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Series: PDE Seminar

We will introduce a recently found channel of energy inequality for outgoing waves, which has been useful for semi-linear wave equations at energy criticality. Then we will explain an application of this channel of energy argument to the energy critical wave maps into the sphere. The main issue is to eliminate the so-called "null concentration of energy". We will explain why this is an important issue in the wave maps. Combining the absence of null concentration with suitable coercive property of energy near traveling waves, we show a universality property for the blow up of wave maps with energy that are just above the co-rotational wave maps. Difficulties with extending to arbitrarily large wave maps will also be discussed. This is joint work with Duyckaerts, Kenig and Merle.

Series: Algebra Seminar

The Brill-Noether varieties of a curve C parametrize embeddings of C of prescribed degree into a projective space of prescribed dimension. When C is general in moduli, these varieties are well understood: they are smooth, irreducible, and have the “expected” dimension. As one ventures deeper into the moduli space of curves, past the locus of general curves, these varieties exhibit intricate, even pathological, behaviour: they can be highly singular and their dimensions are unknown. A first measure of the failure of a curve to be general is its gonality. I will present a generalization of the Brill—Noether theorem, which determines the dimensions of the Brill—Noether varieties on a general curve of fixed gonality, i.e. “general inside a chosen special locus". The proof blends a study of Berkovich skeletons of maps from curves to toric varieties with tropical linear series theory. The deformation theory of logarithmic stable maps acts as the bridge between these ideas. This is joint work with Dave Jensen.

Series: Geometry Topology Seminar

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.

Series: CDSNS Colloquium

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.

Series: Combinatorics Seminar

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.

Friday, March 3, 2017 - 15:05 ,
Location: Skiles 254 ,
Lu Xu ,
School of Mathematics, Jilin University ,
Organizer: Jiaqi Yang

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.

Series: CDSNS Colloquium

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.

Series: Algebra Seminar

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.

Series: Stochastics Seminar

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.

Thursday, March 2, 2017 - 14:00 ,
Location: Skiles 006 ,
Professor Kui Ren ,
University of Texas, Austin ,
Organizer: Sung Ha Kang

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.