"Propagation of Chaos for a simple model of electric conduction with a Gaussian Thermostat"
Time: 2:00 PM
Location: Hill 705
Abstract: We will start from a system of N pointlike particles in a dispersing
billiard under the inuence of an electric field and a Gaussian
thermostat. We will argue that many interesting characteristic of the
system are preserved if one replace the determistic collision between
particles and obstacles with Poisson distributed random collision. We
will then show that the stochastic process obtained in this way propagate
chaos so that the one particle distribution satisfies a Boltzmann-like
equation.
Thursday, May 9th
Alessandro Giuliani, Universit^Sa Degli Studi Roma Tre
"Conformal Invariance and Central Charge In Non-Solvable Ising Models"
Time: 12:00 PM
Location: Hill 705
Abstract: We investigate a non-solvable ferromagnetic two-dimensional
Ising modeXSl with nearest neighbor plus weak finite range interactions.
We rigorously establish two properties of the critical theory: (1) We
prove and compute the existence of a scaling limit for the multipoint
energy correlations, as the lattice spacing "a" goes to zero and the
temperature goes to the critical one, with explicit bounds on the
finite-"a" corrections. (2) We prove that at the critical temperature
the finite size corrections to the free energy are universal, in the
sense that they are exactly independent of the interaction. The
corresponding central charge, defined in terms of the coeffcient of the
first subleading term to the free energy, as proposed by Affleck and
Blote-Cardy-Nightingale, is constant and equal to 1/2. These are two of
the very few cases where the predictions of confor- mal field theory can
be rigorously verified starting from a microscopic non solvable
statistical model.
Joint work with R. Greenblatt and V. Mastropietro.
THERE WILL BE A BROWN BAG LUNCH FROM 1-2PM. PLEASE JOIN US
Thursday, May 2nd
Amit Einav, Cambridge University
"Trends to equilibrium in Boltzmann's equation via Kac's model and the entropy method"
Time: 12:00 PM
Location: Hill 705
Abstract: One of the most influential equations in the kinetic theory of
gases is the so-called Boltzmann equation. While used widely in
practice, its irreversibility poses a huge mathematical difficulty as one
assumes that such equation arises in Newtonian settings where the laws
are all reversible. This raises an interesting mathematical question:
Can one achieve an irreversible system from a reversible one, in
macroscopic time scales? If so, how can it happen? In his 1956 paper,
Marc Kac presented a method to achieve such a thing by considering the
limit of a many particle jump process. Kac considered a model of N
indistinguishable particles, with one dimensional velocities, that
undergo a random binary collision process. Under a special property,
called chaoticity, Kac managed to show that when one takes the number of
particles to infinity, the limit of the first marginal of the N-particle
distribution function satisfies a caricature of the Boltzmann equation.
Kac's hope was that using his model and taking the number of particles to
infinity, one can learn properties of the Boltzmann equation -
specifically, the rate of convergence to equilibrium. In our talk we
will mention the spatially homogeneous Boltzmann equation and describe
Kac's model. We will discuss the so-called spectral gap problem, posed
by Kac in hope to achieve an exponential rate of decay to equilibrium in
the Boltzmann equation, and show why that method has failed. We will
then discuss the entropy and entropy-entropy production ratio problem,
extending Kac's spectral gap problem to a more natural functional, and
explain why in its full generality this method is still inadequate to
deal with the issue. We will also mention McKean model, an extension of
Kac's model where the velocities of the particles are allowed to be
d-dimensional, and mention why even in this more realistic setting the
entropy-entropy production ratio doesn't yield a better result. If we'll
have some time remaining we will briefly discuss possible connections
between the problem at hand and the concept of entropic chaos, one that
plays in interesting role on the entropy-entropy production problem.
Thursday, April 25th
Natan Andrei, Rutgers University
"An Exact Formalism For Quench Dynamics"
Time: 2:00 PM
Location: Hill 705
Abstract: We describe a formulation for studying the quench dynamics of in-
tegrable systems generalizing an approach by Yudson. We study the
evolution of Lieb-Liniger system, a gas of interacting bosons moving
on the continuous in nite line and interacting via a short range poten-
tial. The formalism allows us to quench the system from any initial
state. We nd that for any value of repulsive coupling independently
of the initial state the system asymptotes towards a strongly repulsive
gas, while for any value of attractive coupling, the system forms a
maximal bound state that dominates at longer times. In either case
the system equilibrates but does not thermalize, an e ect that is con-
sistent with prethermalization. If time permits I shall discuss also
the quench dynamics of the XXZ Heisenberg chain and of a mobile
impurity in an interacting Bose gas.
Thursday, April 25th
Vieri Mastropietro, University of Milano
"Conductivity In The XXZ Heisenberg Chain With Next To Nearest Neighbor Interaction"
Time: 12:00 PM
Location: Hill 705
Abstract: We consider a spin chain given by the XXZ model with a weak next to nearest neighbor perturbation which breaks its exact integrability.
We prove that such system has an ideal metallic behavior (in nite
conductivity), by rigorously establishing strict lower bounds on the
zero temperature Drude weight which are strictly positive.
THERE WILL BE A BROWN BAG LUNCH FROM 1-2PM. PLEASE JOIN US
Thursday, April 11th
Tobias Kuna, University of Reading
"The full infinite dimensional moment problem on semi-algebraic sets"
Time: 2:00 PM
Location: Hill 705
Abstract: In the recent years some progress has been made for the full moment problem on semi-algebraic sets. The latter are subsets of the d-dimensional Euclidean space given by polynomial constrains. In collaboration with Maria Infusion and Aldo Rota (Reading, UK) we generalized these results to semi-algebraic subsets of an infinite-dimensional space given by uncountable many constrains. We review the classical results and outline the proof. Then we demonstrate that the result gives new necessary and sufficient condition for the moment problem for random measures, random densities, point processes and other cases.
Thursday, April 11th
Nader Masmoudi, New York University
"Nonlinear stability and high frequency cascade for 2D Euler"
Time: 12:00 PM
Location: Hill 705
Abstract: We prove the global asymptotic stability of shear flows close to planar Couette flow in the 2D inviscid Euler equations.
That is, given an initial perturbation of Couette flow small in a suitable regularity class, the velocity converges strongly (in $L^2$) to a shear flow which is also close to Couette flow.
This phenomenon of strong convergence is commonly referred to as {inviscid damping}. The vorticity is asymptotic to an almost linear evolution and weakly converges as t goes to infinity. This is analogous to nonlinear Landau damping in the Vlasov equations and should most likely be called Orr damping as this was first studied by Orr in 1907.
This is a joint work with J. Bedrossian.
THERE WILL BE A BROWN BAG LUNCH FROM 1-2PM
Thursday, April 4th
Michael Aizenman, Princeton University
"Ergodicity Hypothesis breakdown in Random Schroedinger Operators"
Time: 2:00 PM
Location: Hill 705
Abstract: Under certain situations, Random Schroedinger operators may exhibit
energy regimes (phases) in which extended states are formed from
resonating local quasi-modes. The corresponding eigenstates are
“non-ergodic”, in the sense that they violate a heuristic
version of the equidistribution principle, yet they do not exhibit
Anderson localization. Such phases were initially encountered in the the
random Schroedinger operator on tree graphs (in a joint work with Simone
Warzel), and are also expected to show up in simplified versions of
many-particle systems, which are the subject of ongoing work (with
SW and Mira Shamis).
Thursday, April 4th
Ovidiu Costin, Ohio State University
"Constructive global analysis of differential systems"
Time: 12:00 PM
Location: Hill 705
Abstract: I will present a new method for proving global existence and uniqueness
of solutions of differential systems and for studying their properties
throughout their domain of existence. The method relies on recent results
on the behavior of functions at essential singularities, and it appears
to be quite general. I will illustrate the method on the stability
question of solitons in focusing NLS and on the recent proof of the
Dubrovin conjecture in integrable systems.
Work in collaboration with M. Huang and W. Schlag (U Chicago) and S.
Tanveer (OSU).
THERE WILL BE A BROWN BAG LUNCH FROM 1-2PM. PLEASE JOIN US
Thursday, March 28th
Jared Speck, Massachusetts Institute of Technology
"Stable Big Bang Formation in Near-FLRW Solutions to the Einstein-Scalar Field System"
Time: 2:00 PM
Location: Hill 705
Abstract: I will discuss some results that I recently obtained in collaboration with Igor Rodnianski. Our main result is a proof of stable Big Bang formation in small perturbations of the well-known %spatially flat cosmological Friedmann-Lema^{i}tre-Robertson-Walker (FLRW) solution to the Einstein-scalar field system. The FLRW solution is a special case of a family of spatially homogeneous, isotropic solutions that arise in cosmology. It models a scalar field evolving in a spacetime that expands as $t to infty$ and that collapses as $t downarrow 0.$ In particular, the FLRW solution contains a ``Big Bang'' singularity at $Sigma_0 := lbrace t = 0 rbrace.$ To study the perturbed solutions, we place data on a Cauchy hypersurface $Sigma_1$ that are close to the FLRW data (at time $1$) as measured by a Sobolev norm. No symmetry assumptions are made on the data. We then study the global behavior of the perturbed solution in the emph{collapsing} direction. We first show that the spacetime region of interest can be foliated by a family of spacelike Cauchy hypersurfaces $Sigma_t,$ $t in (0,1],$ of constant mean curvature $- frac{1}{3} t^{-1}.$ We then analyze the behavior of the solution as $t downarrow 0$ and provide a detailed description of its asymptotics. Our main conclusion is that the perturbed solution remains globally close to the FLRW solution and has approximately monotonic behavior. In particular, the perturbed solution also has a Big Bang singularity at $Sigma_0.$ More precisely, as $t downarrow 0,$ various curvature invariants uniformly blow-up and the volume of $Sigma_t$ collapses to $0.$ These blow-up results demonstrate the validity of Penrose's Strong Cosmic Censorship conjecture for the past half of the perturbed spacetimes. We have also shown that the same results hold for the stiff fluid matter model. From the point of view of analysis, our main results can be viewed as a proof of stable blow-up for an open set of solutions to a highly nonlinear elliptic-hyperbolic system. The most important aspect of our analysis is our identification of a new $L^2-$type emph{energy almost-monotonicity inequality} that holds for the solutions under consideration.
Thursday, March 28th
Konstantin Mischaikow, Rutgers University
"Characterizing Complex Spatiotemporal Systems Using Persistent Homology"
Time: 12:00 PM
Location: Hill 705
Abstract: Persistent homology is a fairly new algebraic topological tool that
can be used to quantify the ”shape” of a function. I will describe it
and discuss our attempts to use it to understand phenomena such as
Rayleigh-Benard convection and dense granular Media.
THERE WILL BE A BROWN BAG LUNCH FROM 1-2PM. PLEASE JOIN US.
Thursday, March 14th
Sergio Lukic, Institute for Advanced Study
"How organisms evolve to recognize and repress transposable elements"
Time: 2:00 PM
Location: Hill 705
Abstract: Transposable Elements (TEs) are the most widespread genetic parasites in
eukaryotic genomes. Almost half of the human genome is explicitly derived
from TE DNA. Many of these elements are remnants of past retroviral
infections of the germ line that have kept intact important parts of
their molecular replication machinery. Given the potential detrimental
consequences of TE activity, host mechanisms of defense have evolved to
identify and repress these transposons. For instance, recent progress has
shed light on several molecular mechanisms that employ trans-acting
repressors that recognize cis-acting sequences in a given TE, thereby
blocking its expression. In this talk, I will present novel work
regarding the evolution of these molecular mechanisms by which the host
is able to learn to recognize and silence newly invading TEs. In
particular, I will review basic aspects of the biology of piRNAs and show
how these repressors evolve by means of new mutations that arise after
their targets have invaded the host genome. Second, I will review the
biology of the zinc finger/KAP1 transcriptional repressor complex, and
show evidence that suggests that these repressors have been selected from
existing genetic variation generated before their targets invaded the
host genome. Finally, I will discuss some mathematical work motivated by
the study of the dynamical processes by which the host population is able
to produce and maintain a standing genetic variation of repressors.
Thursday, March 14th
Tatyana Shcherbina, Institute for Low Temperature Physics Ukr.Ac.Sci.
"Universality of the second mixed moment of the characteristic polynomials of the 1D Gaussian band matrices"
Time: 12:00 PM
Location: Hill 705
Abstract: We consider the asymptotic behavior of the second mixed moment of the
characteristic polynomials of the 1D Gaussian band matrices. Assuming
that the width of the band grows faster than $sqrt{N}$, where $N$ is a
matrix size, we show that this asymptotic behavior in the bulk of the
spectrum coincides with those for the Gaussian Unitary Ensemble.
THERE WILL BE A BROWN BAG LUNCH FROM 1-2PM. PLEASE JOIN US
Thursday, March 7th
David Ruelle, IHES
"Hydrodynamic Turbulence as a Problem in Non-equilibrium Statistical Mechanics"
Time: 12:00 PM
Location: Hill 705
Abstract: The problem of hydrodynamic turbulence is reformulated as a
heat flow problem along a chain of mechanical systems which describe
units of fluid of smaller and smaller spatial extent. These units are
macroscopic but have few degrees of freedom, and can be studied by the
methods of (microscopic) non-equilibrium statistical mechanics. The
fluctuations predicted by statistical mechanics correspond to the
intermittency observed in turbulent flows. Specifically, we obtain the
formula $$ zeta_p={pover3}-{1overlnkappa}lnGamma({pover3}+1) $$
for the exponents of the structure functions
($langle|Delta_rv|^pranglesim r^{zeta_p}$). The meaning of the
adjustable parameter $kappa$ is that when an eddy of size $r$ has
decayed to eddies of size $r/kappa$ their energies have a thermal
distribution. The above formula, with $(lnkappa)^{-1}=.32pm.01$ is
in good agreement with experimental data. This lends support to our
physical picture of turbulence, a picture which can thus also be used
in related problems.
Thursday, February 28th
John Stalker, Trinity College Dublin
"How much mass can you fit inside a sphere?"
Time: 2:00 PM
Location: Hill 705
Abstract: For almost half a century various researchers in General Relativity
have studied the question of how much mass can be fit inside a sphere
without creating a black hole or other singularity. The answer, not
surprisingly, depends on the matter model and more specifically on the
restrictions which it places on the energy-stress-momentum tensor. A
variety of special cases are of particular physical interest. I will
discuss a new method, developed with Paschalis Karageorgis, for solving such
problems. This method recovers all previously known results and leads
to a number of new ones.
Thursday, February 28th
Jeff Kahn, Rutgers University
"Thresholds and expectation thresholds"
Time: 12:00 PM
Location: Hill 705
Abstract: Thresholds for increasing properties are a central concern in probabilistic combinatorics and elsewhere. (An increasing property, say F, is a superset-closed family of subsets of some (here finite) set
X; the threshold question for such an F asks, roughly, about how
many random elements of X should one choose to make it likely that
the resulting set lies in F? For example: about how many random
edges from the complete graph Kn are typically required to produce
a Hamiltonian cycle?)
We'll discuss recent progress and lack thereof on a few
threshold-type questions, and try to say something about a ludicrously general conjecture of G. Kalai and the speaker to the effect that there is always a pretty good naive explanation for a threshold being what it is.
THERE WILL BE A BROWN BAG LUNCH FROM 1-2PM. PLEASE JOIN US
Thursday, February 21st
Percy Deift, Courant Institute of Mathematical Sciences
"Toeplitz matrices and Toeplitz determinants under the impetus of the Ising model (Part II)"
Time: 2:00 PM
Location: Hill 705
Abstract: Toeplitz matrices and Toeplitz determinants under the impetus of the
Ising model (Part II)
Thursday, February 21st
Percy Deift, Courant Institute of Mathematical Sciences
"Toeplitz matrices and Toeplitz determinants under the impetus of the Ising model (Part I)"
Time: 12:00 PM
Location: Hill 705
Abstract: The speaker will discuss the development of the Szego Strong Limit
Theorem for Toeplitz determinants, and its later generalizations, in
response to questions arising in the analysis of the Ising model of
statistical mechanics
This is joint work with Alexander Its and Igor Krasovsky.
THERE WILL BE A BROWN BAG LUNCH FROM 1-2PM. PLEASE JOIN US
Thursday, February 14th
A. Shadi Tahvildar-Zadeh, Rutgers University
"Dressing with Simple Poles Can Get You Naked: Integrability and Vesture for Harmonic Maps and Einstein's Equations"
Time: 2:00 PM
Location: Hill 705
Abstract: Vesture refers to the recipe for finding nontrivial exact solutions to a
nonlinear differential equation by “dressing” a known
solution (which
could be rather trivial) with one or more prescribed singularities. This
is in particular possible if the nonlinear equation is completely integrable. We will explain what that means, and provide examples from
the theory of harmonic maps, and General Relativity, to demonstrate
how this procedure works. We will in particular show how to obtain
the nontrivial and physically interesting hyperextreme Kerr metric,
which has a naked ring singularity at its center, from the trivial solution of Einstein Vacuum Equations, namely the Minkowski metric.
This is joint work with Shabnam Beheshti.
Thursday, February 14th
Sagun Chanillo, Rutgers University
"The Shape of Uniformly Rotating White Dwarf Stars"
Time: 12:00 PM
Location: Hill 705
Abstract: The shape of uniformly rotating, self gravitating fluids has occupied
the interest of mathematicians since the days of Newton and Maclaurin. Riemann, Jacobi, Riemann and Poincare and Chandrasekhar also
made significant contributions to the subject. We study the compressible case of fluids with an equation of state given by that for white dwarfs, that is of a degenerate electron gas. The aim in the lecture is
to classify the free boundary of these uniformly rotating masses of compressible, self-gravitating fluids.
This is a joint work with Georg Weiss
and has already appeared in J. Differential Equations, 253(2012).
THERE WILL BE A BROWN BAG LUNCH FROM 1-2PM. PLEASE JOIN US
Thursday, February 7th
Alan Demlow, University of Kentucky
"Convergence of adaptive finite element methods for nonstandard norms"
Time: 2:00 PM
Location: Hill 705
Abstract: Adaptive finite element methods are popular in computational science
and engineering because of their ability to automatically produce efficient solutions to partial differential equations. Numerical experiments have long indicated that such methods converge optimally under reasonable conditions. However, a satisfying theory confirming these practical observations has only been developed over the past
decade. Most such convergence results concern methods for controlling the (global) energy norm of the error, which is easiest to work with theoretically but not always the most relevant in practice. In
this talk I will survey recent progress in understanding convergence
behavior of adaptive methods for controlling “nonstandard” norms of
the error such as local energy and global L2 norms.
Thursday, February 7th
Haim Brezis, Rutgers University/University of Paris VI
"Henri Poincare, a founding father of the modern theory of PDEs"
Time: 12:00 PM
Location: Hill 705
Abstract: I will discuss some contributions of Henri Poincare which have had a
major impact on the development of PDEs in the 20th century; in
particular, the solution of the Laplace equation, the spectral theory
for the Laplacian, and much more. Poincare also had prophetic insights
about the role of PDEs within Mathematics.
THERE WILL BE A BROWN BAG LUNCH FROM 1-2PM. PLEASE JOIN US
Thursday, January 31st
Jae Kyoung Kim, University of Michigan
"Central Problems in Biological Clocks: Identifying Structures of Biochemical Networks and Understanding Functions of Biochemical Networks"
Time: 2:00 PM
Location: Hill 705
Abstract: Biological rhythms control important aspects of cell physiology in-
cluding circadian (daily) events, cell division, embryogenesis and DNA
damage repair. While recent experimental work has identified many
genes and proteins that are involved in biological clocks, identification
of entire biochemical network seems far from complete since current
experimental techniques require tremendous amount of work. On the
other hand, output of the networks, timecourses of genes and proteins
can be easily acquired with advances in technology. I will describe how
to use these timecourse data to reveal biochemical network by using
fixed point of iteration map. Moreover, the structures of biochemical
networks are tightly related with their functions. I will present two
different designs of biochemical network: one is optimized to maintain
a constant period over a wide range of conditions and the other is
optimized to tune their period depending on conditions.
Thursday, January 31st
Wilhelm Schlag, University of Chicago
"Large data dynamics for nonlinear dispersive PDEs"
Time: 12:00 PM
Location: Hill 705
Abstract: We will discuss recent work on wave evolutions for large data. Particular
emphasis will be placed on concentration compactness ideas.
Amongst others, we will describe a result for wave equations from R^3
minus the unit ball into the sphere S^3 where we can show that any solution
approaches the unique harmonic map in its degree class.
Joint work with Cote, Kenig, Lawrie, Nakanishi - in various combinations.
THERE WILL BE A BROWN BAG LUNCH FROM 1-2PM. PLEASE JOIN US
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