## News and Events

### Department of Mathematical Sciences Colloquia and Seminars:

#### Recent Colloquia and Seminars:

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

**Wednesday, November 18, 2015**

12:00 to 1:00 PM, BH 205: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

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.

**Wednesday, October 28, 2015**12:00 to 1:00 PM, BH 205:

Speaker: Ruobing Zhang of Princeton University

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

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.

##### The following is a list of the department's colloquia and seminars for the academic year 2014–2015.

**Wednesday, April 29, 2015**

1:30
to 2:30 PM, room 506 of 215 Lexington Avenue, video-conferenced to BH 206

Speaker:
Edward Belbruno of Orbital Design, Princeton University, and Courant Institute
of Mathematical Sciences

Title:
A new route to Mars Abstract:
A new type of transfer trajectory from the Earth to Mars is described that
employs ballistic capture, unlike all previous transfers. It has an
interesting dynamics from a dynamical systems perspective and provides a
completely different methodology for getting to Mars. A mission is being
considered by NASA to use it.

**Wednesday, January 21, 2015**

12p.m. to 1p.m. Belfer Hall, Room 516

Speaker: Pablo Roldan, University of Maryland

Title: Instabilities in the Restricted Three Body Problem Along Mean Motion Resonances

Abstract: In this talk I will present a recent result related to the stability of the Solar System, namely the preprint: Jacques Fejoz, Marcel Guardia, Vadim Kaloshin and Pablo Roldan: Kirkwood gaps and diffusion along mean motion resonances in the restricted planar three-body problem (2013), available here.

**Wednesday, December 17, 2014**

12p.m. to 1p.m. Belfer Hall, Room 205

Speaker: Mina Teicher, Professor, Department of Mathematics and Gonda Brain Research Center,
Director, Emmy Noether Institute for Mathematics, Bar-Ilan University

Title: How Does the Brain Work?

Abstract: "How does the brain work?" -- is the most intriguing question for the 21st century. In this talk I will elaborate on the question in general and report on a project of synchronization in brain activity and on some medical applications.

**Wednesday, December 10, 2014**

12p.m. to 1p.m. Belfer Hall, Room 205

Speaker: Ernesto Perez-Chavela, Universidad Autonoma Metropolitana Iztapalapa

Title: A geometric approach for relative equilibria solutions in the curved N-body problem with negative curvature

Abstract: We extend the Newtonian potential to spaces of constant curvature. The relative equilibria solutions are motions where the mutual distance among the particles remain constant for all time. In this talk, using the Poincare upper semiplane model of the hyperbolic geometry, in the case of negative curvature, we describe all relative equilibria solutions in the three body problem. We also describe some relative equilibria in the five body problem.

**Wednesday, December 3, 2014**

12p.m. to 1p.m. Belfer Hall, Room 205

Speaker: Jaime Burgos-Garcia, Yeshiva University

Title: Hill's approximations to study the dynamics of satellites in the solar system

Abstract:
G.W. Hill developed his lunar theory as an alternative approach for the
study of the motion of the Moon around the Earth. This approach is a
special case of the three body problem for the dynamics of the Earth and
the Moon with a gravitational perturbation produced by a far away
massive body considered as the Sun. If the perturbation due to other
body is considered, an extension of the former theory is required. In
this talk we introduce the required extension and we show that the
resulting problem produces a new Hamiltonian that inherits some basic
features of the restricted three-and-four-body problems. The main
motivation for this model is the study of the motion of a satellite near
one of the Trojan asteroids of Jupiter. We show some basic results on
this model which is a joint work with Marian Gidea.

**Wednesday, November 5, 2014 **

12p.m. to 1p.m. Belfer Hall, Room 516.

Speaker: Adria Simon, of the Departament de Matematica Aplicada I, Universitat Politeenica de Catalunya

Title: Diffusion through non-transversal transition chains

(joint work with Amadeu Delshams and Piotr Zgliczcynski

Abstract:
The main goal of our work is to understand the geometric mechanism that
gives rise to the instability shown by Colliander et al (2010) in the
Nonlinear Schrodinger Equation with cubic defocusing. It can be seen as a
diffusion mechanism, but it appears that the geometric skeleton of the
system is not the standard for Arnold Diffusion: instead of having a
sequence of non-resonant invariant tori connected along transverse
heteroclinic orbits we have a non-transversal situation

We
expose that the instability (diffusion) can be achieved due to the large
dimension of the system, while we try to generate a scheme for this new
kind of diffusion that could be applied to other infinitely dimensional
Hamiltonian systems.

**September 2014**

Speaker:
Dr. Sven Jarohs, Goethe-Universitat, Frankfurt,
Germany

Title: The moving plane method in nonlocal problems

Abstract: In this talk I present recent results which use the moving plane method in a
nonlocal framework. After giving a short introduction to the fractional
Laplacian, I will explain the main ideas for this example and explain the main
differences to the local case. In particular, I will show that the nonlocal
setting admits symmetry results which are not true for local operators. As part
of the proof idea I will also present the fractional Hopf Lemma. Following this, I will state an
over-determined problem with fractional Laplacian and show that again the nonlocal
case admits more general results than the nonlocal case. Finally I will show
how the main idea easily can be applied to general nonlocal operators of
possible varying order which may also be anisotropic. The presented results are
part of joint works with Mouhamed Moustapha Fall and Tobias Weth.

**April 2014**

Speaker: Tere Seara, Universitat Politecnica de Catalunya, Barcelona, Spain

Title: Oscillatory motions and Arnold diffusion in the restricted planar three body problem

The restricted (elliptic) three body problem considers the motion of a particle with zero mass under the effects of two bodies called primaries which move in elliptic orbits around their center of mass.

The circular case is a particular case of the elliptic case where the primaries move in circular orbits. The first part of this talk proves the existence of oscillatory motions for the restricted planar circular three body problem, that is, we will show that there exist orbits which leave every bounded region but which return infinitely often to some fixed bounded region. Our work does not need to assume any smallness
about the mass ratio.

We show that, for large enough Jacobi constant, there exist transversal intersections between the stable and unstable manifolds of infinity which guarantee the existence of a symbolic dynamics that creates the so called oscillatory orbits. The main achievement is to rigorously prove the transversality of the invariant manifolds without assuming the mass ratio small, since then this transversality can not be checked by using classical perturbation theory respect to the mass ratio.

The second part of the talk extends these results about the invariant manifolds to the elliptic case, taking the mass ratio and the eccentricity of the primaries small enough. As a consequence we obtain orbits of the elliptic restricted three body problem whose angular momentum increases. This behavior is usually called Arnold diffusion.