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Large blow-up sets for -curvature equations.

On a bounded domain of the Euclidean space , , Adimurthi, Robert and Struwe pointed out that, even assuming a volume bound , some blow-up solutions for prescribed -curvature equations without boundary conditions may blow-up not only at points, but also on the zero set of some nonpositive nontrivial polyharmonic function. This is in striking contrast with the two dimensional case (). During this talk, starting from a work in progress with Ali Hyder and Luca Martinazzi, we will discuss the construction of such solutions which involves (possible generalizations of) the Walsh-Lebesgue theorem and some issues about elliptic problems with measure data.

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After providing a mathematical background for some curious optical experiments in the 19th century, we will then describe how these ideas inform our understanding of the Deift conjecture for the Korteweg–de Vries equation. Specifically, they allow us to show that the evolution of almost-periodic initial data need not remain almost periodic. This is joint work with Andreia Chapouto and Monica Visan.

We consider a quantum system consisting of N bosons (particles described by a permutation symmetric wave function) trapped in a volume of order one and interacting through a short range potential, with scattering length of the order 1/N (this is known as the Gross-Pitaevskii regime). First, we will show how non-linear Gross-Pitaevskii theory describes, to leading order, the ground state energy of the gas and the time-evolution resulting from a change of the external fields. In the second part of the talk, I will then explain how Bogoliubov theory predicts the next order corrections.

After a gentle introduction on water waves, I will present an exposition of joint work with Huy Quang Nguyen. We prove rigorously that the classical (small-amplitude irrotational steady periodic) water waves are unstable with respect to long-wave perturbations. That is, the perturbations grow exponentially in time. This instability was first observed heuristically more than half a century ago by Benjamin and Feir. However, a rigorous proof was never found except in the case of finite depth. We provide a completely different and self-contained proof of both the finite and infinite depth cases that retains the physical variables. The proof reduces to an analysis of the spectrum of an explicit operator. The growth is obtained by means of a Liapunov-Schmidt reduction that more or less reduces the analysis to four dimensions.

I will discuss the derivative nonlinear Schrodinger equation, how some inherent instabilities have hindered the study of this equation, and how we were able to demonstrate global well-posedness in the natural scale-invariant space. This is joint work with Ben Harrop-Griffiths, Rowan Killip, and Maria Ntekoume.

The classical Busemann-Petty Problem from the 1950s asked the following tomographic question: Assuming you have two origin-symmetric convex bodies and in the -dimensional Euclidean space satisfying the following volume inequality: for all does it follow that ? The answer is affirmative for and negative whenever . However, if belongs to a certain class of convex bodies, the intersection bodies, then the answer to the Busemann-Petty problem is affirmative in all dimension. Several extensions of this result have been shown in the case of measures on convex bodies, and isomorphic results of the same type have been established. Moreover, the isomorphic Busemann-Petty problem is actually equivalent to the isomorphic slicing problem of Bourgain (1986), which remains open to this day. In this talk, we will introduce the notion of an intersection function, provide a Fourier analytic characterization for such functions, and show some versions of the Busemann-Petty problem in this setting. In particular, we will show that if you have a pair of continuous, even, integrable functions which satisfy , where denotes the Radon transform, then one has that provided that the function is an intersection function. This is based on a joint work with Alexander Koldobsky and Artem Zvavitch

In this work, we prove an extension inequality in the hyperbolic space. The inequality involves the hyperbolic harmonic extension of a function on the boundary and the Fefferman-Graham compactification of the hyperbolic metric. We offer an interpretation of the extension inequality as a conformally invariant generalization of Carleman’s inequality to higher dimensions. In addition to that, we classify all the solutions to the Euler-Lagrange equation of the extension inequality. The proof uses the moving sphere method and relies on the properties of the Fefferman-Graham compactification of the hyperbolic metric.

The surface area measure and related Minkowski problem for log-concave functions.

The Minkowski type problems for convex bodies are fundamental in convex geometry, and have found many important connections and applications in analysis, partial differential equations, etc. It is well-known that the log-concave functions behave rather similar to convex bodies in many aspects, for example the famous Prékopa–Leindler inequality to the (dimension free) Brunn-Minkowski inequality. In this talk, I will present an theory for the log-concave functions, which is analogous to the Brunn-Minkowski theory of convex bodies. In particular, I will explain how to define the sum of log-concave functions, present a variational formula related to the addition, and talk about the corresponding Minkowski problems as well as their solutions.

We study the Stokes eigenvalue problem under Navier boundary conditions in $C^{1,1}$-domains $\Omega\subset \mathbb{R}^3$. Differently from the Dirichlet boundary conditions, zero may be the least eigenvalue. We fully characterize the domains where this happens, showing the related validity/failure of a suitable Poincar\'{e}-type inequality. As application we prove regularity results for the solution of the evolution Navier-Stokes equations under Navier boundary conditions in a class of merely {\em Lipschitz domains} of physical interest, that we call {\em sectors}. This is a joint work with Filippo Gazzola, Politecnico di Milano.

Maintaining the topology of objects undergoing deformations is a crucial aspect of elasticity models. In this talk we consider two different settings in which impermeability is implemented via regularization by a suitable nonlocal functional. The behavior of long slender objects may be characterized by the classic Kirchhoff model of elastic rods. Phenomena like supercoiling which play an essential role in molecular biology can only be observed if self-penetrations are precluded. This can be achieved by adding a self-repulsive functional such as the tangent-point energy. We discuss the discretization of this approach and present some numerical simulations. In case of elastic solids whose shape is described by the image of a reference domain under a deformation map, self-interpenetrations can be ruled out by claiming global invertibility. Given a suitable stored energy density, the latter is ensured by the Ciarlet–Nečas condition which, however, is difficult to handle numerically in an efficient way. This motivates approximating the latter by adding a self-repulsive functional which formally corresponds to a suitable Sobolev–Slobodeckiĭ seminorm of the inverse deformation. This is joint work with Sören Bartels (Freiburg) and Stefan Krömer (Prague).

The Fourier uncertainty principle describes a fundamental phenomenon that a function and its Fourier transform cannot simultaneously localize. Dyatlov and his collaborators (Zahl, Bourgain, Jin, Nonnenmacher) recently introduced a concept of Fractal Uncertainty Principle (FUP). It is a mathematical formulation concerning the limit of localization of a function and its Fourier transform on sets with certain fractal structure. The FUP has quickly become an emerging topic in Fourier analysis and also has important applications to other fields such as wave decay in obstacle scattering. In this talk, we consider the discrete Fourier transform and the fractal sets are given by discrete Cantor sets. We present the FUP in this discrete setting with a much more favorable estimate than the one known before, when the Cantor sets are constructed by a random procedure. This is a joint work with Suresh Eswarathasan.

This talk is focused on the vibrating clamped plate problem, initially formulated by Lord Rayleigh in 1877, and solved by M. Ashbaugh & R. Benguria (Duke Math. J., 1995) and N. Nadirashvili (ARMA, 1995) in 2 and 3 dimensional euclidean spaces. We consider the same problem on both negatively and positively curved spaces, and provide various answers depending on the curvature, dimension and the width/size of the clamped plate.

The Heisenberg uncertainty principle is a fundamental result in quantum mechanics. Related inequalities are the hydrogen and Hardy uncertainty principles and all three belong to the family of geometric inequalities known as the Caffarelli-Kohn-Nirenberg inequalities. In this talk, we present our recent results pertaining to uncertainty principles and CKN inequalities with a particular focus on higher order derivatives and vector-valued cases. Presented works were done jointly with G. Lu, N. Lam and C. Cazacu.

We show that nontrivial solutions to higher and fractional order equations with certain nonlinearity are radially symmetric and nonincreasing on geodesic balls in the hyperbolic space as well as on the entire space . Applying Helgason-Fourier analysis techniques on , we developed a moving plane approach for integral equations on . We also established the symmetry to solutions of certain equations with singular terms on Euclidean spaces. Moreover, we obtained symmetry to solutions of some semilinear equations involving fractional order derivatives.

In this talk, I will present a recent result about blow-up asymptotics in the three-dimensional Brezis-Nirenberg problem. More precisely, for a smooth bounded domain and smooth functions and , consider a sequence of positive solutions to on with zero Dirichlet boundary conditions, which blows up as . We derive the sharp blow-up rate and characterize the location of concentration points in the general case of multiple blow-up, thereby obtaining a complete picture of blow-up phenomena in the framework of the Brezis-Peletier conjecture in dimension . I will also indicate a forthcoming new result parallel to this one for dimension . This is joint work with Paul Laurain (IMJ-PRG Paris and ENS Paris).

We show the existence of determining wavenumber for the Naiver-Stokes equation in both 3D and 2D. Estimates on the determining wavenumber are established in term of the phenomenological Kolmogorov’s dissipation number (3D) and Kraichnan’s number (2D). The results rigorously justify the criticality of Kolmogorov’s dissipation number and Kraichnan’s number.

We study the regularity properties of the second order linear operator in : where , are real valued matrices with constant coefficients, with symmetric and strictly positive. We prove that, if the operator satisfies Hörmander’s hypoellipticity condition, and is a Dini continuous function, then the second order derivatives of the solution to the equation are Dini continuous functions as well. We also consider the case of Dini continuous coefficients ‘s. A key step in our proof is a Taylor formula for classical solutions to that we establish under minimal regularity assumptions on .

n joint work with Almaz Butaev (Cincinnati), we study local versions of uniform domains, which can be identified with the epsilon-delta domains used by Jones to extend Sobolev spaces. We show that a domain is locally uniform if and only if it is an extension domain for the nonhomogeneous (also known as “local”) space of functions of bounded mean oscillation introduced by Goldberg, and denoted by bmo. We also prove analogous results for functions of vanishing mean oscillation.

We report on recent results regarding the limiting absorption principle for multi-dimensional, massless Dirac-type operators (implying absence of singularly continuous spectrum) and continuity properties of the associated spectral shift function. We will motivate our interest in this circle of ideas by briefly describing the connection to the notion of the Witten index for a certain class of non-Fredholm operators. This is based on various joint work with A. Carey, J. Kaad, G. Levitina, R. Nichols, D. Potapov, F. Sukochev, and D. Zanin.

The goal of the talk is to discuss a quantitative version of the Levy-Gromov isoperimetric inequality (joint with Cavalletti and Maggi) as well as a quantitative form of Obata’s rigidity theorem (joint with Cavalletti and Semola). Given a closed Riemannian manifold with strictly positive Ricci tensor, one estimates the measure of the symmetric difference of a set with a metric ball with the deficit in the Levy-Gromov inequality. The results are obtained via a quantitative analysis based on the localisation method via L1-optimal transport. For simplicity of presentation, the talk will present the results in case of smooth Riemannian manifolds with Ricci Curvature bounded below; moreover it will not require previous knowledge of optimal transport theory.

This talk presents a joint work with F. Charro, A. Haj Ali, M. Raihen, and M. Torres on a bifurcation phenomenon in a two-phase, singularly perturbed, free boundary problem of phase transition. We show that the uniqueness of the solution for the two-phase problem breaks down as the boundary data decreases through a threshold value. For boundary values below the threshold, there are at least three solutions, namely, the harmonic solution which is treated as a trivial solution in the absence of a free boundary, a nontrivial minimizer of the functional under consideration, and a third solution of the mountain-pass type. We classify these solutions according to the stability through evolution. The evolution with initial data near a stable solution, such as the trivial harmonic solution or a minimizer of the functional, converges to the stable solution. On the other hand, the evolution deviates away from a non-minimal solution of the free boundary problem.

The difference between the first two eigenvalues of the Dirichlet Laplacian on convex sets of R^n and, respectively S^n, satisfies the same strictly positive lower bound depending on the diameter of the domain. In work with collaborators, we have found that the gap of the hyperbolic space on convex sets behaves strikingly different even if a stronger notion of convexity is employed. This is very interesting as many other features of first two eigenvalues behave in the same way on all three spaces of constant sectional curvature.

Spectrum of Laplacian is an important set of geometric invariants. The talk, largely based on joint work with Peter Li and Ovidiu Munteanu, concerns its structure and size on complete manifolds under various curvature conditions. The focus is on sharp estimates of the bottom spectrum in terms of either Ricci or scalar curvature lower bound.

Many inequalities for trace functional are formulated as concavity/convexity theorems. These generally have an equivalent monotonicity version asserting monotonicity of the functional under some class of completely positive maps. The monotonicty formulation has advantages: (1) Often this has a direct physical interpretation. (2) Often a direct proof of the monotonicity version is simpler than a direct proof of the concavity/convexity version, and the later is always recovered using a simple partial trace argument. (3) Often the monotonicty theorem holds for a broader class of maps, not, necessarily completely positive, and is thus a strictly stronger result. We discus significant examples, some coming from recent joint work with Alexander Mueller-Hermes.

I present some results concerning the size of magnetic fields that support zero modes for the three dimensional Dirac equation and related problems for spinor equations. The critical quantity, is the norm of the magnetic field . The point is that the spinor structure enters the analysis in a crucial way. This is joint work with Rupert Frank at LMU Munich.

Orbifolds are a generalization of manifolds in which various types of singularities may occur. After reviewing the notion of Riemannian orbifolds and their Hodge Laplacians, we will address the question: Does the spectrum of the Hodge Laplacian on p-forms detect the presence of singularities? This question remains open in the case of the Laplace-Beltrami operator (i.e., the case p=0), although many partial results are known. We will show that the spectra of the Hodge Laplacians on functions and 1-forms together suffice to distinguish manifolds from orbifolds with sufficiently large singular set. In particular, these spectra always distinguish low-dimensional orbifolds (dimension at most 3) with singularities from smooth manifolds. We also obtain weaker affirmative results for the spectrum on 1-forms alone and show via counterexamples that some of these results are sharp. (This is based on recent joint work with Katie Gittins, Magda Khalile, Ingrid Membrillo Solis, Mary Sandoval, and Elizabeth Stanhope and work in progress with the same co-authors along with Juan Pablo Rossetti.) Time permitting, we will also make a few remarks concerning the Steklov spectrum on Riemannian orbifolds with boundary. The Steklov spectrum is the spectrum of the Dirichlet-to-Neumann operator, which maps Dirichlet boundary values of harmonic functions to their Neumann boundary values.

In this talk, we will consider a question of stability (i.e. compactness of solutions to perturbed equations) for sign-changing solutions to second-order critical elliptic equations on a closed Riemannian manifold. I will present a stability result obtained in the case of dimensions greater than or equal to 7. I will then discuss the optimality of this result by constructing counterexamples in every dimension. This is a joint work with Bruno Premoselli (Université Libre de Bruxelles, Belgium).

In this talk, I will review recent results on symmetry and symmetry-breaking for optimizing solutions of a general class of nonlinear elliptic PDEs. On one hand, I will discuss a novel approach to prove symmetry by using the so-called Fourier rearrangements, which can be applied to PDEs of arbitrary order (where classical method such as the moving plane method or the Polya-Szegö principle fail short). On the other hand, I will discuss recent results on symmetry-breaking for optimizers by using Fourier methods and the Stein-Tomas inequality. This talk is based on joint work with Tobias Weth and Jeremy Sok.

Various notions of convexity of sets and functions in the Heisenberg group have been studied in the past two decades. In this talk, we focus on the horizontally quasiconvex ($h$-quasiconvex) functions in the Heisenberg group. Inspired by the first-order characterization and construction of quasiconvex envelope by Barron, Goebel and Jensen in the Euclidean space, we obtain a PDE approach to construct the $h$-quasiconvex envelope for a given function $f$ in the Heisenberg group. In particular, we show the uniqueness and existence of viscosity solutions to a non-local Hamilton-Jacobi equation and iterate the equation to obtain the $h$-quasiconvex envelope. Relations between $h$-convex hull of a set and the $h$-quasiconvex envelopes are also investigated. This is joint work with Antoni Kijowski (OIST) and Qing Liu (Fukuoka University/OIST).

In this talk, we will discuss the relation between complex Hessian equation and positivity of algebraic numerical conditions. In particular, we will prove a Naki-Moishezon criterion for Donaldson’s J-equation.

For CR manifolds of real dimension three, we study the Jacobi field equation. Under the condition that the torsion be parallel, we obtain comparison results against a family of homogeneous CR structures. As application, we describe the singularities of contact forms on the the homogeneous structures with finite total Q-prime curvature. This is ongoing joint work with Sagun Chanillo.

A celebrated theorem of Gromov says that given there is an such that if a closed Riemannian manifold satisfies then is diffeomorphic to an infranilmanifold. I will show that the lower sectional curvature bound in Gromov’s theorem can be weakened to the lower Bakry-Emery Ricci curvature bound. I will also discuss the relation of this result to the study of manifolds with Ricci curvature bounded below.

I’ll present some recent results concerning the IVP of 1d cubic NLS at the critical level of regularity. I’ll also exhibit a cascade of energy for the 1D Schrödinger map which is related to NLS through the so called Hasimoto transformation. For higher regularity these two equations are completely integrable systems and therefore no cascade of energy is possible.

Consider an essentially nonbranching metric measure space with the measure contraction property of Ohta and Sturm. We prove a sharp upper bound on the inscribed radius of any subset whose boundary has a suitably signed lower bound on its generalized mean curvature. This provides a nonsmooth analog of results dating back to Kasue (1983) and subsequent authors. We prove a stability statement concerning such bounds and — in the Riemannian curvature-dimension (RCD) setting — characterize the cases of equality. This represents joint work with Annegret Burtscher, Christian Ketterer and Eric Woolgar.

In the upper half-space, the distance function to the boundary is a positive solution to Laplace’s equation that vanishes on the boundary, which can be interpreted as the Green function with pole at infinity for the Laplacian. We are interested in understanding the exact relations between the behavior of the Green function, the structure of the underlying operator, and the geometry of the domain. In joint work with G. David and S. Mayboroda, we obtain a precise and quantitative control of the proximity of the Green function and the distance function on the upper half-space by the oscillation of the coefficients of the operator. The class of the operators that we consider is of the nature of the best possible for the Green function to behave like a distance function. More recently, together with J. Feneuil and S. Mayboroda, we obtain analogous results for domains with uniformly rectifiable boundaries.

Loewner and Nirenberg discussed complete metrics conformal to the Euclidean metric and with a constant scalar curvature in bounded domains in the Euclidean space. The conformal factors blow up on boundary. The asymptotic behaviors of the conformal factors near boundary are known in smooth and sufficiently smooth domains. In this talk, we introduce the logarithm of the distance to boundary as an additional independent self-variable and establish a concise boundary regularity.

NATURAL -MEANS FOR THE -LAPLACIAN IN EUCLIDEAN SPACE AND THE HEISENBERG GROUP

We consider semi-discrete approximations to -harmonic functions based on the natural -means of Ishiwata, Magnanini, and Wadade in 2017 (CVPDE 2017), who proved their local convergence. In the Euclidean case we prove uniform convergence in bounded Lipschitz domains. We also consider adapted semi-discrete approximations in the Heisenberg group and prove uniform convergence in bounded -domains. This talk is based in joint work with András Domokos and Diego Ricciotti (Sacramento) and Bianca Stroffolini (Naples)

Geodetically convex sets in Heisenberg group .

A classification of geodetically convex subsets of Heisenberg group of homogeneous dimension 4 was proved by Monti-Rickly. We extend their result to a higher dimension Heisenberg group. This is ongoing work with my PhD student Anoop Varghese.

I will discuss how uniform Sobolev inequalities obtained from the monotonicity of Perelman’s W-functional can be used to prove curvature pinching theorems on Riemannian manifolds. These are based on scale-invariant integral norms and generalize some earlier pointwise and supercritical integral pinching statements. This is joint work with Guofang Wei and Rugang Ye.

We will talk about Alexandrov-Fenchel type inequalities in the hyperbolic space and their applications in a higher order mass of asymptotically hyperbolic manifolds. The talk is based on a series of work joint with Yuxin Ge, Jie Wu and Chao Xia

The Mean Value Theorem implies that the average value of a subharmonic function in a disk can be bounded from above by the average value on the boundary. What happens if we replace the disk by another domain? Maybe surprisingly, the problem has a relatively clean answer — we discuss a whole range of mean value inequalities for convex domains in IR^n. The extremal domain remains a mystery for most of them. The techniques are an amusing mixture of classical potential theory, complex analysis, a little bit of elliptic PDEs and, surprisingly, the theory of solids from the 1850s.

A manifold is said to be stochastically complete if the free heat semigroup preserves probability. It is well-known that this property is equivalent to nonexistence of nonnegative, bounded solutions to certain (linear) elliptic problems, and to uniqueness of solutions to the heat equation corresponding to bounded initial data. We prove that stochastic completeness is also equivalent to similar properties for certain nonlinear elliptic and parabolic problems. This fact, and the known analytic-geometric characterizations of stochastic completeness, allow to give new explicit criteria for existence/nonexistence of solutions to certain nonlinear elliptic equations on manifolds, and for uniqueness/nonuniqueness of solutions to certain nonlinear diffusions on manifolds.

In this talk we plan to describe some results on the extension problem associated to the sublaplacian on the Heisenberg group The Dirichlet to Neumann map induced by this problem leads to conformally invariant fractional powers of We use the results to prove a version of Hardy’s inequality for such fractional powers. These results are based on my joint work with Luz Roncal.

The Helmholtz decomposition of vector fields is a fundamental tool for analysis of vector fields especially to analyze the Navier-Stokes equations in a domain. It gives a unique decomposition of a (tangential) vector field defined in a domain of an Euclidean space (or a riemannian maniford) into a sum of a gradient field and a solenoidal field with supplemental condition like a boundary condition.It is well-known that such decomposition gives an orthogonal decomposition of the space of vector fields in an arbitrary domain and known as the Weyl decomposition. It is also well-studied that in various domains including the half space, smooth bounded and exterior domain, it gives a topological direct sum decomposition of the space of vector fields for . The extension to the case 　(or ) is impossible because otherwise it would imply the boundedness of the Riesz type operator in (or ) which is absurd. In this talk, we extend the Hemlholtz decomposition in a space of vector fields with bounded mean oscillations (BMO) when the domain of vector field is a smooth bounded domain in an Euclidean space. There are several possible definitions of a BMO space of vector fields. However, to have a topological direct sum decomposition, it turns out that components of normal and tangential to the boundary should be handled separately. This decomposition problem is equivalent to solve the Poisson equation with the divergence of the original vector field as a data with the Neumann data with the normal trace of . The desired gradient field is the gradient of the solution of this Poisson equation. To solve this problem we construct a kind of volume potential so that the problem is reduced to the Neumann problem for the Laplace equation. Unfortunately, taking usual Newton potential causes a problem to estimate necessary norm so we construct another volume potential based on normal coordinate. We need a trace theorem to control norm of the normal trace. This is of independent interest. Finally, we solve the Neumann problem with data in a necessary space. The Helmholtz decomposition for BMO vector fields is previously known only in the whole Euclidean space or the half space so this seems to be the first result for a domain with a curved boundary. This is a joint work with my student Z.Gu (University of Tokyo).

Boundary value problems for -elliptic operators.

We give some background about the regularity of solutions to real and complex elliptic operators, motivating a new algebraic condition (-ellipticity). We introduce this condition in order to solve new boundary value problems for operators with complex coefficients. Results with M. Dindos, and with M. Dindos and J. Li, are discussed

We study the regularization effect of the inhomogeneous Boltzmann equation without cutoff. We obtain a priori estimates for all derivatives of the solution depending only on bounds of its hydrodynamic quantities: mass density, energy density and entropy density. As a consequence, a classical solution to the equation may fail to exist after a certain time T only if at least one of these hydrodynamic quantities blows up. Our analysis applies to the case of moderately soft and hard potentials. We use methods that originated in the study of nonlocal elliptic and parabolic equations: a weak Harnack inequality in the style of De Giorgi, and a Schauder-type estimate.

We study nonexistence results for a coercive semilinear biharmonic equation on the whole . The analysis is made for general solutions without any assumption on their sign nor on their behaviour at infinity. A relevant role is played by some extensions of the Hardy-Rellich inequalities for general functions (not necessarily compactly supported).

We first consider classes of variational problems for densities that repel each other at distance. Examples are given by the minimizers of Dirichlet functional or the Rayleigh quotient over the class of functions attaining some boundary conditions on , and subjected to the constraint As second class of problems, we consider energy minimizers of Dirichlet energies with different metrics with constraint For these problems, we investigate the optimal regularity of the solutions, prove a free-boundary extremality condition, and derive some preliminary results characterising the emerging free boundary.

The Euler elastica problem seeks to minimise the -norm of the curvature of curves under certain boundary conditions. If we replace the -norm with the -norm, then we obtain a variational problem with quite different properties. Nevertheless, even though the underlying functional is not differentiable, it turns out that the solutions of the problem can still be described by differential equations. An analysis of these equations then gives a classification of the solutions.

In this talk we will discuss connections between the geometric and PDE properties of sets. The emphasis is on quantifiable, global results which yield true equivalence between the geometric and PDE notions in very rough scenarios, including domains and equations with singularities and structural complexity. The main result establishes that in all dimensions , a -dimensional set in is regular (rectifiable) if and only if the Green function for elliptic operators is well approximated by affine functions (distance to the hyperplanes). To the best of our knowledge, this is the first free boundary result of this type for lower dimensional sets and the first free boundary result in the classical case without restrictions on the coefficients of the equation.

This is a report of some recent joint work with Eden Prywes and Paul Yang. The main result is a bi-Lipschitz equivalence of a class of metrics on 4-shpere under curvature constraints. The proof involves two steps: first a construction of quasiconformal maps between two conformally related metrics in a positive Yamabe class, followed by the step of applying the Ricci flow to establish the bi-Lipschitz equivalence from such a conformal class to the standard conformal class on 4-sphere.

I will discuss a generalization of my previous result on the localization of extremal maps near a strongly pseudo-convex point.

Finding non-constant harmonic 3-spheres for a closed target manifold N is a prototype of a super-critical variational problem. In fact, the direct method fails, as the infimum of Dirichlet energy in any homotopy class of maps from the 3-sphere to any closed N is zero; moreover, the harmonic map heat flow may blow up in finite time, and even the identity map from the 3-sphere to itself is not stable under this flow. To overcome these difficulties, we propose the normalized harmonic map heat flow as a new tool, and we show that for this flow the identity map from the 3-sphere to itself now, indeed, is stable; moreover, the flow converges to a harmonic 3-sphere also when we perturb the target geometry. While our results are strongest in the perturbative setting, we also outline a possible global theory.

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We will introduce wave maps, an important geometric flow, and discuss, for the case when the target is the sphere, the asymptotic behavior near the ground state (without symmetry) and recent results in the general case (under co-rotational symmetry) in joint work with Duyckaerts, Martel and Merle.

We study the existence of regular solutions of the incompressible stationary Navier-Stokes equations in -dimensional Euclidean space with a given bounded external force of compact support. In dimensions , the existence of such solutions was known. In this paper, we extend it to dimensions . This is a joint work with Zhuolun Yang.

Cheng and Yau derived the well-known gradient estimate for positive harmonic functions and obtained the classical Liouville theorem, which states that any bounded harmonic function is constant in complete noncompact Riemannian manifolds with nonnegative Ricci curvature. I will talk about the CR analogue of Yau’s conjecture. We need to derive the CR volume doubling property, CR Sobolev inequality, and mean value inequality. Then we can apply them to prove the CR analogue of Yau’s conjecture on the space consisting of all pseudoharmonic functions of polynomial growth of degree at most in a complete noncompact pseudohermitian -manifold. As a by-product, we obtain the CR analogue of volume growth estimate and Gromov precompactness theorem.

In 1974, Folland and Stein introduced a generalization of ellipticity known as maximal hypoellipticity. This talk will be an introduction to this concept and some of the ways it generalizes ellipticity.

In this talk, we will consider nonlinear parabolic fractional equations We develop a systematical approach in applying an asymptotic method of moving planes to investigate qualitative properties of positive solutions for fractional parabolic equations. To this end, we derive a series of needed key ingredients such as narrow region principles, and various asymptotic maximum and strong maximum principles for antisymmetric functions in both bounded and unbounded domains. Then we illustrate how these new methods can be employed to obtain asymptotic radial symmetry and monotonicity of positive solutions in a unit ball and on the whole space. Namely, we show that no matter what the initial data are, the solutions will eventually approach to radially symmetric functions. We will also consider the entire positive solutions on a half space, in the whole space, and with indefinite nonlinearity. Monotonicity and nonexistence of solutions are obtained. This is joint work with P. Wang, Y. Niu, Y. Hu and L. Wu.

We begin with the sub-Laplacian on the Heisenberg group and then the twisted Laplacian by taking its inverse Fourier transform with respect to the center of the group. The eigenvalues and the eigenfunctions of the twisted Laplacian are computed explicitly. Then we turn our attention to the product of the twisted Laplacian and its transpose, thus obtaining a fourth order partial differential operator dubbed the twisted bi-Laplacian. The connections between the spectral analysis of the twisted bi-Laplacian and Dirichlet divisors, the Riemann zeta function and the Dixmier trace are explained.

Abstract: Some classical nonlinear semigroups arising in mechanics induce unilateral bounds on solutions. Hamilton–Jacobi equations and 1-d scalar conservation laws are classical examples of such nonlinear effects: solutions spontaneously develop one-sided Lipschitz or semi-concavity conditions. When this occurs the range of the semigroup is unilaterally bounded by a threshold. On the other hand, in practical applications, one is led to consider the problem of time-inversion, so to identify the initial sources that have led to the observed dynamics at the final time. In this lecture we shall discuss this problem answering to the following two questions: On one hand, to identify the range of the semigroup and, given a target, to characterize and reconstruct the ensemble of initial data leading to it. Illustrative numerical simulations will be presented, and a complete geometric interpretation will also be provided. We shall also present a number of open problems arising in this area and the possible link with reinforcement learning.

Abstract: A classical problem that traces back to Helmholtz and Kirchoff is the understanding of the dynamics of solutions to the 2d and 3d Euler equations of an inviscid incompressible fluid, when the vorticity of the solution is initially concentrated near isolated points in 2d or vortex lines in 3d. We discuss some recent result on existence and asymptotic behaviour of these solutions. We describe, with precise asymptotics, interacting vortices and travelling helices. We rigorously establish the law of of motion of of “leapfrogging vortex rings”, originally conjectured by Helmholtz in 1858. This is joint work with Juan Davila, Monica Musso and Juncheng Wei.

EXISTENCE RESULTS FOR THE HIGHER-ORDER -CURVATURE EQUATION.

Abstract: In this talk, we will obtain some existence results for the -curvature equation of arbitrary -th order, where is an integer, on a compact Riemannian manifold of dimension . This amounts to solving a nonlinear elliptic PDE involving the powers of Laplacian called the GJMS operator. The difficulty in determining the explicit form of this GJMS operator together with a lack of maximum principle complicates the issues of existence. This is a joint work with Jérôme Vètois (McGill University).

Abstract: We construct families of optimal Hardy-weights for a subcritical linear second-order elliptic operator using a one-dimensional reduction. More precisely, we first characterize all optimal Hardy-weights with respect to one-dimensional subcritical Sturm-Liouville operators on , , and then apply this result to obtain families of optimal Hardy inequalities for general linear second-order elliptic operators in higher dimensions. This is a joint work with Idan Versano.

On -ellipticity and connections to solvability of elliptic complex valued PDEs.

Abstract: The notion of an elliptic partial differential equation (PDE) goes back at least to 1908, when it appeared in a paper J. Hadamard. In this talk we present a recently discovered structural condition, called -ellipticity, which generalizes classical ellipticity. It was co-discovered independently by Carbonaro and Dragicevic on one hand, and Pipher and myself on the other, and plays a fundamental role in many seemingly mutually unrelated aspects of the theory of elliptic complex-valued PDE. So far, -ellipticity has proven to be the key condition for: (i) convexity of power functions (Bellman functions) (ii) dimension-free bilinear embeddings, (iii) -contractivity and boundedness of semigroups associated with elliptic operators, (iv) holomorphic functional calculus, (v) multilinear analysis, (vi) regularity theory of elliptic PDE with complex coefficients. During the talk, I will describe my contribution to this development, in particular to (vi).

Abstract: Let be a compact manifold of dimension . Scale-invariant Strichartz estimates of the form have only been proved for a few model cases of , most of which are compact globally symmetric spaces. In this talk, we report that the above estimate holds true on an arbitrary compact globally symmetric space equipped with the canonical Killing metric, for all , where denotes the rank of . As an immediate application, we provide local well-posedness results for nonlinear Schrödinger equations of polynomial nonlinearities of degree on any compact globally symmetric space of large enough rank, in all subcritical spaces. We also discuss bilinear Strichartz estimates on compact globally symmetric spaces, and critical and subcritical local well-posedness results for the cubic nonlinearity.

Abstract: Harnack inequality, if it holds, is a useful tool in analysis and probability theory. In this talk, I will discuss scale invariant elliptic Harnack inequality for general diffusions, or equivalently, for general differential operators on metric measure spaces, and show that it is stable under form-comparable perturbations for strongly local Dirichlet forms on complete locally compact separable metric spaces that satisfy metric doubling property. Based on Joint work with Martin Barlow and Mathav Murugan.

Abstract: I will report on joint work with Samuel Perez-Ayala in which we consider the problem of extremizing eigenvalues of the conformal laplacian in a fixed conformal class. This generalizes the problem of extremizing the eigenvalues of the laplacian on a compact surface. I will explain the connection of this problem to the existence of harmonic maps, and to nodal solutions of the Yamabe problem (first noticed by Ammann-Humbert).

Abstract: This is joint work with David Tewodrose (Bruxelles). I will explain that a metric measure space with Euclidean heat kernel are Euclidean. An almost rigidity result comes then for free, and this can be used to give another proof of Colding’s almost rigidity for complete manifold with non negative Ricci curvature and almost Euclidean growth.

Abstract: In this talk, we will describe various open questions and recent progress on the Fourier restriction problem associated to hypersurfaces with varying or vanishing curvature.

Abstract: Spaces of bounded mean oscillation (BMO) are relatively large function spaces that are often used in place of L^\infinity to do basic Fourier analysis. It is not well-understood how geometric properties of the underlying point space enters into the functional analysis of BMO. I will describe recent work with Galia Dafni and Ryan Gibara, where we take some steps towards geometric inequalities. Specifically, we show that the symmetric decreasing rearrangement in n-dimensions is bounded, but not continuous in BMO. The question of sharp bounds remains open.

Abstract: I present a joint work with M. Ritoré and G. Giovannardi related to an area functional for submanifolds of fixed degree immersed in a graded manifold. The expression of this area functional strictly depends on the degree of the manifold, so that, while computing the first variation, we need to keep fixed its degree. We will show that there are isolated surfaces, for which this type of degree preserving variations do not exist: they can be considered higher dimensional extension of the subriemannian abnormal geodesics.

Abstract: In this talk I present some recent results obtained in collaboration with B. Franchi and P. Pansu about Poincaré and Sobolev inequalities for differential forms in Heisenberg groups (some results are new also for Euclidean spaces). For L^p, p>1, the estimates are consequence of singular integral estimates. In the limiting case L^1, the singular integral estimates are replaced with inequalities which go back to Bourgain-Brezis and Lanzani-Stein in Euclidean spaces, and to Chanillo-Van Schaftingen and Baldi-Franchi-Pansu in Heisenberg groups. Also the case p=Q (Q is the homogeneous dimension of the Heisenberg group ) is considered.

Abstract: We discuss Schrödinger-Proca constructions in the context of closed manifolds leading to the Bopp-Podolsky-Schrödinger-Proca and the Schrödinger-Poisson-Proca systems. The goal is to present an introduction to these equations (how we build them, what do they represent) and then to present the result we got on these systems about the convergence of (BPSP) to (SPP) as the Bopp-Podolsky parameter goes to zero.

Entire spacelike constant curvature in Minkowski space.

Abstract: We prove that, in the Minkowski space, if a spacelike, -convex hypersurface with constant curvature has bounded principal curvatures, then is convex. Moreover, if is not strictly convex, after an rigid motion, splits as a product . We also construct nontrivial examples of strictly convex, spacelike hypersurface with constant curvature and bounded principal curvatures. This is a joint work with Changyu Ren and Zhizhang Wang.

Abstract: Caffarelli-Kohn-Nirenberg inequalities are established for the Grushin vector fields and for Iwasawa groups (i.e., the boundary group of a real rank one noncompact symmetric space). For all but one parameter case, this is done by introducing a generalized Kelvin transform which is shown to be an isometry of certain weighted Sobolev spaces. For the exceptional parameter case, the best constant is found for the Grushin vector fields by introducing Grushin cylindrical coordinates and studying the transformed Euler-Lagrange equation.

Abstract: Mean curvature flow (MCF) is a geometric heat equation where a submanifold evolves to minimize its area. A central problem is to understand the singularities that form and what these imply for the flow. I will talk about joint work with Toby Colding on higher codimension MCF, where the flow becomes a complicated system of equations and much less is known.

Abstract: Motivated by Yau’s conjecture, the study of the measure of nodal sets (Zero level sets) for eigenfunctions is interesting. We investigate the measure of nodal sets for Steklov, Dirichlet and Neumann eigenfunctions in the domain and on the boundary of the domain. For Dirichlet or Neumann eigenfunctions in the analytic domains, we show some sharp upper bounds of nodal sets which touch the boundary. We will also discuss some upper bounds of nodal sets for eigenfunctions of general eigenvalue problems. Furthermore, some sharp doubling inequalities and vanishing order are obtained. Part of the talk is based on joint work with Fanghua Lin.

Abstract: In this talk we discuss Hardy and Hardy-Rellich type inequalities, so important in establishing useful properties for differential operators with singular potentials and their PDEs. We recall some well-known and recent results and present some new extensions. We analyze singular potentials with one or various singularities. The tools of our proofs are mainly based on the method of supersolutions, proper transformations and spherical harmonics decomposition. We also focus on the best constants and the existence/nonexistence of minimizers in the energy space. This presentation is partially supported by CNCS-UEFISCDI Grant No. PN-III-P1-1.1-TE-2016-2233.

Abstract: For over twenty years A. Grigor’yan and the speaker have studied heat kernel estimates on manifolds with a finite number of nice ends. Despite these efforts, question remains. In this talk, after giving an overview of what the problem is and what we know, the main difficulty will be explained and recent progresses involving joint work with Grigor’yan and Ishiwata will be explained. They provide results concerning Poincaré inequality in large central balls on such manifold.

Abstract: The Brezis-Nirenberg problem considers elliptic equations whose nonlinearity is associated with critical Sobolev exponents. In this talk we will discuss a recent progress on higher order Brezis-Nirenberg problem on hyperbolic spaces. The existence of solutions relates closely to the study of higher order sharp Hardy-Sobolev-Maz’ya inequalities, which is due to G. Lu and Q. Yang. On the other hand, we obtain a nonexistence result on star-shaped domains. In addition, with the help of Green’s function estimates, we apply moving plane method to establish the symmetry of positive solutions. This is a joint work with Guozhen Lu and Qiaohua Yang..

Abstract: In 1975, Lieb and Thirring conjectured that the kinetic energy of fermions is not smaller than its Thomas-Fermi (semiclassical) approximation, at least in three or higher dimensions. I will discuss a rigorous lower bound with the sharp semiclassical constant and a gradient error term which is normally of lower order in applications. The proof is based on a microlocal analysis and a variant of the Berezin-Li-Yau inequality. This approach can be extended to derive an improved Lieb-Thirring inequality for interacting systems, where the Gagliardo-Nirenberg constant appears in the strong coupling limit.

Abstract: Optimal constants and optimal functions are known in some functional inequalities. The next question is the stability issue: is the difference of the two terms controlling a distance to the set of optimal functions ? A famous example is provided by Sobolev’s inequalities: in 1991, G. Bianchi and H. Egnell proved that the difference of the two terms is bounded from below by a distance to the manifold of the Aubin-Talenti functions. They argued by contradiction and gave a very elegant although not constructive proof. Since then, estimating the stability constant and giving a constructive proof has been a challenge. This lecture will focus mostly on subcritical inequalities, for which explicit constants can be provided. The main tool is based on entropy methods and nonlinear flows. Proving stability amounts to establish, under some constraints, a version of the entropy – entropy production inequality with an improved constant. In simple cases, for instance on the sphere, rather explicit results have been obtained by the « carré du champ » method introduced by D. Bakry and M. Emery. In the Euclidean space, results based on constructive regularity estimates for the solutions of the nonlinear flow and corresponding to a joint research project with Matteo Bonforte, Bruno Nazaret, and Nikita Simonov will be presented.

Rigidity of local minimizers of the functional.

Abstract: In this talk, I will present a result on the rigidity of local minimizers of the functional among all conformally flat metrics in the Euclidean (n + 1)-ball. We prove the metric is flat up to a conformal transformation in some (noncritical) dimensions. We also prove the analogous result in the critical dimension n + 1 = 4. The main method is Frank-Lieb’s rearrangement-free argument. If minimizers exist, this implies a fully nonlinear sharp Sobolev trace inequality. I will also discuss a nonsharp Sobolev trace inequality. This is joint work with Jeffrey Case.

Abstract: We will discuss a mean curvature type flow with the goal to solve isoperimetric problem. The flow is induced from the variational properties associated to conformal Killing fields. Such flow was first introduced in space forms in a previous joint work with Junfang Li, where we provided a flow approach to the classical isoperimetric inequality in space form. Later, jointly with Junfang Li and Mu-Tao Wang, we considered the similar flow in warped product spaces with general base. Under some natural conditions, the flow preserves the volume of the bounded domain enclosed by a graphical hypersurface, and monotonically decreases the hypersurface area. Furthermore, the regularity and convergence of the flow can be established, thereby the isoperimetric problem in warped product spaces can be solved. The flow serves as an interesting way to achieve the optimal solution to the isoperimetric problem.

Abstract: We are interested in a new family of reverse Hardy–Littlewood–Sobolev inequalities which involve a power law kernel with positive exponent and a Lebesgue exponent <1. We characterize the range of parameters for which the inequality holds and present results about the existence of optimizers. A striking open question is the possibility of concentration of a minimizing sequence. This talk is based on joint work with J. Carrillo, M. Delgadino, J. Dolbeault and F. Hoffmann.

Abstract: I will discuss two recent rigidity results for Allen-Cahn: the first is Half Space Theorem which states that if the nodal set lies above a half space then it must be one-dimensional. The second result is the stability of Cabre-Terra saddle solutions in R^8, R^{10} and R^{12}.

Abstract: Concentration compactness principle for functions in on a n-dimensional domain was introduced by Lions in 1985 with the Moser-Trudinger inequality in mind. We will discuss some further refinements after Cerny-Cianchi-Hencl’s improvement in 2013. These refinements unifiy the approach for n=2 and n>2 cases and work for higher order or fractional order Sobolev spaces as well. They are motivated by and closely related to the recent derivation of Aubin’s Moser-Trudinger inequality for functions with vanishing higher order moments on the standard 2-sphere (one may see math.sjtu.edu.cn/conference/2020p&g/videos/20200707_FengboHang_M1.html for that part).

Abstract: We establish an affine isoperimetric inequality using a symmetrization scheme that involves a total of 2n elaborately chosen Steiner symmetrizations at a time. The necessity of this scheme, as opposed to the usual Steiner symmetrization, will be demonstrated with an example. This is a joint work with Dr. Yiming Zhao.

Abstract: In this talk, we will discuss the Gauss image problem, a problem that reconstructs the shape of a convex body using partial data regarding its Gauss map. In the smooth category, this problem reduces to a Monge-Ampere type equation on the sphere. But, we will use a variational argument that works with generic convex bodies. This is joint work with Károly Böröczky, Erwin Lutwak, Deane Yang, and Gaoyong Zhang.

Abstract: The regularity of stable solutions to semilinear elliptic PDEs has been studied since the 1970’s. In dimensions 10 and higher, there exist singular stable energy solutions. In this talk I will describe a recent work in collaboration with Figalli, Ros-Oton, and Serra, where we prove that stable solutions are smooth up to the optimal dimension 9. This answers to an open problem posed by Brezis in the mid-nineties concerning the regularity of extremal solutions to Gelfand-type problems.

Abstract: We consider three-dimensional CR manifolds, which are modelled on the Heisenberg group. We introduce a natural concept of “mass” and prove its positivity under the condition that the scalar curvature is positive and in relation to their (holomorphic) embeddability properties. We apply this result to the CR Yamabe problem, and we discuss extremality of Sobolev-type quotients, giving some counterexamples for “Rossi spheres”. This is joint work with J.H.Cheng and P.Yang.

Abstract: Embedded symmetry within the Heisenberg group is used to couple geometric insight and analytic calculation to obtain a new sharp Stein-Weiss inequality with mixed homogeneity on the line of duality. SL(2,R) invariance and Riesz potentials define a natural bridge for encoded information that connects distinct geometric structures. Insight for Stein-Weiss integrals is gained from vortex dynamics, embedding on hyperbolic space, and conformal geometry. The intrinsic character of the Heisenberg group makes it the natural playing field on which to explore the laws of symmetry and the interplay between analysis and geometry on a manifold.