## Seminars and Colloquia by Series

Friday, March 31, 2017 - 15:05 , Location: Skiles 254 , Lei Zhang , School of Mathematics, GT , Organizer: Jiaqi Yang
In this talk, we will give an introduction to the variational approach to dynamical systems. Specifically, we will discuss twist maps and prove the classical results that area-preserving twist map has Birkhoff periodic orbits for each rational rotation number.
Friday, March 31, 2017 - 15:00 , Location: Skiles 114 , Dan Margalit , Georgia Tech , Organizer: Christine Heitsch
A conversation with Dan Margalit, GT math professor and inaugural CoS Leddy Family Faculty Fellow, who was a tenure-track assistant professor at Tufts University for two years prior to coming to Tech.
Friday, March 31, 2017 - 13:05 , Location: Skiles 005 , Jose Acevedo , School of Mathematics, Georgia Tech , Organizer: Marcel Celaya
Using some classical results of invariant theory of finite reflection groups, and Lagrange multipliers, we prove that low degree or sparse real homogeneous polynomials which are invariant under the action of a finite reflection group $G$ are nonnegative if they are nonnegative on the hyperplane arrangement $H$ associated to $G$. That makes $H$ a test set for the above kind of polynomials. We also prove that under stronger sparsity conditions,  for the symmetric group and other reflection groups, the test set can be much smaller. One of the main questions is deciding if certain intersections of some simply constructed real $G$-invariant varieties are empty or not.
Friday, March 31, 2017 - 11:05 , Location: Skiles 006 , , Auburn University at Montgomery , Organizer: Anton Leykin
Networks, or graphs, can represent a great variety of systems in the real world including neural networks, power grid, the Internet, and our social networks. Mathematical models for such systems naturally reflect the graph theoretical information of the underlying network. This talk explores some common themes in such models from the point of view of systems of nonlinear equations.
Thursday, March 30, 2017 - 15:05 , Location: Skiles 006 , , Columbia University , Organizer: Mayya Zhilova
We consider the problem of studying the limiting distribution of the number of monochromatic two stars and triangles for a growing sequence of graphs, where the vertices are colored uniformly at random. We show that the limit distribution of the number of monochromatic two stars is a sum of mutually independent components, each term of which is a polynomial of a single Poisson random variable of degree 1 or 2. Further, we show that any limit distribution for the number of monochromatic two stars has an expansion of this form. In the triangle case the problem is more challenging, as in this case the class of limit distributions can involve terms with products of Poisson random variables. In this case, we deduce a necessary and sufficient condition on the sequence of graphs such that the number of monochromatic triangles is asymptotically Poisson in distribution and in the first two moments. This work is joint with Bhaswar B. Bhattacharya at University of Pennsylvania.
Thursday, March 30, 2017 - 11:05 , Location: Skiles 006 , Larry Goldstein , University of Southern California , Organizer: Christian Houdre
Charles Stein brought the method that now bears his name to life in a 1972 Berkeley symposium paper that presented a new way to obtain information on the quality of the normal approximation, justified by the Central Limit Theorem asymptotic, by operating directly on random variables. At the heart of the method is the seemingly harmless characterization that a random variable $W$ has the standard normal ${\cal N}(0,1)$ distribution if and only if E[Wf(W)]=E[f'(W)] for all functions $f$ for which these expressions exist. From its inception, it was clear that Stein's approach had the power to provide non-asymptotic bounds, and to handle various dependency structures. In the near half century since the appearance of this work for the normal, the `characterizing equation' approach driving Stein's method has been applied to roughly thirty additional distributions using variations of the basic techniques, coupling and distributional transformations among them. Further offshoots are connections to Malliavin calculus and the concentration of measure phenomenon, and applications to random graphs and permutations, statistics, stochastic integrals, molecular biology and physics.
Thursday, March 30, 2017 - 11:00 , Location: Skiles 202 , Scott Spencer , Georgia Institute of Technology , , Organizer: Scott Spencer
This thesis explores topics from two distinct fields of mathematics. The first part addresses a theme in abstract harmonic analysis, while the focus of the second part is a topic in compressive sensing. The first part of this dissertation explores the application of dominating operators in harmonic analysis by sparse operators. We make use of pointwise sparse dominations weighted inequalities for Calder\'on-Zygmund operators, Hardy-Littlewood maximal operator, and their fractional analogues. Dominating bilinear forms by sparse forms allows us to derive weighted inequalities for oscillatory integral operators (polynomially modulated CZOs) and random discrete Hilbert transforms. The later is defined on sets of initegers with asymptotic density zero, making these weighted inequalitites particulalry attractive. We also discuss a characterization of a certain weighted BMO space by commutators of multiplication operators with fractional integral operators. Compressed sensing illustrates the possibility of acquiring and reconstructing sparse signals via underdetermined (linear) systems.  It is believed that iid Gaussian measurement vectors give near optimal results, with the necessary number of measurements on the order of slog⁡(n/s) -- n is ambient dimension and s is sparsity threshhold.The recovery algorithm used above relies on a certain quasi-isometry property of the measurement matrix.  A surprising result is that the same order of measurements gives an analogous quasi-isometry in the extreme quantization of one-bit sensing.  Bylik and Lacey deliver this result as a consequence of a certain stochastic process on the sphere.  We will discuss an alternative method that relies heavily on the VC-dimension of a class of subsets on the sphere.
Series: PDE Seminar
Wednesday, March 29, 2017 - 15:05 , Location: Skiles 006 , Yue Liu , University Of Texas At Arlington , , Organizer: Yao Yao
In this talk, a mathematical model of long-crested water waves propagating mainly in one direction with the effect of Earth's rotation is derived by following the formal asymptotic procedures. Such a model equation is analogous to the Camassa-Holm approximation of the two-dimensional incompressible and  irrotational Euler equations and has a formal bi-Hamiltonian structure.  Its solution corresponding to physically relevant initial perturbations is more accurate on a much longer time scale. It is shown that the deviation of the free surface can be determined by the horizontal velocity at a certain depth in the second-order approximation.  The  effects of the Coriolis force caused by the Earth rotation and  nonlocal higher nonlinearities on blow-up criteria and wave-breaking phenomena are also investigated. Our refined analysis is approached by applying the method of characteristics and conserved quantities to the  Riccati-type differential inequality.
Wednesday, March 29, 2017 - 14:05 , Location: Skiles 006 , Sudipta Kolay , Georgia Tech , Organizer: Justin Lanier
In this series of talks we will show that every closed oriented three manifold is a branched cover over the three sphere, with some additional properties. In the first talk we will discuss some examples of branched coverings of surfaces and three manifolds, and a classical result of Alexander, which states that any closed oriented combinatorial manifold is always a branched cover over the sphere.
Wednesday, March 29, 2017 - 02:05 , Location: Skiles 005 , Shahaf Nitzan , Georgia Tech , Organizer: Shahaf Nitzan
A Gaussian stationary sequence is a random function f: Z --> R, for which any vector (f(x_1), ..., f(x_n)) has a centered multi-normal distribution and whose distribution is invariant to shifts. Persistence is the event of such a random function to remain positive on a long interval [0,N].  Estimating the probability of this event has important implications in engineering , physics, and probability. However, though active efforts to understand persistence were made in the last 50 years, until recently, only specific examples and very general bounds were obtained. In the last few years, a new point of view simplifies the study of persistence, namely - relating it to the spectral measure of the process. In this talk we will use this point of view to study the persistence in cases where the spectral measure is 'small' or 'big' near zero. This talk is based on Joint work with Naomi Feldheim and Ohad Feldheim.