Seminars and Colloquia by Series

Series: PDE Seminar
Tuesday, February 27, 2018 - 15:00 , Location: Skiles 006 , Xinliang An , University of Toronto , xinliang.an@utoronto.ca , Organizer: Yao Yao
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.
Tuesday, February 27, 2018 - 11:00 , Location: Skiles 006 , Alexander Barvinok , University of Michigan , barvinok@umich.edu , Organizer: Prasad Tetali
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.)
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.
Monday, February 26, 2018 - 11:15 , Location: skiles 005 , Tere M. Seara , Departament de Matemàtiques. Universitat Politècnica de Catalunya (UPC) , Organizer: Livia Corsi
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
Friday, February 23, 2018 - 17:00 , Location: Classroom 105 in the D.M. Smith building , Associate Professor , Swarthmore College , Organizer: Joseph Rabinoff
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.
Friday, February 23, 2018 - 15:05 , Location: Skiles 005 , Robert Hough , SUNY, Stony Brook , robert.bough@stonybrook.edu , Organizer: Prasad Tetali
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.
Friday, February 23, 2018 - 14:00 , Location: none , none , none , Organizer: John Etnyre
Friday, February 23, 2018 - 13:55 , Location: Skiles 269 , Prof. Justin Kakeu , Morehouse University , Organizer: Sung Ha Kang
We use a stochastic dynamic programming approach to address the following question: Can a homogenous resource extraction model (one without extraction costs, without new discoveries, and without technical progress) generate non-increasing resource prices?  The traditional answer to that question contends that prices should exhibit an increasing trend as the exhaustible resource is being depleted over time (The Hotelling rule). In contrast, we will show that injecting concerns for temporal resolution of uncertainty in a resource extraction problem can generate a non-increasing trend in the resource price.  Indeed, the expected rate of change of the price can become negative if the premium for temporal resolution of uncertainty is negative and outweighs both the positive discount rate and the short-run risk premium.  Numerical examples are provided for illustration.

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