Upcoming events:

Event:       Mathematical Physics Seminar
Speaker:   Suddhasattwa Das (Courant Institute of Mathematical Science, New York University) 
Title:         The spectral measure of a dynamical system
Time:         Wednesday, February 19, 2020, 11:00 am
Location:  Yeshiva University, 215 Lexington Ave, 6th Floor, Conference Room 628
Abstract:  Many dynamical systems are described by a flow $\Phi^t$ on an ambient manifold $M$. Instead of the trajectories of this flow, the operator theoretic framework studies the dynamics induced on the space of observables. This gives rise to a unitary group $U^t$ called the Koopman group. It describes the time-evolution of measurements, such as the state-space variables of an ODE. Many problems in theoretical and applied dynamics can be restated in terms of the Koopman group. A fundamental notion for such groups is that of a spectral measure, which is an operator valued, Borel measure on the complex plane. The spectral measure completely characterizes $U^t$ and hence the trajectories of the flow. I will discuss many diverse ways in which the spectral measure manifests itself, such as in the spectral analysis of data generated by the dynamical system, decay of correlations, periodic approximation of dynamical systems, and visibly as coherent spatiotemporal patterns. Each of these topics are of great interest of their own, and thus an accurate determination and computation of the spectral measure is of great value. I will finally describe a data-driven means of approximating the spectral measure, which relies on a number of tools from functional analysis.
Co-authors: Dimitrios Giannakis, Joanna Slawinska.
Paper url : https://arxiv.org/abs/1808.01515
Web page: https://www.yu.edu/ug/math/colloquia-seminars  


Past events:

Event:       Mathematical Physics Seminar
Speaker:   Edward Belbruno (Yeshiva University) 
Title:         A Family of Periodic Orbits to the 3-Dimensional Lunar Problem and Applications
Time:        Wednesday, February 12, 2020, 11:00 am
Location: Yeshiva University, 215 Lexington Ave, 6th Floor, Conference Room 628
Abstract: An interesting family of periodic orbits is found to the 3-dimensional restricted 3-body problem about the smaller primary perpendicular to the orbital plane. These orbits evolve from a family about the larger primary studied by EB in 1981(CMDA, 1981).  Numerical behavior is studied and stability analyzed.  Applications discussed. Co authors: Urs Frauenfelder, Otto van Koert. Published in CMDA Feb 2019.   
Web page: https://www.yu.edu/ug/math/colloquia-seminars  

Event: Mathematical Physics Seminar
Speaker: Yingkai Liu (Yeshiva University) 
Title: Topological Quantum Qudits: Principles and Simulations 
Time: Wednesday, January 29, 2020, 1:30 pm
Location: Yeshiva University, 205 Lexington Ave, 6th Floor, Main Conference Room
Abstract: The research efforts towards a multi-purpose quantum computer have accelerated in the past years on both the hardware and software fronts. A major event in the field was a proposal of a theoretical fault-tolerant quantum computation platform based on topologically protected quantum qudits and quantum gates. In this talk, I will describe these concepts and principles using the originally proposed quantum models as well as newer ones. I will then sketch a general proof of the topological degeneracy for these models. The latter manifests in a 4^g degeneracy of each eigenvalue whenever the quantum models are deployed on surfaces of genus g. It is this degeneracy which delivers the topologically protected qudits. In the second part, I will present a numerical algorithm that enabled us to simulate this extremely unusual phenomenon.  
Web page: https://www.yu.edu/ug/math/colloquia-seminars  

Event: Mathematical Physics Seminar
Speaker: Yiftach Dayan (Technion, Haifa, Israel) 
Title: Random walks on tori and an application to normality of numbers in self-similar sets
Time: Wednesday, January 22, 2020, 12:00 pm
Location: Yeshiva University, 205 Lexington Ave, 6th Floor, Main Conference Room
Abstract: We show that under certain conditions, random walks on a d-dim torus by affine expanding maps have a unique stationary measure. We then use this result to show that given an IFS of contracting similarity maps of the real line with a uniform contraction ratio 1/D, where D is some integer > 1, under some suitable condition, almost every point in the attractor of the given IFS (w.r.t. a natural measure) is normal to base D. Joint work with Arijit Ganguly and Barak Weiss.
Web page: https://www.yu.edu/ug/math/colloquia-seminars  


Event: Mathematical Physics Seminar
Speaker: Shane Kepley (Rutgers University, Department of Mathematics) 
Title: Parameterization of Invariant Manifolds and Connecting Orbits in Celestial Mechanics
Time: Wednesday, December 18, 2019, 12:00 pm
Location: Yeshiva University, 205 Lexington Ave, 6th Floor, Main Conference Room
Abstract: In 2003 Cabre, Fontich, and de la Llave introduced the “Parameterization Method” for proving existence and regularity of invariant manifolds in dynamical systems. In this talk we will discuss techniques which combine the Parameterization Method with tools from topology and numerical analysis to study global dynamics and transport in Celestial Mechanics problems. As an example, we will describe some recent results about homoclinic and collision dynamics in the circular restricted three body problem.
Web page: https://www.yu.edu/ug/math/colloquia-seminars  

 

Event: Mathematical Physics Seminar
Speaker: Tere Seara (Universitat Politecnica de Catalunya, Barcelona, Spain) 
Title: On the breakdown of small amplitude breathers for the reversible Klein-Gordon equation
Time: Wednesday, December 11, 2019, 12:00 pm
Location: TBA
Abstract: Breathers are periodic in time spatially localized solutions of evolutionary PDEs. They are known to exist for the sine-Gordon equation but are believed to be rare in other Klein-Gordon equations. Breathers can be interpreted as homoclinic solutions to a steady solution in an infinite dimensional space. In this talk, we prove an asymptotic formula for the distance between the stable and unstable manifold of the
steady solution when the steady solution has weakly hyperbolic one dimensional stable and unstable manifolds. This formula allows to say that for a wide set of Klein-Gordon equations breathers do not exist. The distance is exponentially small with respect to the amplitude of the breather and therefore classical perturbative techniques cannot be applied. This is a joint work with O. Gomide, M: Guardia and C. Zeng.
Web page: https://www.yu.edu/ug/math/colloquia-seminars  

 

Event: Mathematical Physics Seminar
Speaker: Dan Pirjol (Stevens Institute of Technology)
Title: Large deviations for time-averaged diffusions in the small time limit  
Time: Wednesday,  November 20, 2019, 12:00 pm
Location: Yeshiva University, 205 Lexington Ave, 6th Floor, Main Conference Room 
Abstract: Time integrals of one-dimensional diffusions appear in the statistical mechanics of disordered systems, actuarial science and mathematical finance. The talk presents large deviations properties for the time-average of a diffusion in the small time limit. The result follows from the classical pathwise large deviations result for diffusions obtained by Varadhan in 1967, and the contraction principle. The rate function is expressed as a variational problem, which is solved explicitly. As an application we discuss the short maturity asymptotics of Asian options in mathematical finance. [Based on work with Lingjiong Zhu, Florida State University]
Web page: https://www.yu.edu/ug/math/colloquia-seminars

 

Event: Mathematical Physics Seminar
Speaker: Jacob Shapiro (Columbia University)
Title: The Bulk-Edge Correspondence from the Fredholm Perspective  
Time: Wednesday,  November 13, 2019, 12:00 pm
Location: Yeshiva University, 245 Lexington Ave, Room 601 
Abstract: We present the well known bulk edge correspondence for the integer quantum Hall effect using homotopy theory of Fredholm operators, which allows to extend the proof also to the time-reversal invariant case via a version of the theory for skew-adjoint operators.
Web page: https://www.yu.edu/ug/math/colloquia-seminars

 

Event: Mathematical Physics Seminar
Speaker: Daisy Dahiya (National Institutes of Health)
Title: Computing the quasi-potential for nongradient SDEs  
Time: Wednesday,  November 6, 2019, 12:00 pm
Location: Yeshiva University, 205 Lexington Ave, 6th Floor, Main Conference Room 
Abstract: Quasi-potential is a key function in the Large Deviation Theory that allows one to estimate the transition rates between attractors of the corresponding ordinary differential equation and find the maximum likelihood transition paths. It also gives an estimate of the invariant probability distribution in the neighborhood of the attractor up to the exponential order. Quasi-potential is defined as the solution to a certain action minimization problem. In general, it cannot be found analytically. In this work, we present numerical methods, named the Ordered Line Integral Methods (OLIM), for computing the quasi-potential for nongradient SDEs with a small white noise. OLIM are 1.5-4 times faster as compared to the first quasipotential finder based on the ordered upwind method (OUM) (Cameron 2012), can produce error two to three orders of magnitude smaller, and may exhibit faster convergence. Similar to the OUM, OLIM employ the dynamical programming principle but use a different computational strategy leading to a notable speed-up. A modification of OLIM to compute the quasi-potential for SDEs with varying and anisotropic diffusion term will be presented where we demonstrate the effects of the anisotropy on the quasi-potential and maximum likelihood transition paths for the Maier-Stein model. An application of the method to the Lambda Phage gene regulation model (Aurell and Sneppel, 2002) will be discussed.
Web page: https://www.yu.edu/ug/math/colloquia-seminars


Event: Mathematical Physics Seminar
Speaker: Marcel Guardia  (Universitat Politecnica de Catalunya, Barcelona, Spain) 
Title: Growth of Sobolev norms in the nonlinear Schrödinger equation  
Time: Wednesday,  October 30, 12:00 PM 
Location:Yeshiva University,  2495 Amsterdam Ave, New York, NY 10033, Belfer Hall, BH-825
Abstract: The study of solutions of Hamiltonian PDEs undergoing growth of Sobolev norms H^s (with s\neq 1) as time evolves has drawn considerable attention in recent years. The importance of growth of Sobolev norms is due to the fact that it implies that the solution transfers energy to higher modes. In this talk I will report on recent results in constructing solutions of the cubic nonlinear defocusing Schro\"odinger equation which start close to different invariant objects and achieve, after long time, large finite growth of $H^s$ Sobolev norm.
Web page: https://www.yu.edu/ug/math/colloquia-seminars

 

Event: Mathematical Physics Seminar
Speaker: Edward Belbruno (Yeshiva University)
Title: Equivalence of the Gravitational Three-Body Problem with Schrodinger's Equation: Solving the Three-Body Problem Using Methods of Quantum Mechanics
Time: Wednesday,  September 25, 2019, 12:00 pm
Location: Yeshiva University, 205 Lexington Ave, 6th Floor, Main Conference Room 
Abstract: The three-body problem of celestial mechanics is not solved to this day due to chaotic motions. We show that it can be solved for an interesting class of resonance orbits using methods of quantum mechanics. This is a surprising result since these fields are so different in their methodology. In fact, it seems the quantum mechanics approach is much easier. Real applications are described.
Web page: https://www.yu.edu/ug/math/colloquia-seminars

Event: Mathematical Physics Seminar
Speaker: Marco Lenci  (University of Bologna)
Title: Infinite-volume mixing and the case of one-dimensional maps with an indifferent fixed point  
Time: Wednesday,  August 28, 2019, 12:00 pm
Location: Yeshiva University, 205 Lexington Ave, 6th Floor, Main Conference Room 
Abstract: I will first discuss the question of mixing in infinite ergodic theory, which will serve as a motivation for the introduction of the notions of "infinite-volume mixing". Then I will focus on a prototypical class of infinite-measure-preserving dynamical systems: non-uniformly expanding maps of the unit interval with an indifferent fixed point. I will show how the definitions of infinite-volume mixing play out in this case. As it turns out, the most significant property, and the hardest to verify, is the so-called global-local mixing, corresponding to the decorrelation in time between global and local observables. I will present sufficient conditions for global-local mixing, which will cover the most popular examples of maps with an indifferent fixed point (Pomeau-Manneville and Liverani-Saussol-Vaienti). If time permits, I will also present some peculiar limit theorems that can be derived for these systems out of the property of global-local mixing. 
Web page: https://www.yu.edu/ug/math/colloquia-seminars