# Difference between revisions of "Analysis Seminar"

Analysis Seminar

The seminar will meet Tuesdays, 4:00 p.m. in VV B139, unless otherwise indicated.

# Summer/Fall 2017 Analysis Seminar Schedule

 date speaker title host(s) institution September 8 in B239 Tess Anderson UW Madison A Spherical Maximal Function along the Primes Tonghai September 19 Brian Street UW Madison Convenient Coordinates Betsy September 26 Hiroyoshi Mitake Hiroshima University Derivation of multi-layered interface system and its application Hung October 3 Joris Roos UW Madison A polynomial Roth theorem on the real line Betsy October 10 Michael Greenblatt UI Chicago Maximal averages and Radon transforms for two-dimensional hypersurfaces Andreas October 17 David Beltran Basque Center of Applied Mathematics Fefferman-Stein inequalities Andreas Wednesday, October 18, 4:00 p.m. in B131 Jonathan Hickman University of Chicago Factorising X^n Andreas October 24 Xiaochun Li UIUC Recent progress on the pointwise convergence problems of Schroedinger equations Betsy Thursday, October 26, 4:30 p.m. in B139 Fedor Nazarov Kent State University The Lerner-Ombrosi-Perez bound in the Muckenhoupt Wheeden conjecture is sharp Sergey, Andreas Friday, October 27, 4:00 p.m. in B239 Stefanie Petermichl University of Toulouse Title Betsy, Andreas Wednesday, November 1, 4:00 p.m. in B239 Shaoming Guo Indiana University Title Andreas November 14 Naser Talebizadeh Sardari UW Madison Quadratic forms and the semiclassical eigenfunction hypothesis Betsy November 28 Xianghong Chen UW Milwaukee Title Betsy December 5 Tentatively reserved by Betsy (10/1) Title December 12 Alex Stokolos GA Southern Title Andreas

# Abstracts

### Brian Street

Title: Convenient Coordinates

Abstract: We discuss the method of picking a convenient coordinate system adapted to vector fields. Let X_1,...,X_q be either real or complex C^1 vector fields. We discuss the question of when there is a coordinate system in which the vector fields are smoother (e.g., C^m, or C^\infty, or real analytic). By answering this in a quantitative way, we obtain coordinate charts which can be used as generalized scaling maps. When the vector fields are real this is joint work with Stovall, and continues in the line of quantitative sub-Riemannian geometry initiated by Nagel, Stein, and Wainger. When the vector fields are complex one obtains a geometry with more structure which can be thought of as "sub-Hermitian".

### Hiroyoshi Mitake

Title: Derivation of multi-layered interface system and its application

Abstract: In this talk, I will propose a multi-layered interface system which can be formally derived by the singular limit of the weakly coupled system of the Allen-Cahn equation. By using the level set approach, this system can be written as a quasi-monotone degenerate parabolic system. We give results of the well-posedness of viscosity solutions, and study the singularity of each layers. This is a joint work with H. Ninomiya, K. Todoroki.

### Joris Roos

Title: A polynomial Roth theorem on the real line

Abstract: For a polynomial P of degree greater than one, we show the existence of patterns of the form (x,x+t,x+P(t)) with a gap estimate on t in positive density subsets of the reals. This is an extension of an earlier result of Bourgain. Our proof is a combination of Bourgain’s approach and more recent methods that were originally developed for the study of the bilinear Hilbert transform along curves. This talk is based on a joint work with Polona Durcik and Shaoming Guo.

### Michael Greenblatt

Title: Maximal averages and Radon transforms for two-dimensional hypersurfaces

Abstract: A general local result concerning L^p boundedness of maximal averages over 2D hypersurfaces is described, where p > 2. The surfaces are allowed to have either the traditional smooth density function or a singularity growing as |(x,y)|^{-t} for some 0 < t < 2. This result is a generalization of a theorem of Ikromov, Kempe, and Mueller. Similar methods can be used to show sharp L^p to L^p_a Sobolev estimates for associated Radon transform operators when p is in a certain interval containing 2.

### David Beltran

Title: Fefferman Stein Inequalities

Abstract: Given an operator T, we focus on obtaining two-weighted inequalities in which the weights are related via certain maximal function. These inequalites, which originated in work of Fefferman and Stein, have been established in an optimal way for different classical operators in Harmonic Analysis. In this talk, we survey some classical results and we present some recent Fefferman-Stein inequalities for pseudodifferential operators and for the solution operators to dispersive equations.

### Jonathan Hickman

Title: Factorising X^n.

Question: how many ways can the polynomial $X^n$ be factorised as a product of linear factors? Answer: it depends on the ring... In this talk I will describe joint work with Jim Wright investigating certain exponential sum estimates over rings of integers modulo N. This theory serves as a discrete analogue of the (euclidean) Fourier restriction problem, a central question in contemporary harmonic analysis. In particular, as part of this study, the question of counting the number of factorisations of polynomials over such rings naturally arises. I will describe how these number-theoretic considerations can themselves be approached via methods from harmonic analysis.

### Xiaochun Li

Title: Recent progress on the pointwise convergence problems of Schrodinger equations

Abstract: Recently, Guth, Du and I solved the pointwise convergence problem of Schrodinger equations in two-dimensional case. We proved that the solution to free Schrodinger equation in R^2 converges to its initial data, provided the initial data belongs to H^s for s larger than 1/3. This result is sharp, up to the end point, due to Bourgain's example. The proof relies on the polynomial partitioning method and the decoupling method. In addition, the pointwise convergence problem is closely related to Fourier restriction conjecture.

### Fedor Nazarov

Title: The Lerner-Ombrosi-Perez bound in the Muckenhoupt-Wheeden conjecture is sharp.

Abstract: We show that the upper bound $[w]_{A_1}\log (e+[w]_{A_1})$ for the norm of the Hilbert transform on the line as an operator from $L^1(w)$ to $L^{1,\infty}(w)$ cannot be improved in general. This is a joint work with Andrei Lerner and Sheldy Ombrosi.

Title: Quadratic forms and the semiclassical eigenfunction hypothesis

Abstract: Let $Q(X)$ be any integral primitive positive definite quadratic form in $k$ variables, where $k\geq4$, and discriminant $D$. For any integer $n$, we give an upper bound on the number of integral solutions of $Q(X)=n$ in terms of $n$, $k$, and $D$. As a corollary, we prove a conjecture of Lester and Rudnick on the small scale equidistribution of almost all functions belonging to any orthonormal basis of a given eigenspace of the Laplacian on the flat torus $\mathbb{T}^d$ for $d\geq 5$. This conjecture is motivated by the work of Berry\cite{Berry, Michael} on semiclassical eigenfunction hypothesis.