Colloquia and Seminars

Upcoming Events

Quantitative global-local mixing for accessible skew products

Speaker:  Paolo Giulietti (University of Pisa)
Time:  Wednesday, April 14, 2021, 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:  Skew products are important examples of partially hyperbolic systems. We will present decay of correlations results on skew products that are locally accessible such that the base is hyperbolic and the fiber space is the real line.  We will focus on quantitative estimates with respect to global-local mixing. We will show that the rate of decay of correlations is tied to the  ''low frequency behaviour'' of the spectral measure associated to our global observables.

This is a joint work with Andy Hammerlindl and Davide Ravotti.

Past Events

An effective equation to study Bose gases at all densities

Speaker:  Ian Jauslin (Princeton University)
Time:  Wednesday, April 7, 2021, 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:  I will discuss an effective equation, which is used to study the ground state of the interacting Bose gas. The interactions induce many-body correlations in the system, which makes it very difficult to study, be it analytically or numerically. A very successful approach to solving this problem is Bogolubov theory, in which a series of approximations are made, after which the analysis reduces to a one-particle problem, which incorporates the many-body correlations. The effective equation I will discuss is arrived at by making a very different set of approximations, and, like Bogolubov theory, ultimately reduces to a one-particle problem. But, whereas Bogolubov theory is accurate only for very small densities, the effective equation coincides with the many-body Bose gas at both low and at high densities. I will show some theorems which make this statement more precise, and present numerical evidence that this effective equation is remarkably accurate for all densities, small, intermediate, and large. That is, the analytical and numerical evidence suggest that this effective equation can capture many-body correlations in a one-particle picture beyond what Bogolubov can accomplish. Thus, this effective equation gives an alternative approach to study the low density behavior of the Bose gas (about which there still are many important open questions). In addition, it opens an avenue to understand the physics of the Bose gas at intermediate densities, which, until now, were only accessible to Monte Carlo simulations.

Indicators of quantum chaos and the transition from few- to many-body systems

Speaker:  Lea Santos (Yeshiva University)
Time:  Wednesday, March 24, 2021, 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:  Quantum chaos, especially when caused by particle interactions, is closely related with topics of high experimental and theoretical interest, from the thermalization of isolated systems to the difficulties to reach a localized phase, and the emergence of many-body quantum scars. In this talk, various indicators of quantum chaos will be compared, including level statistics, structure of eigenstates, matrix elements of observables, out-of-time ordered correlators, and the correlation hole (ramp). These indicators are then employed to identify the minimum number of interacting particles required for the onset of strong chaos in quantum systems with short-range and also with long-range interactions.

The essential coexistence phenomenon in Hamiltonian dynamics

Speaker:  Yakov Pesin (Penn State University)
Time:  Wednesday, March 17, 2021, 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:  I will outline a construction of a Hamiltonian system which demonstrates a "KAM-type picture" of coexistence of positive volume Cantor-like set of invariant KAM tori and surrounding "chaotic sea", i.e., an invariant set on which dynamics is ergodic with respect to the Liouville measure and has non-zero Lyapunov exponents.

Ballistic Capture Dynamics and Applications

Speaker:  Francesco Topputo (Politecnico di Milano)
Time:  Wednesday, March 10, 2021, 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:  Ballistic capture is a process through which a massless particle with initial positive Kepler energy can approach and orbit a primary in a totally natural way. By definition, this mechanism can take place when n-body dynamics are considered, n ≥ 3. Ballistic capture may arise in spacecraft, asteroids, and comets motion about moons, planets, and stars. In spacecraft trajectory design, the ballistic capture reduces the relative hyperbolic excess velocity upon arrival, which in turn makes it possible to design low energy transfer. This can be achieved through a wise exploitation of the natural dynamics in the Solar System, and is a paradigm shift compared to its classic Keplerian decomposition. Ballistic capture orbits have been receiving increased attention throughout the past two decades due to their flexibility in providing multiple insertion opportunities, their capability in reducing fuel requirements, and their increased launch windows.

In this lecture, a survey on ballistic capture orbits is given, together with a discussion on their application to practical problems. The talk will focus on the method used to derive the stable sets, which are sets of initial conditions that generate orbits satisfying a simple definition of stability. The methods to compute these sets in the circular and elliptic restricted three-body problems will be shown, as well as their implementation into a three-dimensional, full-ephemeris n-body problem. The manipulation of the stable and unstable sets to achieve orbits with prescribed behavior will be given. Applications involve interplanetary trajectory design, lunar missions, and asteroid retrieval scenarios.

Bio:  Dr. Francesco Topputo is an Associate Professor of Aerospace Systems at Politecnico di Milano, Italy, and holds a position as Visiting Researcher at TU Delft, The Netherlands. His core research activities involve space flight dynamics and control, autonomous navigation, interplanetary CubeSat mission and system design. He has been PI in 9 research projects, and leads a research group composed by 5 PhD students and 4 visiting PhD students. He has authored 45 peer-reviewed articles published in international journals and over 170 works in total. He is Associated Editor at the journals Advances in Space Research (Elsevier) and Astrodynamics (Springer).

Recent developments on the polynomial entropy of dynamical systems

Speaker:  Jean-Pierre Marco (Sorbonne Université - Université Pierre et Marie Curie)
Time:  Wednesday, March 3, 2021, 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:  Topological entropy is a well-known measure of the complexity of dynamical systems. Positive topological entropy can be seen as an indication of "chaotic behavior", which has been widely studied in the past fifty years. However, completely integrable Hamiltonian systems - which are at the very core of the development of perturbation theory - in general have zero topological entropy, while their behavior obviously present different scales of complexity. The geodesic flow on the sphere is certainly much simpler than the flow on a nondegenerate ellipsoid. It turns out that such different behaviors are properly discriminated by a finer invariant, the polynomial entropy. In this talk we will define the polynomial entropy and review some recent results, ranging from integrable Hamiltonian systems to complex dynamics on Kähler manifolds.

Quantum Computing: A Brief Tour With Mathematical Highlights

Speaker:  Jesse Berwald (Quantum Computing, Inc.)
Time:  Wednesday, February 24, 2021, 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:  The race is on to build a quantum computer that provides a clear advantage over classical computers. A quantum computer is a specialized device that leverages the properties of quantum mechanics for computational purposes. Unfortunately, much of the noise surrounding quantum computing can seem like hype, but the achievements are solidly real and the devices steadily improve. Amidst the physical hardware challenges there are many deep and exciting mathematical avenues to explore. I will give a high-level overview of different quantum computing paradigms from the hardware, software, and mathematical perspectives.

Bio:  Jesse Berwald holds a Ph.D. in Mathematics from Montana State University. He held postdoc positions at William & Mary (dynamical systems and topology) and the IMA (TDA) before becoming a data scientist at Target. In 2018 he moved to a role in quantum computing at D-Wave Systems. He is currently a Quantum R & D Engineer at Quantum Computing, Inc., where he develops software for QC-enabled hybrid computation. In his spare time he likes to climb mountains.

Expansions in the mixing local limit theorems for dynamical systems

Speaker:  Kasun Fernando (University of Toronto)
Time:  Wednesday, December 16, 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 this talk, we introduce higher-order expansions in the mixing local limit theorems for Birkhoff sums of chaotic probability preserving dynamical systems. We will discuss assumptions that lead to these expansions and show that general classes of dynamical systems like the Young towers with exponential return times satisfy these. Our techniques even work for unbounded observables with integrability order arbitrarily close to the optimal moment condition required in the iid setting.

This is joint work with Françoise Pène.

Around the stability of elliptic equilibria for real analytic Hamiltonian systems

Speaker:  Bassam Fayad (CNRS-Centre National de la Recherche Scientifique)
Time:  Wednesday, December 2, 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 4 or more degrees of freedom, we give explicit examples of real entire Hamiltonians having an equilibrium with an arbitrary chosen non-resonant frequency vector (with coordinates not all of the same sign) that are Lyapunov unstable and have divergent Birkhoff normal form. In all degrees of freedom larger or equal to 2, we give explicit examples of real entire Hamiltonians having an equilibrium with an arbitrary chosen non-resonant frequency vector and a divergent Birkhoff normal form.

Perturbative methods in space debris dynamics

Speaker:  Alessandra Celletti (University of Rome Tor Vergata)
Time:  Wednesday, November 25, 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:  Space debris are polluting the sky and represent a threat for operative satellites and space missions. Understanding their dynamics is of paramount importance. In particular, it is crucial to understand the location of the equilibria, the existence of regular or chaotic regions, and especially their stable and unstable behavior as some control parameters are varied. These studies might allow to develop mitigation, maintenance and control strategies based on mathematical investigations.

I will present results obtained recently about the dynamics of resonances (tesseral and secular resonances) as well as the stability of space debris in different regions of space around our planet. The stability will be analyzed through (i) normal forms to get the long-term behaviour of the orbits, (ii) Nekhoroshev's theorem to obtain exponential stability times. Using a simulator of breakup events, a classification of space debris after collision or explosion is obtained through a suitable computation of the so-called 'proper elements'.

This talk refers to different works made in collaboration with: I. De Blasi, C. Efthymiopoulos, C. Gales, C. Lhotka, G. Pucacco, T. Vartolomei.

A Feynman-Kac based numerical method for the exit time probability of a class of transport problems

Speaker:  Diego del-Castillo-Negrete (Oak Ridge National Laboratory)
Time:  Wednesday, November 18, 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:  The exit time probability, which gives the likelihood that an initial condition leaves a prescribed region of the phase space of a dynamical system at, or before, a given time, is arguably one of the most natural and important transport problems. Here we present an accurate and efficient numerical method for computing this probability for systems described by non- autonomous (time-dependent) stochastic differential equations (SDEs) or their equivalent Fokker-Planck partial differential equations. The method is based on the direct approximation of the Feynman-Kac formula that establishes a link between the adjoint Fokker-Planck equation and the forward SDE. The Feynman-Kac formula is approximated using the Gauss-Hermite quadrature rules and piecewise cubic Hermite interpolating polynomials, and a GPU accelerated matrix representation is used to compute the entire time evolution of the exit time probability using a single pass of the algorithm. The method is unconditionally stable, exhibits second order convergence in space, first order convergence in time, and it is straightforward to parallelize. Applications are presented to the advection diffusion of a passive tracer in a fluid flow exhibiting chaotic advection, and to the runaway acceleration of electrons in a plasma in the presence of an electric field, collisions, and radiation damping. Benchmarks against analytical solutions as well as comparisons with explicit and implicit finite difference standard methods for the adjoint Fokker-Planck equation are presented.

Diffusion of the Random Lorentz Gas

Speaker:  Christopher Lutsko (Rutgers University)
Time:  Wednesday, November 11, 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 this talk, I will present a result proving an Invariance principle for a point particle moving through an array of randomly distributed spherical scatterers in 3 dimensions (the so-called Lorentz gas). That is, we prove that the particle-trajectory converges in distribution to a Brownian motion, in the diffusive limit, provided we apply a simultaneous Boltzmann-Grad scaling. The goal for this talk, will be to illustrate a flavor of the proof, which relies on a novel coupling method and probabilistic estimates. In the remaining time, I will present two similar results proved using similar methodology, for the random wind-tree process and the 2 dimensional Lorentz gas in the presence of a magnetic field.

All new results presented in this talk are joint with Balint Toth.

Innovation in materials science via K-Theory and Non-Commutative Geometry

Speaker:  Emil Prodan (Yeshiva University)
Time:  Wednesday, November 4, 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:  Any aperiodic structure has an intrinsic degree of freedom, called the phason, which lives on a topological space, whose complexity has basically no limitations. The phason space augments the physical space and supplies virtual dimensions that can be used to access physics that usually occurs in very high dimensions. The Galilean invariant Hamiltonians (in the case of quantum systems) or the dynamical matrices (for classical metamaterials) defined over an aperiodic pattern belong to a specific C*-algebra, which in many cases can be computed explicitly. The K-Theory of this algebra supplies all the topological phases and model Hamiltonians hosted by such systems and the pairing between K-Theory and the so called cyclic cohomology supplies numerical invariants that underly the bulk-boundary principle. In many cases, these invariants can be shown to be robust against disorder.

Ref. [1] supplied an algorithmic method to design phason spaces with pre-defined topology. This together with the principles stated above enabled us to design new topological classes of metamaterials such as quasi-periodic [2], incommensurate [3] and twisted [4]. In this talk I will explain these principles in a general context and then I will exemplify using the three classes I just mentioned. For example, for the twisted bilayer shown in Fig. 1, the phason lives on a 2-torus, leading to a virtual total 4-dimensional space. Furthermore, the algebra that generates the Galilean invariant dynamical matrices is the non-commutative 4-torus. This together with its K-theory explains the spectral butterfly observed in the numerical simulations and gives quantitative description of the features seen in the Integrated Density of States. Furthermore, the full set of topological invariants associated to each of the gaps seen in Fig. 1 can be explicitly computed and the bulk-boundary principle can be stated and verified.
[1] E. Prodan, Y. Shmalo, The K-Theoretic bulk-boundary principle for dynamically patterned resonators, Journal of Geometry and Physics 135, 135-171 (2019).
[2] D. J. Apigo, K. Qian, C. Prodan, E. Prodan, Topological Edge Modes by Smart Patterning, Phys. Rev. Materials 2, 124203 (2018).
[3] W. Cheng, E. Prodan, C. Prodan, Demonstration of dynamic topological pumping across incommensurate acoustic meta-crystals, arXiv:2005.14066 (2020).
[4] M. Rosa, M. Ruzzene, E. Prodan, Topological gaps by twisting, arXiv:2006.10019 (2020).

Instability and Breakup Dynamics of the Leapfrogging Vortex Quartet

Speaker:  Roy Goodman (New Jersey Institute of Technology)
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.

Exponential energy growth due to slow periodic perturbations of quantum-mechanical systems

Speaker:  Dimitry Turaev (Imperial College London)
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.

On the Local Limit Theorem in Dynamical Systems

Speaker:  Zemer Kosloff (Hebrew University of Jerusalem)
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.

Domains of analyticity and expansions of singular perturbations of Hamiltonian systems - Part II (Numerical methods, Gevrey properties of expansions)

Speaker:  Adrian Perez Bustamante (Georgia Institute of Technology)
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.

Domains of analyticity and expansions of singular perturbations of Hamiltonian systems - Part I (rigorous non-perturbative methods and geometry of domain of analyticity)

Speaker:  Rafael de la Llave (Georgia Institute of Technology)
Time:  Wednesday, September 9, 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 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.

Continuation, bifurcation, and disintegration of homoclinic orbits in a restricted four body problem

Speaker:  Jason D Mireles-James (Florida Atlantic University)
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.

Continuation of relative equilibria in the n--body problem to spaces of constant curvature

Speaker:  Pablo Roldan (Yeshiva University)
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

The Kac Model and (Non-)Equilibrium Statistical Mechanics

Speaker:  Federico Bonetto (Georgia Institute of Technology)
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.

Arnold Diffusion and Stochastic Behaviour

Speaker:  Maciej Capiński (AGH University of Science and Technology, Krakow, Poland)
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.

Correlated randomly growing graphs

Speaker:  Miklos Racz (Princeton University)
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.

Perspectives on Mathematical Modeling in Population Biology

Speaker:  Howie Weiss (The Pennsylvania State University)
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.

An approach to solving dx/dt = ? .

Speaker:  Konstantin Mischaikow (Rutgers University)
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.

Effect of non-conservative perturbations on homoclinic and heteroclinic orbits.

Speaker:  Maxwell Musser (Yeshiva University)
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.

Hillel Furstenberg: a YU alumnus wins the Abel prize

Speaker:  Peter Nandori (Yeshiva University)
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.

Multiple Borel Cantelli Lemma

Speaker:  Dmitry Dolgopyat (University of Maryland)
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.

The spectral measure of a dynamical system

Speaker:  Suddhasattwa Das (Courant Institute of Mathematical Science, New York University)
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

A Family of Periodic Orbits to the 3-Dimensional Lunar Problem and Applications

Speaker:  Edward Belbruno (Yeshiva University)
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.

Topological Quantum Qudits: Principles and Simulations

Speaker:  Yingkai Liu (Yeshiva University)
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.

Random walks on tori and an application to normality of numbers in self-similar sets

Speaker:  Yiftach Dayan (Technion, Haifa, Israel)
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.

Parameterization of Invariant Manifolds and Connecting Orbits in Celestial Mechanics

Speaker:  Shane Kepley (Rutgers University, Department of Mathematics)
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.

On the breakdown of small amplitude breathers for the reversible Klein-Gordon equation

Speaker: Tere Seara (Universitat Politecnica de Catalunya, Barcelona, Spain)
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.

Large deviations for time-averaged diffusions in the small time limit

Speaker:  Dan Pirjol (Stevens Institute of Technology)
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]

The Bulk-Edge Correspondence from the Fredholm Perspective

Speaker:  Jacob Shapiro (Columbia University)
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.

Computing the quasi-potential for nongradient SDEs

Speaker:  Daisy Dahiya (National Institutes of Health)
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.

Growth of Sobolev norms in the nonlinear Schrödinger equation

Speaker:  Marcel Guardia (Universitat Politecnica de Catalunya, Barcelona, Spain)
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.

Equivalence of the Gravitational Three-Body Problem with Schrodinger's Equation: Solving the Three-Body Problem Using Methods of Quantum Mechanics

Speaker: Edward Belbruno (Yeshiva University)
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.

Infinite-volume mixing and the case of one-dimensional maps with an indifferent fixed point

Speaker: Marco Lenci  (University of Bologna)
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.

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