# Difference between revisions of "Analysis Seminar"

Analysis Seminar

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

# Abstracts

### Simon Marshall

Integrals of eigenfunctions on hyperbolic manifolds

Let X be a compact hyperbolic manifold, and let Y be a totally geodesic closed submanifold in X. I will discuss the problem of bounding the integral of a Laplace eigenfunction on X over Y, as the eigenvalue tends to infinity. I will present an upper bound for these integrals that is sharp on average, and briefly describe ongoing work with Farrell Brumley in which we attempt to produce eigenfunctions with very large periods.

### Hong Wang

About Falconer distance problem in the plane

If E is a compact set of Hausdorff dimension greater than 5/4 on the plane, we prove that there is a point x\in E such that the set of distances between x and E has positive Lebesgue measure. Our result improves upon Wolff's theorem for dim E> 4/3. This is joint work with Larry Guth, Alex Iosevich and Yumeng Ou.

### Polona Durcik

Singular Brascamp-Lieb inequalities and extended boxes in R^n

Brascamp-Lieb inequalities are L^p estimates for certain multilinear forms on functions on Euclidean spaces. In this talk we consider singular Brascamp-Lieb inequalities, which arise when one of the functions is replaced by a Calderon-Zygmund kernel. We focus on a family of multilinear forms in R^n with a certain cubical structure and discuss their connection to some patterns in positive density subsets in R^n. Based on joint works with V. Kovac and C. Thiele.

### Song-Ying Li

Estimates for the first positive eigenvalue of Kohn Laplacian on a pseudo-Hermitian manifold

In this talk, I will present my recent works with my collaborators on the lower bound and upper bounds estimates for the first positive eigenvalues of Kohn Laplacian and sub-Laplacian on a strictly pseudoconvex pseudo-Hermitian CR manifold, which include CR Lichnerowicz-Obata theorem for the lower and upper bounds for the first positive eigenvalue for the Kohn Laplacian on strictly pseudoconvex hypersurfaces.

### Hanlong Fan

A generalization of the theorem of Weil and Kodaira on prescribing residues

An old theorem of Weil and Kodaira says that: For a K\"ahler manifold X, there exists a closed meromorphic one-form with residue divisor D if and only if D is homologous to zero. In this talk, I will generalize Weil and Kodaira's criterion to non-K\"ahler manifolds.

### Kyle Hambrook

Fourier Decay and Fourier Restriction for Fractal Measures on Curves

I will discuss my recent work on some problems concerning Fourier decay and Fourier restriction for fractal measures on curves.

### Laurent Stolovitch

Equivalence of Cauchy-Riemann manifolds and multisummability theory

We apply the multisummability theory from Dynamical Systems to CR-geometry. As the main result, we show that two real-analytic hypersurfaces in $\mathbb C^2$ are formally equivalent, if and only if they are $C^\infty$ CR-equivalent at the respective point. As a corollary, we prove that all formal equivalences between real-algebraic Levi-nonflat hypersurfaces in $\mathbb C^2$ are algebraic (and in particular convergent). This is a joint work with I. Kossovskiy and B. Lamel.

### Brian Cook

Equidistribution results for integral points on affine homogenous algebraic varieties

Let Q be a homogenous integral polynomial of degree at least two. We consider certain results and questions concerning the distribution of the integral points on the level sets of Q.

### Shaoming Guo

Polynomial Roth theorems in Salem sets

Let P(t) be a polynomial of one real variable. I will report a result on searching for patterns of the form (x, x+t, x+P(t)) within Salem sets, whose Hausdorff dimension is sufficiently close to one. Joint work with Fraser and Pramanik.