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Wednesday, February 28, 2018 - 14:00 ,
Location: Skiles 006 ,
Hyun Ki Min ,
GaTech ,
Organizer: Anubhav Mukherjee

I will introduce the notion of satellite knots and show that a knot in a 3-sphere is either a torus knot, a satellite knot or a hyperbolic knot.

Series: Analysis Seminar

Joint with Guth and Li, recently we showed that the solution to the free Schroedinger equation converges to its initial data almost everywhere, provided that the initial data is in the Sobolev space H^s(R^2) with s>1/3. This is sharp up to the endpoint, due to a counterexample by Bourgain. This pointwise convergence problem can be approached by estimates of Schroedinger maximal functions, which have some similar flavor as the Fourier restriction estimates. In this talk, I'll first show how to reduce the original problem in three dimensions to an essentially two dimensional one, via polynomial partitioning method. Then we'll see that the reduced problem asks how to control the size of the solution on a sparse and spread-out set, and it can be solved by refined Strichartz estimates derived from l^2 decoupling theorem and induction on scales.

Series: PDE Seminar

Black holes are predicted by Einstein's theory of general relativity, and now we have ample observational evidence for their existence. However theoretically there are many unanswered questions about how black holes come into being. In this talk, with tools from hyperbolic PDE, quasilinear elliptic equations, geometric analysis and dynamical systems, we will prove that, through a nonlinear focusing effect, initially low-amplitude and diffused gravitational waves can give birth to a black hole region in our universe. This result extends the 1965 Penrose’s singularity theorem and it also proves a conjecture of Ashtekar on black-hole thermodynamics. Open problems and new directions will also be discussed.

Series: ACO Colloquium

Many hard problems of combinatorial counting can be encoded as problems
of computing an appropriate partition function. Formally speaking, such a
partition function is just a multivariate polynomial with great many
monomials enumerating combinatorial structures of interest. For example,
the permanent of an nxn matrix is a polynomial of degree n in n^2
variables with n! monomials enumerating perfect matchings in the
complete bipartite graph on n+n vertices. Typically, we are interested
to compute the value of such a polynomial at a real point; it turns out
that to do it efficiently, it is very helpful to understand the behavior
of complex zeros of the polynomial. This approach goes back to the
Lee-Yang theory of the critical temperature and phase transition in
statistical physics, but it is not identical to it: thinking of the
phase transition from the algorithmic point of view allows us greater
flexibility: roughly speaking, for computational purposes we can freely
operate with “complex temperatures”.
I plan to illustrate this approach on the problems of computing the
permanent and its versions for non-bipartite graphs (hafnian) and
hypergraphs, as well as for computing the graph homomorphism partition
function and its versions (partition functions with multiplicities and
tensor networks) that are responsible for a variety of problems on
graphs involving colorings, independent sets, Hamiltonian cycles, etc. (This is the first (overview) lecture; two more will follow up on Thursday 1:30pm, Friday 3pm of the week. These two lectures are each 80 minutes' long.)

Monday, February 26, 2018 - 14:00 ,
Location: Skiles 005 ,
Prof. Hyenkyun Woo ,
Korea University of Technology and Education ,
Organizer: Sung Ha Kang

Bio: Hyenkyun Woo is an assistant professor at KOREATECH (Korea University of Technology and Education). He got a Ph.D at Yonsei university. and was a post-doc at Georgia Tech and Korea Institute of Advanced Study and others.

In machine learning and signal processing, the beta-divergence is well known as a similarity measure between two positive objects. However, it is unclear whether or not the distance-like structure of beta-divergence is preserved, if we extend the domain of the beta-divergence to the negative region. In this article, we study the domain of the beta-divergence and its connection to the Bregman-divergence associated with the convex function of Legendre type. In fact, we show that the domain of beta-divergence (and the corresponding Bregman-divergence) include negative region under the mild condition on the beta value. Additionally, through the relation between the beta-divergence and the Bregman-divergence, we can reformulate various variational models appearing in image processing problems into a unified framework, namely the Bregman variational model. This model has a strong advantage compared to the beta-divergence-based model due to the dual structure of the Bregman-divergence. As an example, we demonstrate how we can build up a convex reformulated variational model with a negative domain for the classic nonconvex problem, which usually appears in synthetic aperture radar image processing problems.

Series: CDSNS Colloquium

The so-called Hopf-zero singularity consists in a vector field in $\mathbf{R}^3$ having the origin as a critical point, with a zero eigenvalue and a pair of conjugate purely imaginary eigenvalues. Depending of the sign in the second order Taylor coefficients of the singularity, the dynamics of its unfoldings is not completely understood. If one considers conservative (i.e. one-parameter) unfoldings of such singularity, one can see that the truncation of the normal form at any order possesses two saddle-focus critical points with a one- and a two-dimensional heteroclinic connection. The same happens for non-conservative (i.e. two-parameter) unfoldings when the parameters lie in a certain curve (see for instance [GH]).However, when one considers the whole vector field, one expects these heteroclinic connections to be destroyed. This fact can lead to the birth of a homoclinic connection to one of the critical points, producing thus a Shilnikov bifurcation. For the case of $\mathcal{C}^\infty$ unfoldings, this has been proved before (see [BV]), but for analytic unfoldings it is still an open problem.Our study concerns the splittings of the one and two-dimensional heteroclinic connections (see [BCS] for the one-dimensional case). Of course, these cannot be detected in the truncation of the normal form at any order, and hence they are expected to be exponentially small with respect to one of the perturbation parameters. In [DIKS] it has been seen that a complete understanding of how the heteroclinic connections are broken is the last step to prove the existence of Shilnikov bifurcations for analytic unfoldings of the Hopf-zero singularity. Our results [BCSa, BCSb] and [DIKS] give the existence of Shilnikov bifurcations for analytic unfoldings. [GH] Guckenheimer, J. and Holmes, P., Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Springer-Verlag, New York (1983), 376--396. [BV] Broer, H. W. and Vegter, G., Subordinate Sil'nikov bifurcations near some singularities of vector fields having low codimension. Ergodic Theory Dynam. Systems, 4 (1984), 509--525. [BSC] Baldoma;, I., Castejon, O. and Seara, T. M., Exponentially small heteroclinic breakdown in the generic Hopf-zero singularity. Journal of Dynamics and Differential Equations, 25(2) (2013), 335--392. [DIKS] Dumortier, F., Ibanez, S., Kokubu, H. and Simo, C., About the unfolding of a Hopf-zero singularity. Discrete Contin. Dyn. Syst., 33(10) (2013), 4435--4471. [BSCa] Baldoma, I., Castejon, O. and Seara, T. M., Breakdown of a 2D heteroclinic connection in the Hopf-zero singularity (I). Preprint: https://arxiv.org/abs/1608.01115 [BSCb] Baldoma, I., Castejon, O. and Seara, T. M., Breakdown of a 2D heteroclinic connection in the Hopf-zero singularity (II). The generic case. Preprint: https://arxiv.org/abs/1608.01116

Saturday, February 24, 2018 - 09:30 ,
Location: Helen M. Aderhold Learning Center (ALC), Room 24 (60 Luckie St NW, Atlanta, GA 30303) ,
Wenjing Liao and others ,
GSU, Clemson,UGA, GT, Emory ,
Organizer: Sung Ha Kang

The Georgia Scientific Computing Symposium is a forum for professors,
postdocs, graduate students and other researchers in Georgia to meet in
an informal setting, to exchange ideas, and to highlight local
scientific computing research. The symposium has been held every year
since 2009 and is open to the entire research community. This year, the symposium will be held on Saturday, February 24, 2018, at Georgia State University. More information can be found at: https://math.gsu.edu/xye/public/gscs/gscs2018.html

Series: Other Talks

Degeneracy loci of morphisms between vector bundles have been used in a wide range of situations, including classical approaches to the Brill--Noether theory of special divisors on curves. I will describe recent developments in Schubert calculus, including K-theoretic formulas for degeneracy loci and their applications to K-classes of Brill--Noether loci. These recover the formulas of Eisenbud--Harris, Pirola, and Chan--López--Pflueger--Teixidor for Brill--Noether curves. This is joint work with Dave Anderson and Nicola Tarasca.

Series: Combinatorics Seminar

I will describe two new local limit theorems on the
Heisenberg group, and on an arbitrary connected, simply connected
nilpotent Lie group. The limit theorems admit general driving measures
and permit testing against test functions with an arbitrary
translation on the left and the right. The techniques introduced include
a rearrangement group action, the Gowers-Cauchy-Schwarz inequality, and
a Lindeberg replacement scheme which approximates the driving measure
with the corresponding heat kernel. These
results generalize earlier local limit theorems of Alexopoulos and
Breuillard, answering several open questions. The work on the
Heisenberg group is joint with Persi Diaconis.

Friday, February 23, 2018 - 15:00 ,
Location: Skiles 271 ,
Jiaqi Yang ,
GT Math ,
Organizer: Jiaqi Yang

We will present a rigorous proof of non-existence of homotopically non-trivial invariant circles for standard map:x_{n+1}=x_n+y_{n+1}; y_{n+1}=y_n+\frac{k}{2\pi}\sin(2\pi x_n).This a work by J. Mather in 1984.