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Quick access to titles & abstracts by sessions:

• Algebraic and complex geometry
• Classical harmonic analysis and combinatorics
• Differential geometry
• Life sciences – mathematical modelling and analysis
• Number theory
• PDE: Dissipation and fluid mechanics
• PDE: Inviscid fluid mechanics and general relativity
• Probability theory
• Recent trends in geometric analysis
• Symplectic geometry and dynamical systems

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Algebraic & complex geometry

The setting for this talk is the study of 7-dimensional manifolds with torsion free $G_{2}$-structures. While these are not complex manifolds there are many interactions with complex geometry and the talk will survey some of these. Topics that will be discussed include “$G_{2}$-cobordisms” between Calabi-Yau 3-folds; Kovalev’s twisted connected sum construction which involves of Fano or semi-Fano 3-folds and the adiabatic limits of $G_{2}$-geometry on manifolds with $K3$-fibrations.

Calabi functional is defined as the $L^2$ norm of the scalar curvature and conjecturally its lower bound is achieved by a sequence of the normalized Donaldson-Futaki invariants. It is naturally related to the limit behavior of the Calabi flow. For the Fano manifolds the problem can be reformulated in terms of the Ricci curvature potential. We prove in this situation that the lower bound of the Ricci-Calabi functional is achieved by a sequence of the normalized D-invariants, taking the multiplier ideal sheaves of the appropriate geometric flow. The same argument can be applied to the Dervan-Székelyhidi’s lower bound of the entropy functional.

Due to Kulikov theorem and its applications, one has a good understanding of the degenerations of K3 surfaces and consequently some understanding of compactifications for moduli of K3 surfaces. In this talk, I will discuss some aspects of higher dimensional analogues of these results. Most of the results will concern Hyperkaehler manifolds, where the picture is quite similar to that for K3 surfaces. I will close with some ideas on how to deal with the more subtle Calabi-Yau case.

Certain locally Hermitian symmetric spaces parameterize complex algebraic varieties, such as polarized abelian varieties and K3 surfaces through periods. In this talk, we will see that one of Satake compactifications of locally symmetric spaces, which is different from the Baily-Borel compactification, parameterizes limits of canonical (Ricci-flat) metrics on abelian varieties or K3 surfaces. This in particular involves parameterization of “tropical” varieties by locally symmetric spaces and confirms a conjecture of Kontsevich-Soibelman in the case of K3 surfaces.

Deligne-Illusie proved that the Frobenius pushforward of the de Rham complex is decomposable in the derived category under suitable conditions. It is called the de Rham decomposition theorem, that is the key for an algebraic proof of the E_1 degeneration of the Hodge to de Rham spectral sequence over the field of complex numbers. In their nonabelian Hodge theory in positive characteristic, Ogus-Vologodsky established the de Rham decomposition theorem with coefficients, that generalizes Deligne-Illusie’s result in a far reaching way. In my talk, I shall report a further generalization of Ogus-Vologodsky’s decomposition theorem, that takes care of an intersection condition at infinity. This work was motivated by Gabber’s purity theorem for perverse sheaves, and Zucker, Cattani-Kaplan-Schmid and Kashiwara-Kawai’s works on intersection cohomologies of variations of Hodge structure. This is a joint work with Zebao Zhang.

In this presentation, we discuss some recent progress on the geometry of compact manifolds with RC-positive tangent bundles, including an affirmative answer to an open problem of S.T. Yau on rational connectedness of compact Kahler manifolds with positive holomorphic sectional curvature, and new Liouville type theorems for holomorphic maps and harmonic maps. Several open problems related to the theory of RC-positivity will also be discussed.

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Classical harmonic analysis & combinatorics

Multilinear functionals, and inequalities governing them, arise
in various contexts in harmonic analysis (in connection with Fourier restriction), in partial differential equations (nonlinear interactions) and in additive combinatorics (existence of certain patterns in sets of appropriately bounded density). This talk will focus on an inequality that quantifies a weak convergence theorem of Joly, Métivier, and Rauch (1995) concerning threefold products, and on related inequalities for trilinear expressions involving highly oscillatory factors. Sublevel set inequalities, which quantify the impossibility of exactly solving certain systems of linear functional equations (the frustration of the title), are a central element of the analysis.

We give an overview of some recent results on point configurations in large subsets of the Euclidean space and discuss their connection with multilinear singular integrals.

We consider the question of how large a union of affine subspaces must be depending on the family of affine subspaces constituting the union. In the famous Kakeya problem one considers lines in every direction. Here the position of the lines or higher-dimensional affine subspaces is more general, and accordingly the expected dimension bound is different. We prove that the union of any s-dimensional family of $k$-dimensional affine subspaces is at least $k + s/(k+1)$ -dimensional, and is exactly $k + s$-dimensional if $s$ is at most 1. Partially based on joint work with Tamás Keleti and András Máthé.

A measure is called a frame-spectral measures if we can find a countable set of exponential functions $\{e^{2\pi i \lambda x}:\lambda\in \Lambda\}$ such that it forms a frame in $L^2(\mu)$. i.e. $$ \|f\|_{\mu}^2 \asymp \sum_{\lambda\in \Lambda} |\langle f,e_{\lambda}\rangle_{\mu}|^2 $$ Frames are natural generalization of orthnormal basis. It is known that some singular measures also admit a Fourier frames. However, it is still largely unknown which singular measures are frame-spectral. In this talk, we will explore some of the recent progresses about this problem.

There are two different looking proofs of Vinogradov’s Mean Value Theorem. One was Bourgain-Demeter-Guth’s proof via $\ell^2$ decoupling of the moment curve using harmonic analysis methods and another was Wooley’s proof via nested efficient congruencing using number theoretic methods. We will illustrate the main ideas of how an efficient congruencing proof can be translated into a decoupling proof in the case of $\ell^2$ decoupling for the parabola. We will also mention how to use these ideas to give a new proof of $\ell^2$ decoupling for the moment curve. This talk is based off joint work with Shaoming Guo, Po-Lam Yung and Pavel Zorin-Kranich.

We are going to discuss some incidence problems between points and tubes. Then we discuss how they are related to problems in Fourier analysis. This includes joint work with Larry Guth, Noam Solomon, and with Ciprian Demeter, L. Guth.

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Differential geometry

In this talk I will survey recent work with Kleiner in which we verify two topological conjectures using Ricci flow. First, we classify the diffeomorphism group of every 3-dimensional spherical space form up to homotopy. This proves the Generalized Smale Conjecture and gives an alternative proof of the Smale Conjecture, which was originally due to Hatcher. Second, we show that the space of metrics with positive scalar curvature on every 3-manifold is either contractible or empty. This completes work initiated by Marques.

Our proof is based on a new uniqueness theorem for singular Ricci flows, which I have previously obtained with Kleiner. Singular Ricci flows were inspired by Perelman’s proof of the Poincaré and Geometrization Conjectures, which relied on a flow in which singularities were removed by a certain surgery construction. Since this surgery construction depended on various auxiliary parameters, the resulting flow was not uniquely determined by its initial data. Perelman therefore conjectured that there must be a canonical, weak Ricci flow that automatically “flows through its singularities” at an infinitesimal scale. Our work on the uniqueness of singular Ricci flows gives an affirmative answer to Perelman’s conjecture and allows the study of continuous families of singular Ricci flows leading to the topological applications mentioned above. More details and historical background will be given in the talk.

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When we choose a metric on a manifold we determine the spectrum of the Laplace operator. Thus an eigenvalue may be considered as a functional on the space of metrics. For example the first eigenvalue would be the fundamental vibrational frequency. In some cases the normalized eigenvalues are bounded independent of the metric. In such cases it makes sense to attempt to find critical points in the space of metrics. In this talk we will discuss some results on higher eigenvalue optimization for surfaces.

I will talk on some fusion of a topic on Singer conjecture in L^2 harmonic theory with Seiberg-Witten Bauer-Furuta theory on non-compact complete Riemannian 4-manifolds. We explain their analytic settings, certain results and questions.

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End-point maximal $L^1$-regularity for the parabolic initial boundary value problem is considered. For a parabolic boundary value problem with inhomogeneous Dirichlet and Neumann data, maximal $L^1$-regularity for the initial boundary value problem is established in time end-point case upon the Besov space $\dot B_{p,1}^0(\mathbb R^n_+)$ with $1 < p < \infty$.

We utilize a method of harmonic analysis, in particular, the almost orthogonal properties between the boundary potentials of the Dirichlet and the Neumann boundary data and the Littlewood-Paley dyadic decomposition of unity in the Besov and the Lizorkin-Triebel spaces.

This is a joint work with Prof. Takayoshi Ogawa (Tohoku University).

The equations of affine motion for a 2D incompressible ideal fluid surrounded by vacuum reduce to a globally solvable Hamiltonian system of ordinary differential equations for the deformation gradient constrained to SL(2,R). The evolution of the fluid domain is described by a family ellipses of fixed area. We shall provide a complete description of the dynamic behavior of these domains for perfect fluids and for magnetically conducting fluids. For perfect fluids, the displacement generically becomes unbounded as time tends to infinity, and for magnetically conducting fluids, solutions remain bounded and are generically quasi-periodic.

In this paper we develop a new way to study the global existence and uniqueness for the Navier-Stokes equation (NS) and consider the initial data in a class of modulation spaces $E^s_{p,q}$ with exponentially decaying weights $(s<0, \ 1<p,q<\infty)$ for which the norms are defined by $$ \|f\|_{E^s_{p,q}} = \left(\sum_{k\in \mathbb{Z}^d} 2^{s|k|q}\|\mathscr{F}^{-1} \chi_{k+[0,1]^d}\mathscr{F} f\|^q_p \right)^{1/q}. $$ The space $E^s_{p,q}$ is a rather rough function space and cannot be treated as a subspace of tempered distributions. For example, we have the embedding $H^{\sigma}\subset E^s_{2,1}$ for any $\sigma<0$ and $s<0$. It is known that $H^\sigma$ ($\sigma<d/2-1$) is a super-critical space of NS, it follows that $ E^s_{2,1}$ ($s<0$) is also super-critical for NS.

We show that NS has a unique global mild solution if the initial data belong to $E^s_{2,1}$ ($s<0$) and their Fourier transforms are supported in $ \mathbb{R}^d_I:= \{\xi\in \mathbb{R}^d: \ \xi_i \geq 0, \, i=1,…,d\}$. Similar results hold for the initial data in $E^s_{r,1}$ with $2< r \leq d$. Our results imply that NS has a unique global solution if the initial value $u_0$ is in $L^2$ with ${\rm supp} \, \widehat{u}_0 \, \subset \mathbb{R}^d_I$. This is a joint work with Professors H. G. Feichtinger, K. Gr\”ochenig and Dr. Kuijie Li.

In this talk we give survey what is currently known for Chen’s flow, and discuss some very recent results. Chen’s flow is the biharmonic heat flow for immersions, where the velocity is given by the rough Laplacian of the mean curvature vector. This operator is known as Chen’s biharmonic operator and the solutions to the elliptic problem are called biharmonic submanifolds. The flow itself is very similar to the mean curvature flow (this is essentially the content of Chen’s conjecture), however proving this requires quite different strategies compared to the mean curvature flow. We focus on results available in low dimensions – curves, surfaces, and 4-manifolds. We provide characterisations of finite-time singularities and global analysis. The case of curves is particularly challenging. Here we identify a new shrinker (the Lemniscate of Bernoulli) and use some new observations to push through the analysis. Some numerics is also presented. The work reported on in the talk is in collaboration with Yann Bernard, Matthew Cooper, and Valentina-Mira Wheeler.

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Partial Differential Equations: Inviscid Fluid Mechanics and General Relativity

The gravitational Euler-Poisson system is a classical fluid model describing the motion of self-gravitating gaseous Newton stars. We discuss some recent results on expanding, collapsing and rotating star solutions of the Euler-Poisson system.

In general relativity, the Kerr de Sitter family of solutions to Einstein’s equations with positive cosmological constant are a model of a black hole in the expanding universe. In this talk, I will focus on the stability problem for the expanding region of the spacetime, which can be formulated as a characteristic initial value problem to the future of the cosmological horizons of the black hole. Unlike in the stability of Kerr or Kerr de Sitter black hole exteriors, the solution in the cosmological region does not globally converge to an explicit family of solutions, but displays genuine asymptotic degrees of freedom. I will describe my work on the decay of the conformal Weyl curvature in this setting, and discuss the global construction of optical functions in de Sitter, which are relevant for my approach to this problem in double null gauge.

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The Alfven waves are fundamental wave phenomena in magnetized plasmas and the dynamics of Alfven waves are governed by a system of nonlinear partial differential equations called the MHD system. In the talk, we will focus on the rigidity aspects of the scattering problem for the MHD equations: We prove that the Alfven waves must vanish if their scattering fields vanish at infinities. The proof is based on a careful study of the null structure and a family of weighted energy estimates.

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Probability theory

We discuss the spectrum of high dimensional non-normal matrices under small noisy perturbations. That spectrum can be extremely unstable, as the maximal nilpotent matrix $J_N$ with $J_N(i,j)=1$ iff $j=i+1$ demonstrates. Numerical analysts studied worst case perturbations, using the notion of pseudo-spectrum. Our focus is on finding the locus of most eigenvalues (limits of density of states), as well as studying stray eigenvalues (“outliers”), in the case where the unperturbed matrix is either Toeplitz or twisted Toeplitz. I will describe the background, show some fun and intriguing simulations, and present some theorems and work in progress concerning eigenvectors. No background will be assumed. The talk is based on joint works with Anirban Basak, Elliot Paquette, and Martin Vogel.

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The confinement of quarks is one of the enduring mysteries of modern physics. I will present a rigorous result that shows that if a pure lattice gauge theory at some given coupling strength has exponential decay of correlations under arbitrary boundary conditions, and the gauge group is a compact connected matrix Lie group with a nontrivial center, then the theory is confining. This gives mathematical justification for a longstanding belief in physics about the mechanism behind confinement, which roughly says that confinement is the result of strong coupling behavior plus center symmetry. The proof is almost entirely based in probability theory, making extensive use of the idea of coupling probability measures.

Consider a system of N stochastic differential equations interacting through an N-dimensional matrix J of independent random entries (starting at an initial state whose law is independent of J). We show that the trajectories of a large class of observables which are averaged over the N coordinates of the solution, are universal. That is, for a fixed time interval the limit of such observables as N grows, essentially depends only on the first two moments of the marginal distributions of entries of J.

Concrete settings for which such universality holds include aging in the spherical Sherrington-Kirkpatrick spin-glass and Langevin dynamics for a certain collection of Hopfield networks.

This talk is based on joint works with Reza Gheissari, and with Eyal Lubetzky and Ofer Zeitouni.

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A well known result implies that the rescaled maximal height of a system of N non-intersecting Brownian bridges starting and ending at the origin converges, as N goes to infinity, to the Tracy-Widom GOE random variable from random matrix theory. In this talk I will focus on the same question in case where the top m paths start and end at arbitrary locations. I will present several related results about the distribution of the limiting maximal height for this system, which provides a deformation of the Tracy-Widom GOE distribution: it can be expressed through a Fredholm determinant formula and in terms of Painlevé transcendents; it is connected with the fluctuations of models in the KPZ universality class with a particular initial condition; and it is connected with two PDEs, the KdV equation and an equation derived by Bloemendal and Virag for spiked random matrices. Based on joint work with Karl Liechty and Gia Bao Nguyen.

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Sometimes, it is possible to represent a complicated polytope as a projection of a much simpler polytope. To quantify this phenomenon, the extension complexity of a polytope P is defined to be the minimum number of facets in a (possibly higher-dimensional) polytope from which P can be obtained as a (linear) projection. In this talk, we discuss some results on the extension complexity of random polytopes. For a fixed dimension d, we consider random d-dimensional polytopes obtained as the convex hull of independent random points either in the unit ball ball or on the unit sphere. In both cases, we prove that the extension complexity is typically on the order of the square root of number of vertices of the polytope. Joint work with Matthew Kwan and Yufei Zhao.

In this talk, we discuss interacting particles systems exhibiting a phenomenon known as the condensation of particles. For these systems, particles tend to be condensed at a site because of either sticky or attracting interacting mechanism. A fundamental question for these systems is to describe the behavior of the movement of the condensed site as a suitable scaling limit. We introduce recent results regarding this problem for the zero-range process and the inclusion process. This talk is based on joint works with S. Kim, C. Landim and D. Marcondes.

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Recent trends in geometric analysis

In their celebrated work, B. Andrews and J. Clutterbuck proved the fundamental gap conjecture that the difference of the first two eigenvalues of the Laplacian with Dirichlet boundary condition on a convex domain with diameter D in Euclidean space is greater than or equal to $3\pi^2/D^2$. In several joint works with X. Dai, Z. He, S. Seto, L. Wang (in various subsets), the estimate is generalized, showing the same lower bound holds for convex domains in the unit sphere. In sharp contrast, in recent joint work with T. Bourni, J. Clutterbuck, A. Stancu, X. Nguyen and V. Wheeler, we prove that the product of the fundamental gap with the square of the diameter can be arbitrarily small for convex domains of any diameter in the hyperbolic spaces.

A celebrated result of Boutet de Monvel is that a compact strictly pseudoconvex CR manifold $M$ of dimension $2n+1$ is embeddable as a CR submanifold in $\mathbb{C}^N$ , for some (potentially large) $N$, provided $n\geq 2$. The situation for three-dimensional $M$ ($n=1$) is more subtle: ”Most” such, even real-analytic ones, are not embeddable in this way. Much work has been done over the years to characterize and describe the space of embeddable structures. In this talk, we shall consider the embeddability of families of deformations of a given embedded CR $3$-manifold, and the structure of the space of embeddable CR structures on $S^3$. We discuss a modified version of the Cheng-Lee slice theorem in which the embeddable deformations in the slice can be explicitly characterized (in terms of spherical harmonics). We also introduce a canonical family of embeddable deformations and corresponding embeddings starting with any infinitesimally embeddable deformation of the unit sphere in $\mathbb{C}^2$. The talk is based on joint work with Sean Curry.

It is well-known that the existence of Hermitian-Einstein metrics on holomorphic bundles is intimately tied to the notion of stability. In this talk I will discuss how this correspondence extends to the setting of transverse holomorphic bundles on taut Riemannian foliations. I will further elucidate the relation to higher dimensional instantons on Sasakian manifold and mention some applications.

We will introduce the flat structures on minimal surfaces introduced by Chern and Ricci, respectively.

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The correspondence between Poincaré-Einstein spaces and conformal geometry of the boundaries at infinity is actively pursued. Our subject is its lesser-known analog, and yet also classical because it generalizes the study of invariant metrics on bounded strictly pseudoconvex domains. I will discuss the existence matter and construction of CR invariants through asymptotically complex hyperbolic Einstein metrics.

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The network flow is a system of parabolic differential equations that describes the motion of a family of curves in which each of them evolves under curve-shortening flow. This problem arises naturally in physical phenomena and its solutions present a rich variety of behaviors.

The goal of this talk is to describe some properties of this geometric flow and to discuss an alternative proof of short-time existence for non-regular initial conditions. The methods of our proof are based on techniques of geometric microlocal analysis that have been used to understand parabolic problems on spaces with conic singularities. This is joint work with Jorge Lira, Rafe Mazzeo, and Alessandra Pluda.

A famous theorem of Lichnerowicz states that if a closed spin manifold carries a Riemannian metric of positive scalar curvature, then the A-hat genus of the manifold vanishes. We will describe various generalizations of this result, as well as some other classical results concerning positive scalar curvature, to the case of foliations. A typical example is Connes’ theorem which states that if the A-hat genus of a compact foliated manifold with spin leaves does not vanish, then there is no metric with positive scalar curvature along the leaves.

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Symplectic geometry & dynamical systems

Weinstein symplectic manifolds is one of the basic objects in symplectic topology, similar to Stein complex manifolds in the high-dimensional complex analysis. The arborealization program initiated by David Nadler aims to describe Weinstein manifolds as cotangent bundles of complexes, called arboreal spaces, which are more general than smooth manifolds, and yet have simple standard local chart description. This allows to state symplectic topological questions about Weinstein manifolds as problems in differential topology of arboreal spaces. In the talk I’ll describe the program and its current status. This is a joint work with Daniel Álvarez-Gavela and David Nadler.

In the 60s and 70s, there was a flurry of activity concerning the question of whether or not various subgroups of homeomorphism groups of manifolds are simple, with beautiful contributions by Fathi, Kirby, Mather, Thurston, and many others. A funnily stubborn case that remained open was the case of area-preserving homeomorphisms of surfaces. For example, for balls of dimension at least 3, the relevant group was shown to be simple by work of Fathi in 1980; but, the answer in the two-dimensional case, asked in the 70s, was not known. I will explain recent joint work proving that the group of compactly supported area preserving homeomorphisms of the two-disc is in fact not a simple group; this answers the ”Simplicity Conjecture” in the affirmative. Our proof uses new spectral invariants, defined via periodic Floer homology, that I will introduce: these recover the Calabi invariant of monotone twists.

Convex surface theory and bypasses are extremely powerful tools for analyzing contact 3-manifolds. In particular they have been successfully applied to many classification problems. After briefly reviewing convex surface theory in dimension three, we explain how to generalize many of their properties to higher dimensions. This is joint work with Yang Huang.

Families of moduli spaces in symplectic Gromov-Witten theory and gauge theory are often manifolds that have “thin” compactifications, in the sense that the boundary of the generic fiber has codimension at least two. In this talk we discuss a notion of a relative fundamental class for such thinly compactified families. It associates to each fiber, regardless whether it is regular or not, an element in its Cech homology in a way that is consistent along paths. The invariants defined by relative fundamental classes agree with those defined by pseudo-cycles, and the relative fundamental class is equal to the virtual fundamental class defined by Pardon via implicit atlases in all cases when both are defined. We give some examples of this construction, discuss some of its properties, and its benefits. This talk is based on joint work with Tom Parker.