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Series: ACO Student Seminar

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.

Series: Algebra Seminar

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.

Series: Stochastics Seminar

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.

Series: School of Mathematics Colloquium

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.

Series: Dissertation Defense

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

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

Series: Analysis Seminar

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.

Series: PDE Seminar

In 1944, L.D. Landau first discovered explicit (-1)-homogeneous solutions of 3-d stationary incompressible Navier-Stokes equations (NSE) with precisely one singularity at the origin, which are axisymmetric with no swirl. These solutions are now called Landau solutions. In 1998 G. Tian and Z. Xin proved that all solutions which are (-1) homogeneous, axisymmetric with one singularity are Landau solutions. In 2006 V. Sverak proved that with just the (-1)-homogeneous assumption Landau solutions are the only solutions with one singularity. He also proved that there are no such solutions in dimension greater than 3. Our work focuses on the (-1)-homogeneous solutions of 3-d incompressible stationary NSE with finitely many singularities on the unit sphere.In this talk we will first classify all (-1)-homogeneous axisymmetric no-swirl solutions of 3-d stationary incompressible NSE with one singularity at the south pole on the unit sphere as a two dimensional solution surface. We will then present our results on the existence of a one parameter family of (-1)-homogeneous axisymmetric solutions with non-zero swirl and smooth on the unit sphere away from the south pole, emanating from the two dimensional surface of axisymmetric no-swirl solutions. We will also present asymptotic behavior of general (-1)-homogeneous axisymmetric solutions in a cone containing the south pole with a singularity at the south pole on the unit sphere. We also constructed families of solutions smooth on the unit sphere away from the north and south poles.This is a joint work with Professor Yanyan Li and Li Li.

Series: Math Physics Seminar

We consider the two-dimensional BCS functional with a radial pair interaction. We show that the translational symmetry is not broken in a certain temperature interval below the critical temperature. For this purpose, we first introduce the full BCS functional and the translation invariant BCS functional. Our main result states that theminimizers of the full BCS functional coincide with the minimizers of the translation invariant BCS functional for temperatures in the aforementioned interval. In the case of vanishing angular momentum our results translate to the three dimensional case. Finally, we will explain the strategy and main ideas of the proof. This is joint work with Andreas Deuchert, Christian Hainzl and Michael Loss.