Schedule for: 21w5139 - Interaction Between Partial Differential Equations and Convex Geometry (Online)

A brief introduction to IASM with important logistical information, technology instruction, and opportunity for participants to ask questions.

In this talk, motivated by the potential applications, we will focus on the existence of solution to exterior Dirichlet problem for degenerate complex $k$-Hessian equations with prescribed decay. We construct a sequence of approximate solutions and establish the a-priori estimates for those approximate solutions. This is a joint work with Prof. Xinan Ma and Dekai Zhang.

We prove an inequality that unifies previous works of the authors on the properties of the Radon transform on convex bodies including the Busemann-Petty problem and a slicing inequality for arbitrary functions. Joint work with A.Giannopoulos and A.Zvavitch.

In this lecture we discuss projective spray geometry. For a manifold $M$, let $\mathcal{GDW}(M)$ denotes the class of all sprays on $M$ satisfying that the rate of change of the Douglas curvature along a geodesic is tangent to the geodesic. We show that $\mathcal{GDW}(M)$ is closed under projective changes. We know that there are infinitely many sprays of scalar curvature which cannot be induced by any (not necessary positive definite) Finsler metric. We show that every spray of scalar curvature on $M$ belong to $\mathcal{GDW}(M)$, generalizing Sakaguchi result previously known only when the spray is induced by a positive definite Finsler metric.

The $L_p$-Minkowski problem is an extension of the classical Minkowski problem. It can be formulated as the Monge-Ampere equation $$\det (D^2 u+uI) = f(x) u^{p-1} \ \ \text{on}\ S^n . $$ The corresponding functional is related to the Blaschke-Santalo inequality. Accordingly the problem can be divided into the sub-critical case when $p>-n-1$, the critical case when $p=-n-1$, and the supercritical case when $p<-n-1$. There is a rich phenomenon on the existence and multiplicity of solutions. In this talk we will review the development of the $L_p$-Minkowski problem, and report a recent result on the existence of solutions in the supercritical case by Qiang Guang, Qi-Rui Li and myself.

In this talk, we discuss the Minkowski problem in the sphere and the problem of prescribing the centro-affine curvature in the Euclidean space. The two problems are equivalent to solving the Euler-Lagrangian equations of a Monge-Ampere type functional in the sphere or Euclidean space. The solutions are obtained by using a Gauss curvature type flow together with min-max principle or some topological argument. The talk is based on recent joint work with Qiang Guang and Xu-Jia Wang.

In this talk, we discuss critical maps of a horizontal energy functional for maps from a sub-Riemannian manifold to a Riemannian manifold. These critical maps are referred to as subelliptic harmonic maps. In terms of the subelliptic harmonic map heat flow, we investigate the existence problem for subelliptic harmonic maps. Under the assumption that the target Riemannian manifold has non-positive sectional curvature, we can establish some Eells-Sampson type existence results, and also some Hartman type results for the flow.

The Dirac operator was introduced by physists as a “square root” of the Laplacian operator, it has close relationships with the geometry and topology of manifolds. In this talk, we will introduce some recent results on the geometric analysis of Riemannian spin manifolds related to Dirac operators.

In this talk, I will discuss stability and index estimates for compact and noncompact capillary surfaces. A classical result in minimal surface theory says that a stable complete minimal surface in $\mathbf{R}^3$ must be a plane. We show that, under certain curvature assumptions, a strongly stable capillary surface in a 3-manifold with boundary has only three possible topological configurations. In particular, we prove that a strongly stable capillary surface in a half-space of $\mathbf{R}^3$ which is minimal or has the contact angle less than or equal to $\pi/2$ must be a half-plane. We also give index estimates for compact capillary surfaces in 3-manifolds by using harmonic one-forms. This is joint work with Aiex and Saturnino.

In this talk, we will discuss the $L_p$ Brunn-Minkowski theory associated with the $C$-coconvex sets for $p\in [0, 1) $ and the $L_p$ Minkowski problem associated with the $C$-coconvex sets for $p\in\mathbf{R} $. In particular, we will talk about our solution to an open problem raised by Schneider in [Schneider, Adv. Math., 332 (2018), pp. 199-219]. This talk is based on a joint work with Jing Yang and Deping Ye.

I will discuss a family of discrete Brunn-Minkowski type inequalities. As particular cases, this family includes the four functions theorem of Ahlswede and Daykin, a result due to Klartag and Lehec, and other variants, both known and new. Two proofs will be outlined, the first is an elementary short proof and the second is a transport proof which extends a result due to Gozlan, Roberto, Samson and Tetali, and which implies stronger entropic versions of our inequalities. Partly based on joint work with Diana Halikias and Bo'az Klartag

In this talk, I will first review the Bernstein theorem for a class of fourth order equations of Monge-Ampere type, including the affine maximal surface equation and Abreu’s equation. The famous affine Bernstein theorem, also called Chern’s conjecture, was first proved by Trudinger-Wang in dimension two in 2000. Then I will present a new proof without using Caffarelli-Gutierrez’s regularity for the linearized Monge-Ampere equation. The new idea is to establish the interior estimates of the fourth order equations by partial Legendre transform.

Let ${\mbox{Conv}_{{sc}}({\mathbb R}^n)}$ be the space of lower semicontinuous, super-coercive, convex functions $u:{\mathbb R}^n\to (-\infty, \infty]$. A functional $\operatorname{Z}\colon{\mbox{ Conv}_{{ sc}}({\mathbb R}^n)}\to {\mathbb R}$ is called a valuation if \begin{equation*}\label{valuation} \operatorname{ Z}(u_1) +\operatorname{ Z} (u_2)=\operatorname{ Z}(u_1\vee u_2)+\operatorname{Z}(u_1\wedge u_2) \end{equation*} for all $u_1,u_2\in {\mbox{ Conv}_{{ sc}}({\mathbb R}^n)} $ such that the pointwise maximum $u_1\vee u_2$ and the pointwise minimum $u_1\wedge u_2$ are in $ {\mbox{ Conv}_{{ sc}}({\mathbb R}^n)}$. Recently, the Hadwiger theorem was established on ${\mbox{ Conv}_{{ sc}}({\mathbb R}^n)}$, and functional intrinsic volumes were introduced. The functional intrinsic volumes are the continuous, rotation and epi-translation invariant valuations on ${\mbox{ Conv}_{{ sc}}({\mathbb R}^n)}$. For $\zeta\in C(0,\infty)$ with bounded support and a suitable growth condition at $0$ and $0\le j\le n$, the $j$th functional intrinsic volume (depending on $\zeta$) is defined as $$\operatorname{V}_{{j}{\zeta}}(u):=\int_{{\mathbb R}^n} \zeta(|\nabla u(x)|)\big[{{\operatorname{D}}^2} u(x)\big]_{n-j}{\,\mathrm{d}} x$$ for smooth convex functions $u:{\mathbb R}^n\to {\mathbb R}$, where $[{{\operatorname{D}}^2} u]_k$ is the $k$th elementary function of the eigenvalues of the Hessian matrix ${{\operatorname{D}}^2} u$. We present a new representation of functional intrinsic volumes using a new special class of mixed Monge--Amp\`ere measures. (Based on joint work with Andrea Colesanti and Fabian Mussnig)

In this talk we will take a look at the recent progress in the asymptotic theory of random polytopes in non-Euclidean geometries, with a focus on spherical convex geometry. We investigate the asymptotic behavior of the expectation and variance for the volume of random polytopes, that is, the convex hull of an inscreasing number of random points chosen inside or on the boundary of a convex body. In particular I will present results on the asymptotic normality of the volume of random polytopes, which was recently established in joint work with Daniel Rosen and Christoph Thäle.

Given an Orlicz function $M:\mathbf{R}\to[0,\infty)$ (i.e., an even convex function such that $M(0)=0$ and $M(t)>0$ for every $t\neq 0$) and $R>0$, we consider the Orlicz ball $$B_M^n(nR):=\{x\in\mathbf{R}^n\,:\,\sum_{i=1}^nM(x_i)\leq nR\}.$$ We will give some estimates on how a random vector $X_n$ uniformly distributed on $B_M^n(nR)$ concentrates on a thin shell of radius, $\sqrt{n\textrm{Var} Z}$, where $Z$ is a random variable defined by $M$. The proof of these estimates relies on Large Deviation Principles proved by Petrov, and provides the asymptotic value of the isotropic constant of $B_M^n(nR)$ as $n$ tends to infinity.

The framework of $L_p$ operations for functions including the $L_{p,s}$ convolution sum and the $L_{p,s}$ Asplund sum for functions when $p>0$ will be presented. Particularly, the $L_{p,s}$ convolution summations contain the $L_{p,s}$ supremal-convolution when $p\geq1$ and the $L_{p,s}$ inf-sup-convolution when $p\in (0, 1)$, respectively. Moreover, the $L_{p,s}$ Asplund summation is constructed by the $L_p$ averages of bases for $s$-concave functions. Based on the properties of these summations for functions, we establish the $L_p$-Borell-Brascamp-Lieb inequalities for the $L_{p,s}$ supremal-convolution when $p\geq1$. Furthermore, the integral formula for $L_{p,s}$ mixed quermassintegral in terms of the $L_{p,s}$ Asplund summation has also been discovered via tackling the variation formula of quermassintegral for functions when $p\geq 1$. This talk is based on the joint works with Dr. Michael Roysdon.

We introduce a new deformed Hermitian Yang-Mills flow in the supercritical case and show the existence of solution to the deformed Hermitian Yang-Mills equation which was solved by Collins-Jacob-Yau by the continuity method. This is a joint work with Dekai Zhang.

Trudinger-Moser inequality can be considered as a limiting Sobolev inequality corresponding to end-point phenomena for the Sobolev imbeddings. The proofs of the optimal versions of the Trudinger-Moser type inequalities are usually based on the standard symmetric rearrangement of functions. In this talk, we review some rearrangement-free arguments in studying the best constants and optimizers of the Trudinger-Moser type inequalities.

A unified approach used to generalize classical Brunn-Minkowski type inequalities to $L_p$ Brunn-Minkowski type inequalities, called the $L_p$ transference principle, is refined. As illustrations of the effectiveness and practicability of this method, several new $L_p$ Brunn-Minkowski type inequalities concerning the mixed volume, moment of inertia, quermassintegral, projection body and capacity are established. This talk is based on the joint work with Du Zou.

We improve known upper bounds for the minimal dispersion of a point set in the unit cube and its inverse. Some of our bounds are sharp up to logarithmic factors. The talk is partially based on a joint work with G. Livshyts.

In 1885, Fedorov characterized the three-dimensional lattice tiles. They are parallelotopes, hexagonal prisms, rhombic dodecahedra, elongated dodecahedra, or truncated octahedra. Through the works of Minkowski, Voronoi, Delone, Venkov and McMullen, we know that, in all dimensions, every translative tile is a lattice tile. Recently, Mei Han, Kirati Sriamorn, Qi Yang and Chuanming Zong have made a series of discoveries in multiple tilings in two and three dimensions. In particular, in three dimensions, they proved that, if a convex body can form a two, three or fourfold translative tiling, it must be a lattice tile (a parallelohedron). In other words, it must be a parallelotope, a hexagonal prism, a rhombic dodecahedron, an elongated dodecahedron, or a truncated octahedron. In this talk, we will report this progress.

In this talk, we will discuss extensions of the classical Rogers-Shephard inequality, and the Zhang inequality to the measure theoretic setting. The Rogers-Shephard inequality may be seen as a reverse form of the celebrated Brunn-Minkowski inequality, and Zhang's inequality is a reverse form of the Petty-Projection inequality, a very deep and important affine isoperimetric inequality. Both inequalities have the same feature, in that they provide a classification of the n-dimensional simplex in a volumetric way. The heart of this talk consists of the Rogers-Shephard inequality for radially non-increasing measure, a class of measure which contains the class of all measures whose densities are log-concave, and the Zhang inequality for radially non-decreasing measure, which contains the class of measures whose densities are geometric convex functions (those which fix the origin and are non-negative). If time permits, we will discuss a measure theoretic extension of the notion of the covargiogram function, and of a measure theoretic version of the projection body operator. Based on a joint work with D. Alonso-G, M. H. Cifre, J. Yepes-N. and A. Zvavitch, and on a separate work with D. Langharst and A. Zvavitch.

The classical Minkowski problem asks to find a convex body $K$ in $\mathbb{R}^n$ having a prescribed surface-area measure, which boils down to solving the Monge-Amp\`ere equation on the unit sphere $S^{n-1}$. Existence and regularity were extensively studied by Minkowski, Alexandrov, Lewy, Nirenberg, Cheng--Yau, Pogorelov, Caffarelli and many others; uniqueness (up to translation) is an immediate consequence of the classical Brunn--Minkowski inequality. An analogous $L^p$ version for general $p \in \mathbb{R}$ (with the classical case corresponding to $p=1$) was suggested and publicized by E.~Lutwak. The $L^p$-Minkowski problem is related to numerous other fields: a) Non-linear PDE: it is a Monge-Amp\`ere--type equation on $S^{n-1}$; b) Geometric Flows: it describes self-similar solutions to the anisotropic $\alpha$-power-of-Gauss-curvature flow (for $\alpha = \frac{1}{1-p}$); c) Calculus of Variations: it is the Euler-Lagrange equation for the $L^p$-Minkowski functional; d) Brunn--Minkowski theory: it is related to a strengthening of the classical Brunn--Minkowski inequality. The case $p \geq 1$ is well understood, but the case $p < 1$ is much more challenging due to a lack of a corresponding $L^p$-Brunn--Minkowski theory. In particular, no uniqueness is possible in general, but it was conjectured by B\"or\"oczky--Lutwak--Yang--Zhang that for \emph{origin-symmetric} convex bodies $\mathcal{K}_e$, uniqueness in the $L^p$-Minkowski problem should hold for all $p \in [0,1)$. Equivalently, the $L^0-$ (log-)Minkowski functional should have a unique global minimum on $\mathcal{K}_e$, the log-Brunn-Minkowski inequality should hold on $\mathcal{K}_e$, and the anisotropic Gauss-curvature flow should have a unique origin-symmetric self-similar solution. We report on recent progress towards this conjecture. In particular, we resolve the isomorphic version of the log-Minkowski problem, and extend the results of Brendle--Choi--Daskalopoulos on the uniqueness of self-similar solutions to the power-of-Gauss-curvature flow from the isotropic to the pinched anisotropic case (for origin-symmetric solutions). Our main new tool is an interpretation of the problem as a spectral question in centro-affine differential geometry.

A function valued valuation is an additive map defined on convex bodies and taking values in a function space. We are interested in such valuations since they are not only extensions of real, vector, tensor, convex body (Minkowski addition), and star body (radial addition) valued valuations, but are also restricted from some transforms of functions, e.g., the Legendre transform, the Laplace transform, and the Fourier transform. In this talk, I will show some classifications of affine valuations which characterize some fundamental affine operators in convex geometry, e.g., volumes, Euler characteristics, support functions, \(L_p\) projection bodies and \(L_p\) moment bodies (polar \(L_{-p}\) intersection bodies). By such classifications, we also get some Euler-type relations.

One of the classical Sylvester questions asks for the probability that four points $X_1, X_2, X_3, X_4$ independently and uniformly distributed in some plane convex figure $K\subset\mathbf R^2$ form a triangle. Blaschke answered it showing that for any convex figure $K$, \begin{align}\label{2328} \frac{35}{12\pi^2}\leq\mathbf P[\mathrm{conv}(X_1,X_2,X_3, X_4) \text{ is triangle}]\leq\frac{1}{3}. \end{align} The lower bound is achieved if and only if $K$ is an ellipse, and the upper one -- if and only if $K$ is a triangle. It is straightforward that \[ \mathbf P[\mathrm{conv}(X_1,X_2,X_3, X_4) \text{ is triangle}]=4\frac{\mathbf E\, \mathrm S(\mathrm{conv}(X_1,X_2,X_3))}{\mathrm S(K)}, \] where $\mathrm S(\cdot)$ denotes the area of a plane figure. Therefore~\eqref{2328} is equivalent to \[ \frac{35}{48\pi^2}\leq\frac{\mathbf E\, \mathrm S(\mathrm{conv}(X_1,X_2,X_3))}{\mathrm S(K)}\leq\frac{1}{12}, \] which gives the optimal lower and upper bounds for the normalized average area of the random triangle inside a convex figure. If we have only two random points inside $K$, they form a random segment with some random length. Thus it is natural to ask for the optimal bounds of the normalized average length of this segment. While the area of the random triangle is normalized by the area of $K$, the length of the random segment should be normalized by the perimeter of $K$ denoted by $P(K)$. To answer this question, we will show that for any convex figure $K\subset\mathbb R^2$ with non-empty interior, \begin{align}\label{1316} \frac{7}{60}<\frac{\mathbf E \|X_1-X_2\|)}{\mathrm P(K)}<\frac{1}{6}. \end{align} The both lower and upper bounds are optimal. We will also generalize~\eqref{1316} to any dimension. Based on the joint paper: G. Bonnet, A. Gusakova, Ch. Thäle, and D. Zaporozhets, ``Sharp inequalities for the mean distance of random points in convex bodies'', Adv. Math., 386 (2021)

In this talk, we introduce the canonical connection instead of Levi-Civita connection to study the smooth maps between almost Hermitian manifolds, especially, the pseudoholomorphic ones. By using the Bochner formulas, we obtian the \(C^2\)-estimate, Liouville type theorems of pseudoholomorphic maps,pseudoholomorphicity of pluriharmonic maps, Simons integral inequality and bounds of half norm of parallel canonical second fundamental form of pseudoholomorphic isometry.

It is shown that for any sufficiently regular even Minkowski valuation $\Phi$ which is homogeneous and intertwines rigid motions, and for any convex body $K$ in a smooth neighborhood of the unit ball, there exists a sequence of positive numbers $(\gamma_m)_{m=1}^\infty$ such that $(\gamma_m\Phi^m K)_{m=1}^\infty$ converges to the unit ball with respect to the Hausdorff metric.

It is well known that the Schwarz symmetrization of a function diminishes its Dirichlet integral. This is the so-called Polya-Szego Principle. A similar symmetrization with respect to quermassintegral, which has been introduced by Talenti and Tso, diminishes the Hessian integral. In the same spirit, Alvino-Ferone-Lions-Trombetti introduced a convex symmetrization which diminishes the anisotropic Dirichlet integral. In this talk, I will present a joint work with Della Pietra and Gavitone, on convex symmetrization with respect to mixed volumes or anisotropic curvature integrals and the corresponding Polya-Szego-type principle for the anisotropic Hessian integral.

Define centroaffine invariant arc length and curvature functions of a curve in affine n-space. Consider the properties and relations of the curves in affine space and Semi-Euclidean space. Using these notions and conclusions, by solving certain differential equations, we give some examples and classifications of the curves in affine 2-space and 3-space.

The (B)-property is a certain concavity property of measures. It is conjectured that every even log-concave measure has the (B)-property, and this conjecture was confirmed in dimension 2 as a corollary of the log-Brunn-Minkowski inequality. However, in dimension at least 3 there are very few known examples of measures with the (B)-property. These include only the Gaussian measure (proved by Cordero-Erausquin, Fradelizi and Maurey) and certain Gaussians mixtures (proved by Eskenazis, Nayar and Tkocz). In this talk we prove that a large class of measures have the (B)-property. This includes all rotation invariant log-concave measures, thereby creating the first truly non-Gaussian examples. For the proof we present a new Brascamp-Lieb type inequality for rotation invariant measures, as well as a new weighted Poincaré-type inequality. Based on a joint work with Dario Cordero-Erausquin

The intrinsic volumes are fundamental geometric functionals defined on sufficiently nice subsets of Euclidean space, in particular on convex sets, where they are known as quermassintegrals. They can be defined by considering the volume of an epsilon extension of the set. H. Weyl discovered that their value on a Riemannian submanifold of Euclidean space is, remarkably, an intrinsic invariant of the metric independent of the ambient space. We will consider the setting of normed spaces, where the Holmes-Thompson intrinsic volumes extend the Euclidean classical ones, and describe the extent to which Weyl's result remains valid in the Finsler category. Based on a joint work with T. Wannerer.

The classical Isoperimetric inequality for affine surface area states that among all convex bodies with fixed volume, the Euclidean ball has the maximal affine surface area. On the other hand, affine surface area vanishes on polytopes. For a convex body K, we define IS(K) to be the maximum of affine surface areas among all convex bodies contained in K. One still has the isoperimetric inequality: IS(K) is less equal than IS(B) where B is a ball with the same volume as K. We manage to show that unlike affine surface area, IS(K) is bounded below IS(B) with a multiplicative factor n^{-5/6-o(1)}. In other words, it is proportional to IS(B). The same result extends to the lp and dual version of affine surface area. And the proof is based on the thin-shell phenomenon. This is a joint work with O. Giladi, C. Schuett, and E.M. Werner

I will review some aspects of the basic theory of affine $p$-Rayleigh quotients, the affine Poincaré inequality and the affine $p$-Laplace operator. I'll present also several open questions regarding affine invariant differential equations involving the affine Laplacian.

In this talk we first introduce the $L_p$-Minkowski problem, and then discuss the existence of complete, noncompact convex hypersurfaces whose $p$-curvature function is prescribed on a domain in the unit sphere. This noncompact $L_p$-Minkowski problem is related to the solvability of Monge-Ampere type equations subject to certain boundary conditions depending on the value of $p$. The talk is based on a joint work with Yong Huang.

We present links between Mahler's conjecture on log-concave functions and entropy-transport inequalities. This enables us to give a transport proof of Mahler's conjecture for unconditional functions and in particular a new simple proof of the conjecture in dimension 1. Work in collaboration with Nathael Gozlan and Simon Zugmeyer.

In this talk, we will introduce the extended Musielak-Orlicz-Gauss image problem. Such a problem aims to characterize the Musielak-Orlicz-Gauss image measure $\widetilde{C}_{G,\Psi,\lambda}(\Omega,\cdot)$ of convex body $\Omega$ in $\mathbb{R}^{n+1}$ containing the origin (but the origin is not necessary in its interior). In particular, we provide solutions to the extended Musielak-Orlicz-Gauss image problem based on the study of suitably designed parabolic flows, and by the use of approximation technique. Our parabolic flows involve two Musielak-Orlicz functions and hence contain many well-studied curvature flows related to Minkowski type problems as special cases. This is a joint work with Qi-Rui Li, Weimin Sheng and Deping Ye.

In this talk we will discuss a number of inequalities in Convex Geometry inspired by sumsets estimates in additive combinatorics. We will discuss the notion of supermodularity and as application will show that the volume is supermodular to arbitrary order with respect to Minkowski summation on the space of convex bodies. Second, we will explore sharp constants in the convex geometry analogues of some classical sumset estimates including the Plünnecke-Ruzsa inequality and the Ruzsa triangle inequality. This is a joint work with Matthieu Fradelizi and Mokshay Madiman.