Seminars and Colloquia by Series

Thursday, March 17, 2016 - 11:00 , Location: Skiles 006 , Sijue Wu , University of Michigan , sijue@umich.edu , Organizer: Michael Damron
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.
Thursday, March 10, 2016 - 16:05 , Location: Skiles 005 , Rodolfo Torres , University of Kansas , Organizer: Michael Lacey
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.
Tuesday, March 8, 2016 - 11:00 , Location: Skiles 006 , Prof. Dr. Yiming Long , Nankai University , Organizer: Molei Tao
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.
Thursday, March 3, 2016 - 16:05 , Location: Skiles 005 , Peter Trapa , University of Utah , Organizer:
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.
Thursday, March 3, 2016 - 11:00 , Location: Skiles 006 , Daniel Fiorilli , University of Ottawa , daniel.fiorilli@uottawa.ca , Organizer: Michael Damron
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.
Monday, February 29, 2016 - 16:05 , Location: Skiles 005 , Hongkai Zhao , University of California, Irvine , Organizer: Haomin Zhou
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.
Thursday, February 25, 2016 - 11:05 , Location: Skiles 006 , Stefan Siegmund , TU Dresden , stefan.siegmund@tu-dresden.de , Organizer: Michael Damron
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.
Monday, February 22, 2016 - 16:00 , Location: Skiles 005 , Michael Loss , School of Mathematics, Georgia Tech , Organizer:
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.
Thursday, February 11, 2016 - 16:05 , Location: Skiles 005 , Rachel Kuske , University of British Columbia , Organizer: Christine Heitsch
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.
Tuesday, February 9, 2016 - 15:30 , Location: Skiles 005 , Patrick Gerard , Université Paris-Sud , patrick.gerard@math.u-psud.fr , Organizer: Michael Damron
In the world of Hamiltonian partial differential equations, complete integrability is often associated to rare and peaceful dynamics, while wave turbulence rather refers to more chaotic dynamics. In this talk I will first try to give an idea of these different notions. Then I will  discuss the example of the cubic Szegö equation, a nonlinear wave toy model which surprisingly displays both properties. The key is a Lax pair structure involving Hankel operators from classical analysis, leading to the inversion of large ill-conditioned matrices. .

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