik Pal.

Quantum unique ergodicity for Hilbert moular forms and shrinking sets
 

Ivan Doubovik, University of Lille

Algebraicity Results for $L$-functions on Exceptional Groups
 

Bryan Hu, University of California San Diego
 

We discuss algebraicity results for $L$-functions associated to automorphic forms on quaternionic exceptional groups. Quaternionic modular forms (QMFs) are automorphic forms on these groups whose real components lie in the quaternionic discrete series. QMFs have a robust theory of Fourier coefficients developed by Gross-Wallach, Gan-Gross-Savin, and Pollack. Recently Pollack established that the space of cuspidal QMFs has a basis whose Fourier coefficients are algebraic numbers. We combine this theory with Rankin-Selberg integrals due to Hundley and Gurevich-Segal, along with our recent work on exceptional Eisenstein series, to establish algebraicity results that give evidence for Deligne's conjecture on the critical values of $L$-functions. Along the way, we develop exceptional differential operators to mirror the classical picture of Maass-Shimura operators and nearly holomorphic modular forms.

Surfaces Associated to Zeros of Automorphic $L$-functions
 

Debmalya Basak, University of Illinois Urbana Champaign

Assuming the Riemann Hypothesis, Montgomery established results concerning pair correlation of zeros of the Riemann zeta function. Rudnick and Sarnak extended these results to automorphic $L$-functions and all level correlations. We show that automorphic $L$-functions exhibit additional geometric structures related to the correlation of their zeros. This is joint work with Cruz Castillo and Alexandru Zaharescu.

Zeros of twisted spinor zeta function associated to Siegel modular form of degree 2 on critical line
 

Lalit Vaishya, Indian Statistical Institute, Delhi Center

Polynomials with Restricted Digits
 

Naomi Bazlov, Technion - Israel Institute of Technology
 

The problem of primes with restricted digits has been pondered and studied since the mid twentieth century, and has seen much progress in recent years thanks to the work of mathematicians such as James Maynard. But what if the ring of integers is replaced with $\mathbb{F}_q[T]$? In this talk we will look at function field analogues of questions about primes with restricted digits. We will consider both classical approaches, such as the circle method, and more modern ideas. I will present some very recent results about asymptotic behaviour of squarefree polynomials of degree n with coefficients in a restricted subset of $\mathbb{F}_q$.

Classification of exceptional hyperspherical data and their dual
 

Guodong Tang, National University of Singapore
 

Abstract: The recent joint work of Ben-Zvi, Sakellaridis and Venkatesh (BZSV) proposed a duality of the relative Langlands problem though the concept of hyperspherical data $(G, H, \iota, S)$. This talk will focus on the construction of its dual data and the joint work of Wan, Zhang, and myself on the exceptional case though $G = F_4$.

Spectral reciprocity and explicit hybrid subconvexity bound for triple product $L$-functions
 

Xinchen Mao, University of Bonn
 

Subconvexity estimation is one of the most important and challenging problem in the theory of analytic number theory and $L$-functions. In my talk, following the estimations and ideas in Michel-Venkatesh and Hu-Michel-Nelson, we establish a spectral reciprocity formula which links twisted first moment of different $L$-functions. As an application, we can give an explicit hybrid subconvexity bound for triple product $L$-functions in the level aspect, allowing joint ramifications and conductor dropping range.

Explicit formula for Bessel models over an odd GSpin group
 

Yu Xin, Bar-Ilan University

Serre's Conjecture and Anabelian Geometry
 

Benjamin Steklov, Goethe University Frankfurt

$n$ cohomology of limits of discrete series and GGP type branching laws
 

Jin Lee, Duke University
 

We show that a totally degenerate limit of discrete series representation admits a choice of $n$ cohomology group that is nonvanishing at a canonically defined degree. We then show that the combinatorial complexes used by Soergel to compute these cohomology groups satisfies Serre duality. We conclude that this produces two n cohomology groups, each for the TDLDS of $U(n+1)$ and $U(n)$, which are nonvanishing at the same degree, suggesting Gan Gross Prasad type branching laws for the TDLDS of unitary groups of any rank.