Georgia Institute of Technology College of Sciences

The Mean Ergodic Theorem of von Neumann proves the existence of limits of (time) averages for any cyclic group K = {U^n : n \in Z} acting on some Hilbert space H via powers of a unitary transformation U. Subsequent generalizations apply to so-called _multiple_ ergodic averages when Z is replaced by an arbitrary amenable group G, provided the image group K is nilpotent (Walsh's ergodic 2014 theorem for Z; generalization to G amenable by Zorin-Kranich). In this talk we survey a framework for mean convergence of polynomial group actions based on continuous model theory. We prove mean convergence of unitary polynomial Z-actions, and discuss how the full framework accomodates the most recent results mentioned above and allows generaling them.

Topics: local Hausdorff dimension, local Hausdorff measure, diffusion on compact metric spaces, prospective further research.

Explosive technological advances lead to exponential growth of massive data-sets in health-related fields. Of particular important need is an innovative, robust and adaptive acquisition of intrinsic features and metric structure hidden in the massive data-sets. For example, the hidden low dimensional physiological dynamics often expresses itself as atime-varying periodicity and trend in the observed dataset. In this talk, I will discuss how to combine two modern adaptive signal processing techniques, alternating diffusion and concentration of frequency and time(ConceFT), to meet these needs. In addition to the theoreticaljustification, a direct application to the sleep-depth detection problem,ventilator weaning prediction problem and the anesthesia depth problemwill be demonstrated. If time permits, more applications likephotoplethysmography and electrocardiography signal analysis will be discussed.

I'll discuss joint work with my brother Jeff Giansiracusa in which we introduce an exterior algebra and wedge product in the idempotent setting that play for tropical linear spaces (i.e., valuated matroids) a very similar role as the usual ones do for vector spaces. In particular, by working over the Boolean semifield this gives a new perspective on matroids.

Long-time behavior of "generic" 2d Euler solutions is expected to be governed by conserved quantities and simple variational principles related to them. Proving or disproving this from the dynamics is a notoriously difficult problem which remains unsolved. The variational problems which arise from these conjectures are interesting by themselves and we will present some results concerning these problems.

Schistosoma is a parasitic worm that circulates between human and snail hosts. Multiple biological and ecological factors contribute to its spread and persistence in host populations. The infection is widespread in many tropical countries, and WHO has made control of schistosomiasis a priority among neglected tropical diseases.Mathematical modeling is widely used for prediction and control analysis of infectious agents. But host-parasite systems with complex life-cycles like Schistosoma, pose many challenges. The talk will outline the basic biology of Schistosoma, and the principles employed in mathematical modeling of macro parasites. We shall review conventional approaches to Schistosomiasis starting with the classical work of MacDonald, and discuss their validity and implications. Then we shall outline more detailed “stratified worm burden approach”, and show how combining mathematical and computer tools one can explore real-world systems and make reliable predictions for long term control outcomes and the problem of elimination.

In this talk, I will explain how the gradient flow structure of reversible Markov chains (that was discovered by Maas and Mielke independently in 2011) and the Sandier-Serfaty approach to convergence of gradient flows can be combined to study scaling limits for interacting particle systems on lattices. The exposition will be focused on the case of the simple exclusion process on the discrete torus. Joint work with Marielle Simon (INRIA Lille).

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 $s \log(n/s)$ - $n$ is ambient dimension and $s$ is the sparsity threshold. 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.

Log Brunn-Minkowski conjecture was proposed by Boroczky, Lutwak, Yang and Zhang in 2013. It states that in the case of symmetric convex sets the classical Brunn-MInkowski inequality may be improved. The Gaussian Brunn-MInkowski inequality was proposed by Gardner and Zvavitch in 2007. It states that for the standard Gaussian measure an inequality analogous to the additive form of Brunn_minkowski inequality holds true for symmetric convex sets. In this talk we shall discuss a derivation of an equivalent infinitesimal versions of these inequalities for rotation invariant measures and a few partial results related to both of them as well as to the classical Alexander-Fenchel inequality.

The Peierls barrier is an observable which characterizes whether the the set minimizers with a prescribed frequency of a periodic variational problem form a continuum or have gaps. In solid state physics Peierls barrier characterizes whether ground states with a fixed density are pinned or are able to slide. The Peierls barrier is a microscopic explanation of static friction. Remarkably, in dynamical systems, Peierls barrier appears also as characterizing whether KAM circles break down into Cantor sets. Hence, the Peierls barrier has been investigated both by physicists and by mathematicians using a variety of methods. We plan to cover the basic definitions of the variational models and some of the basic results obtainedfrom the 80's. Continuation of last week's seminar

The Georgia Scientific Computing Symposium (GSCS) 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 at Emory University. The format of the day-long symposium is a set of invited presentations, poster sessions and a poster blitz, and plenty of time to network with other attendees. Invited speakers include: Michele Benzi, Mathematics and Computer Science, Emory University; Steven Hamilton, Radiation Transport Group, Oak Ridge National Laboratory; Alexandra Smirnova, Mathematics and Statistics, Georgia State University; Phanish Suryanarayana, School of Civil & Environmental Engineering, Georgia Institute of Technology; Molei Tao, Mathematics, Georgia Institute of Technology; Qing Zhang, Mathematics, University of Georgia. Poster sessions will be held during the lunch and afternoon breaks.