Seminars and Colloquia by Series

Cellular Legendrian contact homology for surfaces

Series
Geometry Topology Seminar
Time
Monday, March 6, 2017 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Dan Rrutherford – Ball State University
This is joint work with Mike Sullivan. We consider a Legendrian surface L in R5 or more generally in the 1-jet space of a surface. Such a Legendrian can be conveniently presented via its front projection which is a surface in R3 that is immersed except for certain standard singularities. We associate a differential graded algebra (DGA) to L by starting with a cellular decomposition of the base projection to R2 of L that contains the projection of the singular set of L in its 1-skeleton. A collection of generators is associated to each cell, and the differential is determined in a formulaic manner by the nature of the singular set above the boundary of a cell. Our cellular DGA is equivalent to the Legendrian contact homology DGA of L whose construction was carried out in this setting by Etnyre-Ekholm-Sullivan with the differential defined by counting holomorphic disks in C2 with boundary on the Lagrangian projection of L. Equivalence of our DGA with LCH is established using work of Ekholm on gradient flow trees. Time permitting, we will discuss constructions of augmentations of the cellular DGA from two parameter families of functions.

Dynamical Structures near the Solitons of the Supercritical gKDV Equations

Series
CDSNS Colloquium
Time
Monday, March 6, 2017 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Dr. Jiayin Jin – Georgia Tech
We classify the local dynamics near the solitons of the supercritical gKDV equations. We prove that there exists a co-dim 1 center-stable (unstable) manifold, such that if the initial data is not on the center-stable (unstable) manifold then the corresponding forward(backward) flow will get away from the solitons exponentially fast; There exists a co-dim 2 center manifold, such that if the intial data is not on the center manifold, then the flow will get away from the solitons exponentially fast either in positive time or in negative time. Moreover, we show the orbital stability of the solitons on the center manifold, which also implies the global existence of the solutions on the center manifold and the local uniqueness of the center manifold. Furthermore, applying a theorem of Martel and Merle, we have that the solitons are asymptotically stable on the center manifold in some local sense. This is a joint work with Zhiwu Lin and Chongchun Zeng.

Z-flows in the random environment

Series
Combinatorics Seminar
Time
Friday, March 3, 2017 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Tomasz Łuczak – Adam Mickiewicz University –
In the talk we state, explain, comment, and finally prove a theorem (proved jointly with Yuval Peled) on the size and the structure of certain homology groups of random simplicial complexes. The main purpose of this presentation is to demonstrate that, despite topological setting, the result can be viewed as a statement on Z-flows in certain model of random hypergraphs, which can be shown using elementary algebraic and combinatorial tools.

Poincar\'e Mechanism in Multi-scaled Hamiltonian Systems

Series
Dynamical Systems Working Seminar
Time
Friday, March 3, 2017 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 254
Speaker
Lu Xu – School of Mathematics, Jilin University
My talk is about the quasi-periodic motions in multi-scaled Hamiltonian systems. It consists of four part. At first, I will introduce the results in integrable Hamiltonian systems since what we focus on is nearly-integrable Hamiltonian system. The second part is the definition of nearly-integrable Hamiltonian system and the classical KAM theorem. After then, I will introduce that what is Poincar\'e problem and some interesting results corresponding to this problem. The last part, which is also the main part, I will talk about the definition and the background of nearly-integrable Hamiltonian system, then the persistence of lower dimensional tori on resonant surface, which is our recent result. I will also simply introduce the Technical ingredients of our work.

Structured matrix computations via algebra

Series
Algebra Seminar
Time
Friday, March 3, 2017 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Lek-Heng Lim – University of Chicago
We show that in many instances, at the heart of a problem in numerical computation sits a special 3-tensor, the structure tensor of the problem that uniquely determines its underlying algebraic structure. In matrix computations, a decomposition of the structure tensor into rank-1 terms gives an explicit algorithm for solving the problem, its tensor rank gives the speed of the fastest possible algorithm, and its nuclear norm gives the numerical stability of the stablest algorithm. We will determine the fastest algorithms for the basic operation underlying Krylov subspace methods --- the structured matrix-vector products for sparse, banded, triangular, symmetric, circulant, Toeplitz, Hankel, Toeplitz-plus-Hankel, BTTB matrices --- by analyzing their structure tensors. Our method is a vast generalization of the Cohn--Umans method, allowing for arbitrary bilinear operations in place of matrix-matrix product, and arbitrary algebras in place of group algebras. This talk contains joint work with Ke Ye and joint work Shmuel Friedland.

Nonlocal transport in bounded domains

Series
CDSNS Colloquium
Time
Friday, March 3, 2017 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Diego Del Castillo-Negrete – Oak Ridge National Lab.
The study of nonlocal transport in physically relevant systems requires the formulation of mathematically well-posed and physically meaningful nonlocal models in bounded spatial domains. The main problem faced by nonlocal partial differential equations in general, and fractional diffusion models in particular, resides in the treatment of the boundaries. For example, the naive truncation of the Riemann-Liouville fractional derivative in a bounded domain is in general singular at the boundaries and, as a result, the incorporation of generic, physically meaningful boundary conditions is not feasible. In this presentation we discuss alternatives to address the problem of boundaries in fractional diffusion models. Our main goal is to present models that are both mathematically well posed and physically meaningful. Following the formal construction of the models we present finite-different methods to evaluate the proposed non-local operators in bounded domains.

End point localization in log gamma polymer model

Series
Stochastics Seminar
Time
Thursday, March 2, 2017 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Vu-Lan Nguyen – Harvard University
As a general fact, directed polymers in random environment are localized in the so called strong disorder phase. In this talk, based on a joint with Francis Comets, we will consider the exactly solvable model with log gamma environment,introduced recently by Seppalainen. For the stationary model and the point to line version, the localization can be expressed as the trapping of the endpoint in a potential given by an independent random walk.

Nonlinear Quantitative Photoacoustic Tomography with Two-photon Absorption

Series
Applied and Computational Mathematics Seminar
Time
Thursday, March 2, 2017 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Professor Kui Ren – University of Texas, Austin
Two-photon photoacoustic tomography (TP-PAT) is a non-invasive optical molecular imaging modality that aims at inferring two-photon absorption property of heterogeneous media from photoacoustic measurements. In this work, we analyze an inverse problem in quantitative TP-PAT where we intend to reconstruct optical coefficients in a semilinear elliptic PDE, the mathematical model for the propagation of near infra-red photons in tissue-like optical media, from the internal absorbed energy data. We derive uniqueness and stability results on the reconstructions of single and multiple coefficients, and perform numerical simulations based on synthetic data to validate the theoretical analysis.

Tropical geometry of algebraic curves

Series
School of Mathematics Colloquium
Time
Thursday, March 2, 2017 - 11:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Sam Payne – Yale University
The piecewise linear objects appearing in tropical geometry are shadows, or skeletons, of nonarchimedean analytic spaces, in the sense of Berkovich, and often capture enough essential information about those spaces to resolve interesting questions about classical algebraic varieties. I will give an overview of tropical geometry as it relates to the study of algebraic curves, touching on applications to moduli spaces.

Do Minkowski averages get progressively more convex?

Series
Analysis Seminar
Time
Wednesday, March 1, 2017 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Artem Zvavitch – Kent State University –
For a compact subset $A$ of $R^n$ , let $A(k)$ be the Minkowski sum of $k$ copies of $A$, scaled by $1/k$. It is well known that $A(k)$ approaches the convex hull of $A$ in Hausdorff distance as $k$ goes to infinity. A few years ago, Bobkov, Madiman and Wang conjectured that the volume of $A(k)$ is non-decreasing in $k$, or in other words, that when the volume deficit between the convex hull of $A$ and $A(k)$ goes to $0$, it actually does so monotonically. While this conjecture holds true in dimension $1$, we show that it fails in dimension $12$ or greater. Then we consider whether one can have monotonicity of convergence of $A(k)$ when its non-convexity is measured in alternate ways. Our main positive result is that Schneider’s index of non-convexity of $A(k)$ converges monotonically to $0$ as $k$ increases; even the convergence does not seem to have been known before. We also obtain some results for the Hausdorff distance to the convex hull, along the way clarifying various properties of these notions of non-convexity that may be of independent interest.Joint work with Mokshay Madiman, Matthieu Fradelizi and Arnaud Marsiglietti.

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