Seminar Series: Arithmetic Applications of Fourier Analysis

Claudia Alfes (Bielefeld): Traces of CM values and geodesic cycle integrals of modular functions

In this talk we give an introduction to the study of generating series of the traces of CM values and geodesic cycle integrals of different modular functions. First we define modular forms and harmonic Maass forms. Then we briefly discuss the theory of theta lifts that gives a conceptual framework for such generating series. We end with some applications of the theory: It can be used to obtain results on the vanishing on the central derivative of the $L$-series of elliptic curves and to obtain rationality results for cycle integrals of certain meromorphic functions.

Regis de la Breteche (Paris): Higher moments of primes in arithmetic progressions

Since the work of Barban, Davenport and Halberstam, the variances of primes in arithmetic progressions have been widely studied and continue to be an active topic of research. However, much less is known about higher moments. Hooley established a bound on the third moment in progressions, which was later sharpened by Vaughan for a variant involving a major arcs approximation. Little is known for moments of order four or higher, other than the conjecture of Hooley. In this talk I will discuss recent joint work with Daniel Fiorilli on weighted moments of moments in progressions.

Michael Christ (UC Berkeley): On quadrilinear implicitly oscillatory integrals

The title refers to multilinear functionals $\int_B \prod_{j\in J} (f_j\circ\varphi_j)$ where $B\subset {\mathbb R}^D$ is a ball, $J$ is a finite index set, $\varphi_j:B\to {\mathbb R}^d$ are $C^\omega$ submersions, $d<D$, and $f_j$ are measurable. The goal is majorization by a product of negative order Sobolev norms of $f_j$, under appropriate hypotheses on the mappings $\varphi_j$. Inequalities of this type are closely related to sublevel inequalities $\big|\big\{x\in B: |\sum_{j\in J} a_j(x)\,(g_j\circ\varphi_j)(x)|<\varepsilon\big\}\big| = O(\varepsilon^c)$ where the coefficients satisfy $a_j\in C^\omega$. I will state results of this type with $(|J|,D,d) = (4,2,1)$ for the multiplicative inequality and $= (3,2,1)$ for the additive inequality, discuss connections between the two, and indicate some elements of proofs.

Ciprian Demeter (Bloomington): Restriction of exponential sums to hypersurfaces

We discuss moment inequalities for exponential sums with respect to singular measures, whose Fourier decay matches those of curved hypersurfaces. Our emphasis will be on proving estimates that are sharp with respect to the scale parameter $N$, apart from $N^\epsilon$ losses. Joint work with Bartosz Langowski.

Alexandra Florea (Columbia): The Ratios Conjecture over function fields

I will talk about some recent joint work with H. Bui and J. Keating where we study the Ratios Conjecture for the family of quadratic L-functions over function fields. I will also discuss the closely related problem of obtaining upper bounds for negative moments of L-functions, which allows us to obtain partial results towards the Ratios Conjecture in the case of one over one, two over two and three over three L-functions.

Rachel Greenfeld (UCLA): Decidability and periodicity of translational tilings

Let $G$ be a finitely generated abelian group, and $F_1,...,F_J$ be finite subsets of $G$. We say that $F_1,...,F_J$ tile $G$ by translations, if $G$ can be covered by translated copies of $F_1,...,F_J$, without any overlaps. Given some finite sets $F_1,...,F_J$ in $G$, can we decide whether they admit a tiling of $G$? Suppose that they do tile $G$, do they admit a periodic tiling? A well known argument of Hao Wang ('61), shows that these two questions are closely related. In the talk, we will discuss this relation, and present some results, old and new, about the decidability and periodicity of translational tilings, in the case of a single tile ($J=1$) as well as in the case of a multi-tileset ($J>1$). The talk is based on an ongoing project, joint with Terence Tao.

Shaoming Guo (UW Madison): Some recent progress on the Bochner-Riesz problem

I will report some recent progress on the Bochner-Riesz conjecture. We observe that recent tools developed to study the Fourier restriction conjecture, including wave packet decompositions, broad-narrow analysis, the polynomial methods, polynomial Wolff axioms, etc., work equally well for the Bochner-Riesz problem. This is joint work with Changkeun Oh, Hong Wang, Shukun Wu and Ruixiang Zhang.

Jonathan Hickman (Edinburgh): The helical maximal function

The circular maximal function is a singular variant of the familiar Hardy--Littlewood maximal function. Rather than take maximal averages over concentric balls, we take maximal averages over concentric circles in the plane. The study of this operator is closely related to certain GMT packing problems for circles, as well as the theory of the Euclidean wave propagator. A celebrated result of Bourgain from the mid 80s showed that the circular maximal function is bounded on $L^p$ if and only if $p > 2$. In this talk I will discuss a higher dimensional variant of Bourgain's theorem, in which the circles are replaced with space curves (such as helices) in $R^3$. Our main theorem is that the resulting helical maximal operator is bounded on $L^p$ if and only if $p > 3$. The proof combines a number of recently developed Fourier analytic tools, and in particular a variant of the Littlewood--Paley theory for functions frequency supported in a neighbourhood of a cone. Joint work with David Beltran, Shaoming Guo and Andreas Seeger.

James Maynard (Oxford): Half-isolated zeros and zero-density estimates

We introduce a new zero-detecting method which is sensitive to the vertical distribution of zeros of the zeta function. This allows us to show that there are few 'half-isolated' zeros, and allows us to improve the classical zero density result to $N(\sigma,T)\ll T^{24(1-\sigma)/11+o(1)}$ if we assume that the zeros of the zeta function are restricted to finitely many vertical lines (and so gives new results about primes in short intervals under this assumption). This relies on a new variant of the Turan power sum method, which might be of independent interest to harmonic analysts. This is joint work with Kyle Pratt.

Mariusz Mirek (Rutgers): Pointwise ergodic theorems for bilinear polynomial averages

We shall discuss the proof of pointwise almost everywhere convergence for the non-conventional (in the sense of Furstenberg and Weiss) bilinear polynomial ergodic averages. This is joint work with Ben Krause and Terry Tao: arXiv:2008.00857. We will also talk about recent progress towards establishing Bergelson's conjecture.

Paul Nelson (Zurich): The orbit method, microlocal analysis and applications to L-functions

I will describe how the orbit method can be developed in a quantitative form, along the lines of microlocal analysis, and applied to local problems in representation theory and global problems concerning automorphic forms. The local applications include asymptotic expansions of relative characters. The global applications include moment estimates and subconvex bounds for L-functions. These results are the subject of two papers, the first joint with Akshay Venkatesh:
https://arxiv.org/abs/1805.07750
https://arxiv.org/abs/2012.02187

Sarah Peluse (Princeton/IAS): Bounds for subsets of $\mathbb{F}_p^n \times \mathbb{F}_p^n$ without L’s

I will discuss the difficult problem of proving reasonable bounds in the multidimensional generalization of Szemer\’edi’s theorem, and describe a proof for such bounds for sets lacking nontrivial configurations of the form $(x,y), (x,y+z), (x,y+2z), (x+z,y)$ in the finite field model setting.

Ian Petrow (London): Relative trace formulas for GL(2) and analytic number theory

The Petersson/Kuznetsov formula is a classical tool in analytic number theory with striking applications in the analytic theory of L-functions. It is the primitive example of a relative trace formula, and acts as a spectral summation device tying together some basic families of automorphic forms. In this talk I will discuss some of these families, and how varying the test function in the relative trace formula can pick out other families of automorphic forms of interest. Along these lines I will describe some past joint work with M.P. Young, some work of Y. Hu, and some current/future work joint between all three of us.

Yiannis Sakellaridis (Johns Hopkins University): Plancherel formula, intersection complexes, and local L-functions

In the theory of automorphic forms, L-functions (and their special values) are usually realized by various types of period integrals. It is now understood that the local L-factors associated to a period represent a Plancherel density for a homogeneous space. I will start by reviewing the conjectural relationship between local Plancherel formulas and local L-factors. Then, I will talk about joint work with Jonathan Wang, which shows that, on certain singular spaces, the test function whose Plancherel density is an L-factor is related to an intersection cohomology complex. The talk will be fairly elementary, e.g., I will not assume knowledge of intersection cohomology.

Will Sawin (Columbia): Sums in progressions to squarefree mouduli among polynomials over a finite field

There are many problems about counting special types of numbers (primes or other numbers with special factorizations) in arithmetic progressions, or summing arithmetic functions in arithmetic progressions. These all have analogues polynomials over a finite field. Recently I proved, by a geometric method, strong bounds for these analogues (approaching level of distribution 1 and square-root cancellation as the size of the finite field goes to infinity). I will explain how these bounds relate to those obtained from a simpler approach using the Riemann hypothesis (i.e. by using Fourier analysis on the multiplicative group) and how we can deduce, using a classical probability-theoretic method, a result that applies to every factorization type at once.

Betsy Stovall (UW Madison): Fourier restriction to the sphere is extremizable more often than not

We will sketch a proof that the $L^p \to L^q$ Fourier extension inequality associated to the $d$-sphere possesses extremizers whenever $p < q < (d+2)p’/d$. This is joint work with Taryn Flock.

Maryna Viazovska (EPFL): Fourier interpolation

This lecture is about Fourier uniqueness and Fourier interpolation pairs. Suppose that we have two subsets X and Y of the Euclidean space. Can we reconstruct a function f from its restriction to the set X and the restriction of its Fourier transform to the set Y? We are interested in the pairs (X,Y) such that the answer to the question above is affirmative. I will give an overview of recent progress on explicit constructions and existence results for Fourier interpolation pairs and corresponding interpolation formulas.

Jim Wright (Edinburgh): Exponential sums and oscillatory integrals: a unified approach

In joint work with Gian Maria Dall'Ara, we have a simple argument which is powerful enough to effectively treat oscillatory integrals defined over general locally compact topological fields whose phase is a general polynomial of many variables. Our bounds have an interesting self-improving feature.

Matthew Young (Texas A&M): Large sieve inequalities for families of automorphic forms

The quality of a large sieve inequality for a family of automorphic forms (or $L$-functions) is a tangible way to measure how well the family is understood. For many $GL_1$ and $GL_2$ families, we have optimal large sieve inequalities; the $GL_1$ family is the classical large sieve, and many $GL_2$ families are covered by work of Deshouillers-Iwaniec. In higher rank, our knowledge is highly incomplete. In this talk, I will discuss some recent progress on the $GL_3$ spectral large sieve.

Po-Lam Yung (Australian National University and the Chinese University of Hong Kong): A formula for Sobolev seminorms involving weak L^p

I will discuss some joint work with Haim Brezis and Jean Van Schaftingen, where a new formula was proved for the $W^{1,p}$ seminorm of any compactly supported smooth function on $R^n$. The formula involves the weak $L^p$ norm of a modified difference quotient on the product space $R^n \times R^n$, and was partly inspired by the BBM formula by Bourgain, Brezis and Mironescu regarding fractional Sobolev seminorms. A similar formula for the $L^p$ norm of any $L^p$ function on $R^n$ has been obtained in a recent paper with Qingsong Gu. The talk will conclude with some applications of this circle of ideas, that remedies the failures of certain critical Gagliardo-Nirenberg type embeddings.

Ruixiang Zhang (IAS): A stationary set method for estimating oscillatory integrals

Given a polynomial $P$ of constant degree in $d$ variables and consider the oscillatory integral $$I_P = \int_{[0,1]^d} e(P(\xi)) \mathrm{d}\xi.$$ Assuming $d$ is also fixed, what is a good upper bound of $|I_P|$? In this talk, I will introduce a "stationary set" method that gives an upper bound with simple geometric meaning. The proof of this bound mainly relies on the theory of o-minimal structures. As an application of our bound, we obtain the sharp convergence exponent in the two dimensional Tarry's problem for every degree via additional analysis on stationary sets. Consequently, we also prove the sharp $L^{\infty} \to L^p$ Fourier extension estimates for every two dimensional Parsell-Vinogradov surface whenever the endpoint of the exponent $p$ is even. This is joint work with Saugata Basu, Shaoming Guo and Pavel Zorin-Kranich.