Upcoming events:

Event:       Mathematical Physics Seminar
Speaker:  Roy Goodman (New Jersey Institute of Technology) 
Title:         Instability and Breakup Dynamics of the Leapfrogging Vortex Quartet
Time:        Wednesday,  October 28, 1:30-2:30 pm (EST)
Virtual Tea Time:  After the lecture we will host a `virtual tea time' to allow participants to interact informally with the speaker and with each other. 
Location:  Zoom Videoconference https://yeshiva-university.zoom.us/j/94745652228
Abstract:  We investigate the stability of a one-parameter family of periodic solutions of the four-vortex problem known as `leapfrogging' orbits. These solutions, which consist of two pairs of identical yet oppositely-signed vortices, were known to Gröbli (1877) and Love (1883), and can be parameterized by a dimensionless parameter α related to the geometry of the initial configuration. Simulations by Acheson (2000) and numerical Floquet analysis by Tophøj and Aref (2012) both indicate, to many digits, that the bifurcation occurs when 1/α=ϕ^2, where ϕ is the golden ratio. These numerical studies indicated a sequence of behaviors that emerge as this parameter is further decreased, leading to the disintegration of the leapfrogging orbit into a pair of dipoles that escape to infinity along transverse rays.
This study has two objectives. The first is to rigorously explain the origin of this remarkable bifurcation value. The second is to understand the sequence of transitions in the phase space of the system that allows for the emergence of the various behaviors. While the first objective is essentially linear, finding the answer requires applying several tricks from the classical mechanics toolkit. The second objective is inherently nonlinear, and our approach involves both analysis and numerics. In particular, we make use of the recently developed technique of Lagrangian descriptors to visualize the phase space structures, including invariant manifolds. This work forms the dissertation research of my recently-graduated student Brandon Behring.
Web page: https://www.yu.edu/ug/math/colloquia-seminars


Past events:

Event:       Mathematical Physics Seminar
Speaker:   Dimitry Turaev (Imperial College London)
Title:         Exponential energy growth due to slow periodic perturbations of quantum-mechanical systems
Time:        Wednesday,  October 21, 1:30-2:30 pm (EST)
Virtual Tea Time:  After the lecture we will host a `virtual tea time' to allow participants to interact informally with the speaker and with each other. 
Location:  Zoom Videoconference https://yeshiva-university.zoom.us/j/94745652228
Abstract:  One of the basic principles of quantum mechanics is that quantum numbers are preserved when the parameters of the system change sufficiently slowly. In particular, a slow cyclic (time-periodic) variation of the Hamiltonian operator is expected to return the system to the initial eigenstate (up to a phase change) after each period, for many periods. We show that this principle is violated if an additional quantum integral is created and destroyed during each cycle of the variation of the Hamiltonian. The adiabatic processes with this property lead to a non-trivial permutation of the eigenstates (of the instantaneous Hamiltonian) after each cycle and to the exponential growth of energy. We also show how the state of a quantum-mechanical system can be controlled with an arbitrarily good precision by an arbitrarily slow change of system parameters.
Web page: https://www.yu.edu/ug/math/colloquia-seminars

Event:       Mathematical Physics Seminar
Speaker:   Zemer Kosloff (Hebrew University of Jerusalem)
Title:         On the local limit theorem in dynamical systems 
Time:        Wednesday, September 23, 2020, 1:30-2:30 pm (EST)
Virtual Tea Time:  After the lecture we will host a `virtual tea time' to allow participants to interact informally with the speaker and with each other. 
Location:  Zoom Videoconference https://yeshiva-university.zoom.us/j/94745652228
Abstract:  In 1987, Burton and Denker proved the remarkable result that in every aperiodic dynamical system (including irrational rotations for example) there is a square integrable, zero mean function such that its corresponding time series satisfies a CLT.  Subsequently, Volny showed that one can find a function which satisfies the strong (almost sure) invariance principle. All these constructions resulted in a non-lattice distribution. 
In a joint work with Dalibor Volny we show that there exists an integer valued cocycle which satisfies the local limit theorem.
Web page: https://www.yu.edu/ug/math/colloquia-seminars

Event:       Mathematical Physics Seminar
Speaker:   Adrian Perez Bustamante (Georgia Institute of Technology)
Title:         Domains of analyticity and expansions of  singular perturbations of Hamiltonian   systems - Part II (Numerical methods, Gevrey properties of expansions) 
Time:        Wednesday, September 16, 2020, 1:30-2:30 pm (EST)
Virtual Tea Time:  After the lecture we will host a `virtual tea time' to allow participants to interact informally with the speaker and with each other. 
Location:  Zoom Videoconference https://yeshiva-university.zoom.us/j/94745652228
Abstract:  We consider the problem of following quasi-periodic  tori in perturbations of Hamiltonian systems which involve friction and external forcing. 
In a first goal, we use different numerical methods (Pade extensions, Newton continuation till boundary, extrapolations, etc.) to obtain numerically the domain of convergence. We also study the properties of the asymptotic series of the solution. In a second goal we study rigorously the(divergent) series of formal expansions of the torus  obtained using Lindstedt method.   We show that, for some systems in the literature,  the series is Gevrey. 
We hope that the method can be of independent interest: We develop KAM estimates for the divergent series. In contrast with the regular KAM method, we loose control of all the domains, so that there is no convergence, but we can generate enough control to show that the series is Gevrey. 
This is joint work with R. Calleja and R. de la Llave
Web page: https://www.yu.edu/ug/math/colloquia-seminars

Event:       Mathematical Physics Seminar
Speaker:   Rafael de la Llave (Georgia Institute of Technology)
Title:         Domains of analyticity and expansions of  singular perturbations of Hamiltonian   systems - Part I (rigorous non-perturbative methods and geometry of domain of analyticity) 
Time:        Wednesday, September 9, 2020, 1:30-2:30 pm (EST)
Location:  Zoom Videoconference https://yeshiva-university.zoom.us/j/94745652228
Virtual Tea Time:  After the lecture we will host a `virtual tea time' to allow participants to interact informally with the speaker and with each other. 
Abstract:  In Celestial Mechanics, Engineering systems, one has  often to consider Hamiltonian mechanical systems subject to a small non-conservative forces  (small internal friction and small forces).
From the point of view of persistence of periodic  and quasi-periodic orbits, this is a very singular limit  since we expect that few such orbits persist (in Hamiltonian  systems, positive measures of quasi-periodic orbits  persist). We consider the case when the friction is proportional  to the velocity, which appears often in real applications. In such a case, the evolution takes the symplectic form into a multiple of itself and this remarkable geometric structure (called conformally symplectic)  gives rise to  cancellations  that lead to non-perturbative results and algorithms. 
Given an analytic family  of systems with friction and forcing (conformally symplectic), so that the unperturbed system is Hamiltonian and has a KAM torus, we show 
that there is a complex domain D   of the perturbative parameter for which there is a KAM torus. 
The domain D  does not  include any ball centered at the origin nor any sector of width bigger than $\pi/a$  (a is an integer  related to the perturbation expansion of the family). We give arguments to show that these domains may be optimal in "generic" systems. 
This is joint work with R. Calleja and A. Celletti.
Web page: https://www.yu.edu/ug/math/colloquia-seminars


YEAR  2019-2020

Event:       Mathematical Physics Seminar
Speaker:  Jason D Mireles-James (Florida Atlantic University)
Title:         Continuation, bifurcation, and disintegration of homoclinic orbits in a restricted four body problem
Time:        Wednesday, June 17, 2020, 11:00 am (EST)
Location:  Zoom Videoconference   https://yeshiva-university.zoom.us/j/96836645415
Abstract:  This talk is concerned with homoclinic phenomena in a two parameter family of equilateral restricted four body problems.  I'll begin by reviewing what is known about the equilibrium (libration point) structure of the problem, where the sharpest results are due to Barros and Leandro (2014). Then, following recent work with S. Kepley, I'll discuss a family of six ``fundamental'' homoclinic orbits which appear to play an important role in organizing homoclinic phenomena in the problem in the special case of equal masses.  Finally I'll discuss some recent work with W. Hetebrij which tries to understand, through a series of deliberate numerical explorations, what becomes of these fundamental homoclinic orbits as parameters are varied.  The computations utilize a number of standard tools including numerical continuation schemes for periodic and homoclinic orbits orbits, as well as formal series expansions for center stable/unstable manifolds.
Web page: https://www.yu.edu/ug/math/colloquia-seminars

Event:       Mathematical Physics Seminar
Speaker:   Pablo Roldan (Yeshiva University)
Title:         Continuation of relative equilibria in the n--body problem to spaces of constant curvature
Time:        Wednesday, May 27, 2020, 11:00 am (EST)
Location:  Zoom Videoconference   
Abstract:  The curved n--body problem is a natural extension of the planar Newtonian n--body problem to surfaces of non-zero constant curvature. We prove that all non-degenerate relative equilibria of the planar problem can be continued to spaces of constant curvature κ, positive or negative, for small enough values of this parameter. We also compute the extension of some classical relative equilibria to curved spaces using numerical continuation. In particular, we extend Lagrange's triangle configuration with different masses to both positive and negative curvature spaces. This is joint work with A. Bengochea, C. Garcia-Azpeitia and E. Perez-Chavela.
Attached figure: Relative equilibrium corresponding to different masses on a hyperboloid (negative curvature). orbneg123_0.pdf
Web page: https://www.yu.edu/ug/math/colloquia-seminars

Event:       Mathematical Physics Seminar
Speaker:   Federico Bonetto (Georgia Institute of Technology)
Title:         The Kac Model and (Non-)Equilibrium Statistical Mechanics
Time:        Wednesday, May 20, 2020, 11:00 am (EST)
Location:  Zoom Videoconference   
Abstract:  In 1959 Mark Kac introduced a simple model for the evolution of a gas of hard spheres undergoing elastic collisions. The main simplification consisted in replacing deterministic collisions with random Poisson distributed collisions. It is possible to obtain many interesting results for this simplified dynamics, like estimates on the rate of convergence to equilibrium and validity of the Boltzmann equation. The price paid is that this system has no space structure. I will review some classical results on the Kac model and report on an attempt to reintroduce some form of space structure and non-equilibrium evolution in a way that preserve the mathematical tractability of the system.
Web page: https://www.yu.edu/ug/math/colloquia-seminars

Event:       Mathematical Physics Seminar
Speaker:   Maciej Capiński (AGH University of Science and Technology, Krakow, Poland)
Title:          Arnold Diffusion and Stochastic Behaviour 
Time:         Wednesday, May 13, 2020, 11:00 am (EST)
Location:  Zoom Videoconference   
Abstract:   We will discuss a construction of a stochastic process on energy levels in perturbed Hamiltonian systems. The method follows from shadowing of dynamics of two coupled horseshoes. It leads to a family of stochastic processes, which converge to a Brownian motion with drift, as the perturbation parameter converges to zero. Moreover, we can obtain any desired values of the drift and variance for the limiting Brownian motion, for appropriate sets of initial conditions. The convergence is in the sense of the functional central limit theorem. We give an example of such construction in the planar elliptic restricted three-body problem. 
Web page: https://www.yu.edu/ug/math/colloquia-seminars

Event:       Mathematical Physics Seminar
Speaker:   Miklos Racz (Princeton University)
Title:          Correlated randomly growing graphs
Time:         Wednesday, May 6, 2020, 11:00 am (EST)
Location:  Zoom Videoconference   
Abstract:   I will introduce a new model of correlated randomly growing graphs and discuss the questions of detecting correlation and estimating aspects of the correlated structure. The model is simple and starts with any model of randomly growing graphs, such as uniform attachment (UA) or preferential attachment (PA). Given such a model, a pair of graphs $(G_1, G_2)$ is grown in two stages: until time $t_{\star}$ they are grown together (i.e., $G_1 = G_2$), after which they grow independently according to the underlying model. We show that whenever the seed graph has an influence in the underlying graph growth model---this has been shown for PA and UA trees and is conjectured to hold broadly---then correlation can be detected in this model, even if the graphs are grown together for just a single time step. We also give a general sufficient condition (which holds for PA and UA trees) under which detection is possible with probability going to $1$ as $t_{\star} \to \infty$. Finally, we show for PA and UA trees that the amount of correlation, measured by $t_{\star}$, can be estimated with vanishing relative error as $t_{\star} \to \infty$. This is based on joint work with Anirudh Sridhar.
Web page: https://www.yu.edu/ug/math/colloquia-seminars

Event:       Mathematical Physics Seminar
Speaker:   Howie Weiss (The Pennsylvania State University)
Title:          Perspectives on Mathematical Modeling in Population Biology
Time:         Wednesday, April 29, 2020, 11:00 am (EST)
Location:  Zoom Videoconference   
Abstract:  In this talk, we discuss five types of mathematical models that provide insights into biological questions. We illustrate each type, some with an example from our recent research. Along the way, we show that models can be more useful when they don’t fit the data, than when they do.
Web page: https://www.yu.edu/ug/math/colloquia-seminars

Event:       Mathematical Physics Seminar
Speaker:   Konstantin Mischaikow (Rutgers University)
Title:          An approach to solving dx/dt = ? . 
Time:         Wednesday, April 22, 2020, 11:00 am (EST)
Location:  Zoom Videoconference   
Abstract:  The life sciences provide archetypical examples of nonlinear systems for which an accurate understanding dynamics is essential, but for which models derived from first principles are not available. This implies that an analytic expression of a nonlinearity is typically chosen based on heuristics or simplicity of evaluation.  As a consequence parameters do not have an intrinsic physical basis, but bifurcation theory tells us that in general the invariant sets of a dynamical system are  parameter dependent.  Furthermore experimental measurements of variables tend to be quantified on log scales.  This is not a setting for which the classical theory of dynamical systems was designed to address. With these challenges in mind I will outline an approach to dynamics based on order theory and algebraic topology that allows us to consider large classes of differential equations over large regions of parameter space and derive rigorous results.  I will use gene regulatory networks to provide a concrete example of how this approach can be applied.
Web page: https://www.yu.edu/ug/math/colloquia-seminars

Event:       Mathematical Physics Seminar
Speaker:   Maxwell Musser (Yeshiva University)
Title:          Effect of non-conservative perturbations on homoclinic and heteroclinic orbits. 
Time:         Wednesday, April 1, 2020, 11:00 am (EST)
Location:  Zoom Videoconference  https://yeshiva-university.zoom.us/j/112170926
Abstract:  The motivation of this work comes from astrodynamics. Consider a spacecraft traveling between the Earth and the Moon. Assume that the spacecraft follows a zero-cost orbit by coasting along the hyperbolic invariant manifolds associated to periodic orbits near the equilibrium points, at some fixed energy level. We would like to make a maneuver  (impulsive or low thrust) in order to jump to the hyperbolic invariant manifold corresponding to a different energy level. Mathematically, turning on the thrusters amounts to a adding a small, non-conservative, time-dependent perturbation to the original system. Given such an explicit perturbation, we would like to estimate its effect on the orbit of the spacecraft. We study this question in the context of two simple models: the pendulum-rotator system, and the planar circular restricted three-body problem. Homoclinic/heteroclinic excursions can be described via the scattering map, which gives the future asymptotics of an orbit as a function of the past asymptotics. We add a time-dependent, non-conservative perturbation, and provide explicit formulas, in terms of convergent integrals, for the perturbed scattering map. This is based on joint work with M. Gidea and R. de la Llave.
Web page: https://www.yu.edu/ug/math/colloquia-seminars

Event:       Mathematical Physics Seminar
Speaker:   Peter Nandori (Yeshiva University)
Title:          Hillel Furstenberg: a YU alumnus wins the Abel prize
Time:         Wednesday, March 25, 2020, 11:00 am (EST)
Location:  Zoom Videoconference  https://yeshiva-university.zoom.us/j/568203799
Abstract:  The Abel prize is awarded annually by the Norwegian Academy of Science and Letters and is considered the most prestigious award in mathematics. In 2020, the prize goes to Hillel Furstenberg (Hebrew University of Jerusalem)​ and Gregory Margulis (Yale University) for "pioneering the use of methods from probability and dynamics in group theory, number theory and combinatorics". Furstenberg graduated with B.A. and M.S. degrees from Yeshiva University in 1955. In this talk, we review some of the fascinating results of Professor Furstenberg. We will discuss his ergodic theoretical approach to some problems in number theory (Furstenberg multiple recurrence and correspondence principle) and the Furstenberg criterion for the positivity of the largest Lyapunov exponent. The first part of the talk will be accessible for advanced undergraduate students.
Web page: https://www.yu.edu/ug/math/colloquia-seminars

Event:       Mathematical Physics Seminar
Speaker:   Dmitry Dolgopyat (University of Maryland) 
Title:         Multiple Borel Cantelli Lemma
Time:        Wednesday, March 4, 2020, 11:00 am
Location:  Yeshiva University, 215 Lexington Ave, 6th Floor, Conference Room 628
Abstract:  The classical Borel Cantelli Lemma provides necessary and sufficient conditions for infinitely many rare events to occur. However, when the infinite sequence of events does occur, the Borel Cantelli Lemma does not tell us how well separated in time whose occurrences are. In this talk we discuss the question when a fixed number r of rare events happen at the same scale for chaotic systems. This problem is intermediate between the standard Borel Cantelli regime and Poisson regime. The talk is based on the joint work with Bassam Fayad and Sixu Liu.
Web page: https://www.yu.edu/ug/math/colloquia-seminars  

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  

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