## News and Events

### Department of Mathematical Sciences Colloquia and Seminars:

#### Recent and Upcoming Colloquia and Seminars

The following is a list of the department's colloquia and seminars for the academic year 2015–2016.

**November 18, 2015**

12:00 to 1:00 PM

Speaker: Aaron Golden of Department of Mathematical Sciences, Yeshiva University, and Albert Einstein College of Medicine

Title: Evolutionary Learning and Einstein's Theory of General Relativity

Belfer Hall, 2495 Amsterdam Avenue, Room 205

Abstract: Evolutionary learning algorithms provide an extremely powerful means of parsing and characterizing complex datasets, and these techniques form part of the bedrock of Artificial Intelligence. Perhaps the most well known of these algorithms attempts to replicate the processes associated with cognitive/biological learning, termed 'neural networks', and of these, the Self Organizing Map - or SOM - has been of particular interest to researchers for decades. This unsupervised algorithm implements a mathematical model that simulates the process of topographic mapping of input stimuli (i.e. vectors) from a set of sensors within the retina to the cerebral cortex (i.e. a network of neurons), with neurons being trained to identify specific input stimuli types. Originally developed by Teuvo Kohonen in 1982, this somewhat elementary but immensely powerful algorithm has permeated disciplines as diverse as meteorology,
quantitative finance and astronomy. Our group in particular has developed
several variants designed to identify unique biological signals embedded in
genomes. Despite its highly attractive `sui generis' property, the algorithm's
evolutionary learning is to some extent biased by several user defined
constraints and heuristics, whose optimum selection can be a non-trivial
problem in itself. In this talk I will describe how ideas and concepts first
proposed a century ago by Einstein can be applied to this evolutionary learning
algorithm, with the goal of developing a truly holistic implementation of the
Self Organizing Map.

**October 28, 2015**12:00 to 1:00 PM

Speaker: Ruobing Zhang of Princeton University

Title: Positivity of Non-local Curvature and Topology
of Locally Conformally Flat Manifolds

Belfer Hall, 2495 Amsterdam Avenue, Room 205

Abstract: In this talk, we focus on the geometry of compact conformally flat manifolds (M^n,g) with positive scalar curvature. It is a
classical result by R. Schoen and S. Yau that the universal cover of M^n is
conformally embedded in n such that M^n is a Kleinian manifold. Moreover, the
limit set of the Kleinian group has Hausdorff dimension <(n-2)/2. If
additionally we assume that the non-local curvature Q_{2\gamma}>0 for some
1<\gamma<2, we prove that the Hausdorff dimension of the limit set is
less than <(n-2\gamma)/2. In fact, the above upper bound is sharp. As
applications, we obtain some topological rigidity and classification theorems
for dimensions less than 6.

**December 9, 2015**

12:00 to 1:00 PM

Speaker: Amadeu Delshams (Polytechnic University of Catalonia, Barcelona)

Title: Global Instability through non-transverse heteroclinic chains,
with an application to the periodic cubic defocusing NLS equation

Belfer Hall, 2495 Amsterdam Ave, Room 205

Abstract: We introduce a new mechanism for global instability in dynamical systems, based on the shadowing of a sequence of invariant tori connected along non-transverse heteroclinic orbits, under some geometric restrictions. This mechanism can be readily applied to systems of large dimensions, like infinite-dimensional Hamiltonian systems, particularly the periodic cubic defocusing nonlinear Schrodinger (NLS) equation. This is a joint work with A. Simon and P. Zgliczynski.

**December 16, 2015**

12:00 to 1:00 PM

Speaker: Anatole Katok

Title: Geometry and Dynamics of Surfaces of Negative Curvature

Belfer Hall, 2495 Amsterdam Ave, Room 205

Abstract: Study of Riemannian metrics of negative curvature on compact closed surfaces brings into a focus interaction among several core areas of mathematics: Riemannian geometry, dynamical systems, complex analysis and variational calculus. In this talk I will discuss several results, both old and recent, that use this interaction and illustrate it in interesting and often unexpected ways. I will start with definition and properties of principal global characteristics of such metrics: topological entropy, conformal coefficient, entropy with respect to the smooth (Liouville) measure, Lyapunov characteristic exponents, and connect those with more familiar geometric characteristic such as area, diameter, systole, average curvature. I will describe both rigidity and flexibility phenomena, that play important role in several areas of analysis and geometry, in this context. Finally, I will mention several easily understandable open problems and discuss conceptual difficulties that appear in attempts to solve them.

**January 28, 2016 **12:00 to 1:00pm

Speaker: Jason Mireles James (Florida Atlantic University)

Title: An introduction to computer aided proof in nonlinear analysis

215 Lexington Avenue, Room 506

Abstract: Over the last several decades the role of the digital computer as a tool for nonlinear analysis has been growing steadily. I will discuss some functional analytic tools which,

when combined with standard truncation error analysis and deliberate control of round off errors, can be used in order to obtain computer assisted existence proofs for solutions of nonlinear equations. I will give a brief overview of the history of these methods and also discuss some newer developments.

**March 2, 2016 **

12:00 to 1:00 PM

Speaker: Boris Hasselblatt (Tufts University)

Title: Statistical properties of deterministic systems by elementary means

Belfer Hall, 2495 Amsterdam Avenue, Room 205

Abstract: The Maxwell-Boltzmann ergodic hypothesis aimed to lay a foundation under statistical mechanics, which is at a microscopic scale a deterministic system. Similar complexity was discovered by Poincaré in celestial mechanics and by Hadamard in the motion of a free particle in a negatively curved space. We start with a guided tour of the history of the subject from various perspectives and then discuss the central mechanism that produces pseudorandom behavior in these deterministic systems, the Hopf argument. It has been known to extend well beyond the scope of its initial application in 1939, and we show that it also leads to much stronger conclusions: Not only do time averages of observables coincide with space averages (which was the purpose for making the ergodic hypothesis), but any finite number of observables will become decorrelated with time. That is, the Hopf argument does not only yield ergodicity but mixing, and often mixing of all orders.

**March 9, 2016 **

12:00 to 1:00pm

Speaker: Rafael de la Llave (Georgia Tech)

Title: Perturbations of quasi-periodic orbits: From Theory to computations

Abstract: Since the time if Hyparco, it was known that the motion of celestial bodies is approximately given by epicycles. In modern language, expressed in Fourier series of a few frequencies, which we now call quasi-periodic.The advent of celestial mechanics raised the question of whether these motions are compatible with the laws of mechanics.The resonances in quasi-periodic motion could destroy each other. A great lore of formal computations and expansions was developed for 300 years. On the other hand, the first proofs of persistence of quasi-periodic motions appeared only in the 1950's (Kolmogorov-Arnold-Moser theory). With the use of computers and the need of applications, it has become important to know whether KAM theory can yield results applicable to problems of technological interest. We will describe several results by many people that show that one can develop proofs that lead to efficient algorithms and which give confidence in the results. Perhaps more importantly, the implementations permit to explore the boundaries and lead to beautiful conjectures.

**April 6, 2016 **

12:00 to 1:00pm

Speaker: Federico Rodriguez Hertz (Penn State)

Title: Stationary measures, P-invariant measures and invariant measures for group actions

Abstract: For actions of non-amenable groups, existence of an invariant measure often has important implications. In this lecture we shall discuss examples of these implications as well as results on existence of invariant measures. These results involve the notion of random dynamics, stationary measures and P-invariant measures when the group is a semisimple Lie group. We will try to maintain the discussion at an elementary level, and present the notions mostly through simple examples. This is based on ongoing work with Aaron Brown and Zhiren Wang.

**April 13, 2016**

12:00 to 1:00pm

Speaker: Jiuyi Zhu (Johns Hopkins University)

Title: Nodal geometry of Steklov eigenfunctions

Abstract: The eigenvalue and eigenfunction problem is fundamental and essential in mathematical analysis. The Steklov problem is an eigenvalue problem with its spectral parameter at the boundary of a compact Riemannian manifold. Recently the study of Steklov eigenfunctions has been attracting much attention. We consider the quantitative properties: Doubling inequality and nodal sets. We obtain the sharp doubling inequality for Steklov eigenfunctions on the boundary and interior of manifolds using delicate Carleman estimates. We can ask Yau's type conjecture for the Hausdorff measure of nodal sets of Steklov eigenfunctions on the boundary and interior of the manifold. I will describe some recent progress about this challenging direction. Part of work is joint with C. Sogge and X. Wang.

**April 20, 2016**

12:00 to 1:00pm

Speaker: Svetlana Katok (Penn State)

Title: Reduction theory for Fuchsian groups and coding of geodesics

Abstract: I will discuss a method of coding of geodesics on quotients of the hyperbolic plane by Fuchsian groups using boundary maps and “reduction theory”. These maps are piecewise fractional-linear given by generators of the Fuchsian group, and the orbit of a point under the boundary map defines its boundary expansion. For compact surfaces they are generalizations of the Bowen-Series map, and for the modular surface are related to a family of (a,b)-continued fractions. For the natural extensions of the boundary maps Zagier’s Reduction Theory Conjecture (RTC) holds: for the appropriate open sets of parameters they have attractors with finite rectangular structure to which (almost) every point is mapped after finitely many iterations. The RTC is used for representing the geodesic flow as a special flow over a cross-section of “reduced” geodesics parametrized by the attractor.

When a boundary expansion has a so-called “dual”, the coding sequences are obtained by juxtaposition of the boundary expansions of the end points of the corresponding geodesic, and the set of coding sequences is a sofic shift. This was proved for the modular group and generalizes for Fuchsian groups that satisfy the RTC. The talk is based on joint works with Ilie Ugarcovici.

**May 4, 2016 **

12:00 to 1:00pm

Speaker: Omri Sariq (Weizmann Institute of Science)

Title: Symbolic dynamics for chaotic dynamical systems

Abstract: "Chaotic" maps have strong sensitivity to initial conditions: orbits of nearby initial conditions can diverge exponentially fast from one another. This makes it difficult to visualize the orbit structure of the system geometrically. Symbolic dynamics is a powerful technique for visualizing the orbit structure combinatorially. Recently, the scope of this technique was extended to a large collection of dynamical systems.The talk is meant for a general audience and will not assume any prior knowledge of dynamics.