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Series: School of Mathematics Colloquium

We consider the following problem. Does there exist an absolute constant C such that for every natural number n, every integer 1 \leq k \leq n, every origin-symmetric convex body L in R^n, and every measure \mu with non-negative even continuous density in R^n, \mu(L) \leq C^k \max_{H \in Gr_{n-k}} \mu(L \cap H}/|L|^{k/n}, where Gr_{n-k} is the Grassmannian of (n-k)-dimensional subspaces of R^n, and |L| stands for volume? This question is an extension to arbitrary measures (in place of volume) and to sections of arbitrary codimension k of the hyperplace conjecture of Bourgain, a major open problem in convex geometry. We show that the above inequality holds for arbitrary origin-symmetric convex bodies, all k and all \mu with C \sim \sqrt{n}, and with an absolute constant C for some special class of bodies, including unconditional bodies, unit balls of subspaces of L_p, and others. We also prove that for every \lambda \in (0,1) there exists a constant C = C(\lambda) so that the above inequality holds for every natural number, every origin-symmetric convex body L in R^n, every measure \mu with continuous density and the codimension of sections k \geq \lambda n. The latter result is new even in the case of volume. The proofs are based on a stability result for generalized intersections bodies and on estimates of the outer volume ratio distance from an arbitrary convex body to the classes of generalized intersection bodies.

Series: School of Mathematics Colloquium

In this talk, I will survey the recent understandings on
the motion of water waves obtained via rigorous
mathematical tools, this includes the evolution of smooth initial
data and some typical singular
behaviors. In particular, I will present our recently results on
gravity water waves with angled crests.

Series: School of Mathematics Colloquium

Decomposition techniques such as atomic, molecular, wavelet and wave-packet expansions provide a multi-scale refinement of Fourier analysis and exploit a rather simple concept: “waves with very different frequencies are almost invisible to each other”. Starting with the classical Calderon-Zygmund and Littlewood-Paley decompositions, many of these useful techniques have been developed around the study of singular integral operators. By breaking an operator or splitting the functions on which it acts into non-interacting almost orthogonal pieces, these tools capture subtle cancelations and quantify properties of an operator in terms of norm estimates in function spaces. This type of analysis has been used to study linear operators with tremendous success. More recently, similar decomposition techniques have been pushed to the analysis of new multilinear operators that arise in the study of (para) product-like operations, commutators, null-forms and other nonlinear functional expressions. In this talk we will present some of our contributions in the study of multilinear singular integrals, function spaces, and the analysis of nanostructure in biological tissues, not all immediately connected topics, yet all centered on some notion of almost orthogonality.

Series: School of Mathematics Colloquium

The closed geodesic problem is a classical topic of dynamical systems,
differential geometry and variational analysis, which can be chased back
at least to Poincar\'e. A famous conjecture claims the existence of
infinitely many distinct closed geodesics on every compact Riemaniann
manifold. But so far this is only proved for the 2-dimentional case. On
the other hand, Riemannian metrics are quadratic reversible Finsler
metrics, and the existence of at least one closed geodesic on every
compact Finsler manifold is well-known because of the famous work of
Lyusternik and Fet in 1951. In 1973 A. Katok constructed a family of
remarkable Finsler metrics on every sphere $S^d$ which possesses
precisely $2[(d+1)/2]$ distinct closed geodesics. In 2004, V. Bangert
and the author proved the existence of at least $2$ distinct closed
geodesics for every Finsler metric on $S^2$, and this multiplicity estimate on $S^2$
is sharp by Katok's example. Since this work, many new results on the
multiplicity and stability of closed geodesics have been established. In
this lecture, I shall give a survey on the study of closed geodesics on
compact Finsler manifolds, including a brief history and results
obtained in the last 10 years. Then I shall try to explain the most
recent results we obtained for the multiplicity and stability of closed
geodesics on compact simply connected Finsler manifolds, sketch the
ideas of their proofs, and then propose some further open problems in
this field.

Series: School of Mathematics Colloquium

Unitary representations of Lie groups appear in many guises in
mathematics: in harmonic analysis (as generalizations of classical
Fourier analysis); in number theory (as spaces of modular and
automorphic forms); in quantum mechanics (as "quantizations" of
classical mechanical systems); and in many other places. They have
been the subject of intense study for decades, but their
classification has only recently emerged. Perhaps
surprisingly, the classification has inspired connections with
interesting geometric objects (equivariant mixed Hodge modules on
flag varieties). These connections have made it possible to extend
the classification scheme to other related settings.
The purpose of this talk is to
explain a little bit about the history
and motivation behind the study of unitary representations and offer
a few hints about the algebraic and geometric ideas which enter into
their study. This is based on joint work with Adams, van Leeuwen,
and Vogan.

Series: School of Mathematics Colloquium

While the fields named in the title seem unrelated, there is a strong
link between them. This amazing connection came to life during a meeting
between Freeman Dyson and Hugh Montgomery at the Institute for Advanced
Study. Random matrices are now known to predict many number theoretical
statistics, such as moments, low-lying zeros and correlations between
zeros. The goal of this talk is to discuss this connection, focusing on
number theory. We will cover both basic facts about the zeta functions
and recent developments in this active area of research.

Series: School of Mathematics Colloquium

One of the simplest and most natural ways of representing geometry and information in three and higher dimensions is using point clouds, such as scanned 3D points for shape modeling and feature vectors viewed as points embedded in high dimensions for general data analysis. Geometric understanding and analysis of point cloud data poses many challenges since they are unstructured, for which a global mesh or parametrization is difficult if not impossible to obtain in practice. Moreover, the embedding is highly non-unique due to rigid and non-rigid transformations.
In this talk, I will present some of our recent work on geometric understanding and analysis of point cloud data. I will first discuss a multi-scale method for non-rigid point cloud registration based on the Laplace-Beltrami eigenmap and optimal transport. The registration is defined in distribution sense which provides both generality and flexibility. If time permits I will also discuss solving geometric partial differential equations directly on point clouds and show how it can be used to “connect the dots” to extract intrinsic geometric information for the underlying manifold.

Series: School of Mathematics Colloquium

From theoretical to applied, we present curiosity driven research which
goes beyond classical dynamical systems theory and (i) extend the notion
of chaos to actions of topological semigroups, (ii) model how the human
bone renews, (iii) study transient dynamics as it occurs e.g. in
oceanography, (iv) understand how to protect houses from hurricane
damage. The talk introduces concepts from topological dynamics,
mathematical biology, entropy theory and mechanics.

Series: School of Mathematics Colloquium

The Caffarelli-Kohn-Nirenberg inequalities form a two parameter family of inequalities that interpolate between Sobolev's inequality and Hardy's inequality. The functional whose minimization yields the sharp constant is invariant under rotations. It has been known for some time that there is a region in parameter space where the optimizers for the sharp constant are {\it not} radial. In this talk I explain this and related problems andindicate a proof that, in the remaining parameter region, the optimizers are in fact radial. The novelty is the use of a flow that decreases the functional unless the function is a radial optimizer. This is joint work with Jean Dolbeault and Maria Esteban.

Series: School of Mathematics Colloquium

There have been many recent advances for analyzing the complex
deterministic behavior of systems with discontinuous dynamics.
With the identification of new types of nonlinear phenomena
exploding in this realm, one gets the feeling that almost anything
can happen. There are many open questions about noise-driven and
noise-sensitive phenomena in the non-smooth context, including the
observation that noise can facilitate or select "regular" dynamics,
thus clarifying the picture within the seemingly endless sea of
possibilities. Familiar concepts from smooth systems such as escapes,
resonances, and bifurcations appear in unexpected forms, and we gain
intuition from seemingly unrelated canonical models of biophysics,
mechanics, finance, and climate dynamics. The appropriate strategy
is often not immediately obvious from the area of application or model
type, requiring an integration of multiple scales techniques,
probabilistic models, and nonlinear methods.