# Prairie Analysis Seminar 2016

### Location and Dates:

Department of Mathematics
University of Kansas
Lawrence, Kansas
September 16-17, 2016

### Principal Lecturer

• Tatiana Toro, University of Washington

Talk: PDEs on non-smooth domains

Abstract: In these lectures we will discuss the relationship between the boundary regularity of the solutions to elliptic second order divergence form partial differential equations and the geometry of the boundary of the domain where they are defined. While in the smooth setting tools from classical PDEs are used to address this question, in the non-smooth setting techniques from harmonic analysis and geometric measure theory are needed to tackle the problem. The goal is to present an overview of the recent developments in this active area of research.

### Invited Speakers

• Arpad Benyi, Western Washington University

Talk: Smoothing of bilinear commutators

Abstract: We will explore the smoothing effect of commutators of certain classes of bilinear operators with functions in appropriate spaces. In particular, we will consider the (weighted) compactness of commutators of bilinear Calderón-Zygmund operators with functions of continuous mean oscillation as well as the boundedness of commutators of bilinear pseudo-differential operators that fall outside the Calderón-Zygmund theory with Lipschitz functions. The talk is based on recent works with W. Damian (Helsinki), K. Moen (Alabama), V. Naibo (Kansas State), T. Oh (Edinburgh) and R.H. Torres (Kansas).

• Alex Iosevich, University of Rochester

Talk: Analytic and combinatorial aspects of the distribution of simplexes in thin subsets of Euclidean space

Abstract: The basic question we ask is, how large does the Hausdorff dimension of a compact subset of $${\Bbb R}^d$$ needs to be to ensure that this set contains a positive proportion of all possible $$k$$-dimensional simplexes? This problem, which has its origins in Falconer and Mattila's work on distance sets and Furstenberg-Katznelson-Weiss work on simplexes in subset of $${\Bbb R}^d$$ of positive density involves delicate Fourier analytic methods and has significant bearing on incidence results in the discrete setting. We shall describe some results results, connections with combinatorics and some future directions.

### Contributed Talks

There will be time allocated for short contributed talks by participants. Priority will be given to graduate students and those in early stages of their careers. Participants interested in giving a contributed talk and/or receiving financial support should visit the registration page.