In 1961, Grunbaum asked whether the centroid $c(K)$ of a convex body $K$ is the centroid of at least $n + 1$ different $(n − 1)$-dimensional sections of $K$ through $c(K)$. A few years later, Loewner asked to find the minimum number of hyperplane sections of $K$ passing through $c(K)$ whose centroid is the same as $c(K)$.

We give an answer to these questions for $n \geq 5$. In particular, we construct a convex body which has only one section whose centroid coincides with the centroid of the body. Joint work with S. Myroshnychenko and V. Yaskin.

An origin-symmetric convex body $K$ in $\mathbb{R}^n$ is said to be in the John position if the maximal volume ellipsoid contained in it is the Euclidean ball. One of the most celebrated theorems in geometric functional analysis is John's theorem, which says that if a convex body $K$ is in John position, then there are a collection of points $u_1,\dots u_m \in S^{n-1} \cap \partial K$ and positive scalars $c_1,\dots, c_m$ for which $I_n = \sum_{j=1}^m c_j u_j \otimes u_j$, where $I_n$ is the identity operator in $\mathbb{R}^n$.  This theorem has numerous consequences such as an estimate for the Banach-Mazur distance to Euclidean space  and the reverse isoperimetric inequality due to Ball. 

Recently, much attention has been given to translating notions from convex geometry and geometric functional analysis to the world of log-concave functions.  In particular, over the last 10 years, extensions of these celebrated theorems (in some forms) have been translated to the world of log-concave functions.

In this talk, we focus on a complementary set of positions, called "maximal-intersection positions," originally introduced by Artstein-Avidan and Katzin in 2016. This collection of position includes the John (and its dual the Lowner position) as particular cases.  The extension of this notion to the world of functions is seen through the following extremal problem for the convolution of a pair of functions: given a pair of integrable log-concave functions $f$ and $g$, find

$\sup_{(T,b) \in SL_n(\mathbb{R}) \times \mathbb{R^n}} \int f(x) g(T^{-1}x-T^{-1}b)dx.$

This is based on a joint work with Steven Hoehner. 

We consider an elliptic operator, double divergence form operator L*, which is the formal adjoint of the elliptic operator in non-divergence from L. An important example of a double divergence form equation is the stationary Kolmogorov equation for invariant measures of a diffusion process. We are concerned with the regularity of weak solutions of L*u=0.

We will also discuss some applications and parabolic counterparts.

An elastic surface resists not only changes in curvature but also tangential stretches and shears.  In classical plate and shell theories, e.g., due to von Karman, the latter two strain measures are approximated infinitesimally.  We motivate our approach via the phenomenon of wrinkling in highly stretched elastomers.  We postulate a novel, physically reasonable class of stored-energy densities, and we prove various existence theorems based on the direct method of the calculus of variations.

We compute asymptotics of  eigenvalues approaching the bottom of the continuous spectrum, and associated resonances, for Schrödinger operators in dimension two for which the potential depends on a parameter.  We distinguish persistent eigenvalues, which have associated resonances,  from disappearing ones, which do not.  We illustrate the significance of this distinction by computing corresponding scattering phase asymptotics and numerical Breit--Wigner peaks.  While we concentrate on the case of the circular well for 
illustrative and computational purposes, we also prove some of our results for more general potentials, using recent results on low-energy resonance expansions.

This talk is based on joint work with Kiril Datchev and Colton Griffin,  and is part of a larger project with K. Datchev.

In the 1960's, T. Kato posed a conjecture about finding the domain and some crucial estimates for the square root of elliptic partial differential operators in divergence form. The question attracted lots of interest because of the applications that it would have, and it turned out to be fairly tough to prove: only after around 40 years and joint efforts from different areas in Analysis (mainly PDE, Functional Analysis and Harmonic Analysis), it was finally solved by Auscher, Hofmann, Lacey, McIntosh and Tchamitchian in 2002.

In this talk, we will illustrate the main changes and new difficulties that arise if we want to solve Kato's problem for operators in non-divergence form instead. We will present a partial solution of the problem which already faces, at least to some extent, some of the difficulties inherent in the non-divergence setting, like the lack of symmetry of the operator and the need to use weights. This is a joint work with L. Escauriaza and S. Hofmann.

We consider the 2-dim water wave problem -- the free boundary problem of the Euler equation with gravity and possibly surface tension -- of finite depth linearized at a uniformly monotonic shear flow \(U(x_2)\). Our main focuses are eigenvalue distribution and inviscid damping. We first prove that in contrast to finite channel flow and gravity waves, the linearized capillary gravity wave has two unbounded branches of eigenvalues for high wave numbers. They may bifurcate into unstable eigenvalues through a rather degenerate bifurcation. Under certain conditions, we provide a complete picture of the eigenvalue distribution. Assuming there are no singular modes (i.e. embedded eigenvalues), we obtain the linear inviscid damping. We also identify the leading asymptotic terms of velocity and obtain stronger decay for the remainders. The linearized gravity waves will also be discussed briefly if time permits. This is a joint work with Xiao Liu.

The work is devoted to the studies of the existence of
solutions of an integro-differential equation in the case of the double
scale anomalous diffusion with the sum of the two negative Laplacians
raised to two distinct fractional powers in \(\mathbb{R}^d\), \(d=4,5\). The proof of the
existence of solutions is based on a fixed point technique. Solvability
conditions for the non-Fredholm elliptic operators in unbounded domains
are used.

https://umsystem.zoom.us/j/94101463494?pwd=NDJaR21PUCtVM0tQWUt0YlNFTmw0UT09

Meeting ID: 941 0146 3494
Passcode: 714934

Extending the Sobolev theory of quasiconformal and quasiregular maps to subdomains of the complex plane motivates our investigation of Sobolev regularity of singular integral operators on domains. We introduce new paraproducts which lead to higher order T1-type testing conditions. A special case provides weighted Sobolev estimates for the compressed Beurling transform which imply quantitative Sobolev estimates for the Beltrami resolvent. This is joint work with Francesco Di Plinio and Brett D. Wick.

In this talk we will survey some old and new results on maximum principles for P-functions and their applications to the study of partial differential equations. More precisely, we will show how one can employ the maximum principle in problems of physical or geometrical interest, in order to get the shape of some free boundaries, isoperimetric inequalities, symmetry results, convexity results and Liouville type results. In the first part of the talk  we'll be mainly focused on some overdetermined problems, while in the second part of the talk we'll present our contributions to some Monge-Ampere type problems and eventually discuss some open problems.