Mathematics Department - Mathematical Physics Seminar - Spring 2017

# Mathematical Physics Seminar - Spring 2017

### Organizer(s)

Joel Lebowitz, Michael Kiessling

## Thursday, April 27th

### " The Geometry of Rotating Neutron Stars and Relativistic Stars"

Time: 12:00 PM
Location: Hill 705
Abstract: In earlier work with Georg Weiss we classified the free boundary of white dwarf stars. Here ideas from Newtonian Physics and Special Relativity sufficed. Pulsars or Neutron stars are highly dense stars and require General Relativity for their description. Pulsars rotate exceptionally fast and mainly emit X-rays and were first observed in the late 1960's by radio astronomers Anthony Hewish, Martin Ryle, and Jocelyn Bell. We will describe a system of equations, a consequence of the Einstein equation, that describes the equilibrium state of a pulsar. We will next use these equations to investigate an appropriate notion of a free boundary associated with the pulsar.

This is joint work with Georg Weiss.

## Thursday, April 27th

### "Bogoliubov Corrections to the Hartree Dynamics"

Time: 2:00 PM
Location: Hill 705
Abstract: We consider the dynamics of N bosons in the mean-field limit. It is well known that to leading order the N-body Schroedinger dynamics can be approximated by the Hartree equation. This talk is about the Bogoliubov approximation, which is the next-to-leading order correction that also describes the fluctuations around the mean-field Hartree state. While the Hartree approximation gives convergence in the sense of reduced densities, the Bogoliubov approximation gives convergence in the sense of the L^2 norm of the N-body Hilbert space. Here, I will present a rigorous derivation of this approximation in the most simple mean-field scaling limit.

## Thursday, April 20th

### "Derivation of the Vlasov equation"

Time: 2:00 PM
Location: Hill 705
Abstract: The rigorous derivation of the Vlasov equation from Newtonian mechanics of N Coulomb-interacting particles is still an open problem. In the talk I will present recent results, where an N-dependent cutoff is used to make the derivation possible. The cutoff is removed as the particle number goes to infinity. Our result holds for typical initial conditions, only. This is, however, not a technical assumption: one can in fact prove deviation from the Vlasov equation for special initial conditions for the system we consider.

## Thursday, April 6th

### "Higher Correlation inequalities"

Time: 12:00 PM
Location: Hill 705
Abstract: The Fortuin-Kasteleyn-Ginibre (FKG) inequality asserts that any two increasing functions on a distributive lattice are positively correlated. This inequality is an important tool in many problems involving probability, combinatorics, and statistical mechanics.

Several years ago, building on ideas of D. Richards, I proposed a generalization of this inequality to any number of functions. I will recall this generalization, explain some ideas and connections, and describe the progress so far on the problem. The last part is easy, there has been almost no progress!

## Thursday, March 30th

### " Linear stability of Schwarzschild spacetime"

Time: 12:00 PM
Location: Hill 705
Abstract: I will discuss recent work, joint with Pei-Ken Hung and Mu-Tao Wang, on the linear stability of the Schwarzschild spacetime. Our method employs Hodge decomposition to split linearized solutions into closed and co-closed portions, respectively identified with even-parity and odd-parity solutions in the physics literature. For each portion, we derive Regge-Wheeler type equations for decoupled, gauge-invariant quantities at the linearized connection level. With the choice of an appropriate gauge, decay estimates on these decoupled quantities are used to establish decay of the linearized metric coefficients of the solution.

## Thursday, March 23rd

### "On a deterministic approach to the Vlasov limit for point particles with unregularized Coulomb/Newton interactions"

Time: 2:00 PM
Location: Hill 705
Abstract: I will discuss some work in progress, in collaboration with Michael Kiessling, about the Vlasov limit for systems of point particles with unregularized inverse square interactions (both repulsive and attractive). Recently, we have been able to obtain uniform bounds (with respect to N, the number of particles) for the force acting on each particle and its time derivative, under certain assumptions about the positioning of the N particles in phase space. We have also been able to prove that these "good" arrangements of particle are generic, and are preserved by the dynamics at least for some time. The new N-uniform bounds are a significant step forward toward a rigorous justification of Vlasov limit for these systems, which has been a long-standing open problem.

## Thursday, March 23rd

### "Quantum kinetic v.s. weak turbulence"

Time: 12:00 PM
Location: Hill 705
Abstract: Although many studies have been carried on to understand the Zakharov weak turbulence equation for capillary waves since its derivation in the 60’s, the question about the existence and uniqueness of solutions to the equation remains unanswered, due to the complexity of the equation. After the production of the first Bose-Einstein Condensates (BECs), there has been an explosion of physics research on the quantum kinetic theory associated to BECs and their thermal clouds. In our talk, we will briefly describe the connection between quantum kinetic and weak turbulence theories. We will also discuss our result trying to answer the existence and uniqueness questions of the Zakharov equation, based on our recent tools developed for quantum kinetic equations.

The talk will be based on the results described on the blog: On the Mathematics of Dispersive Waves and Quantum Gases.

BROWN BAG LUNCH BETWEEN 1:00 - 2:00PM

## Thursday, March 9th

### "Locality and the Uncertainty Principle: A Bell-type argument in the original EPR variables"

Time: 12:00 PM
Location: Hill 705
Abstract: In their 1935 paper, Einstein, Podolsky and Rosen considered two particles moving in opposite direction and with total momentum equal to zero. Then they argued that, if there is no action at a distance, measuring the position or the velocity of on elf the particle would allow us to know the corresponding value for the other particle. Yet ordinary quantum mechanics does not ascribe a precise position or velocity to either particle. Therefore, concluded Einstein Podolsky and Rosen, the quantum description of reality is incomplete.

In his 1951 book on quantum theory, Bohm reformulated that argument in terms of spin variables and it is using those variables (or the polarizations of photons) that Bell proved that some sort of actions at a distance actually do exist.

We will show that, thanks to a little known no hidden variable theorem, one can reach the same conclusion using the original position and momentum variables of Einstein, Podolsky and Rosen.

## Thursday, February 16th

### "A localization result for Dirac operators"

Time: 12:00 PM
Location: Hill 705
Abstract: The symbol map of a Fredholm Operator is carrying essential topological and geometrical information about the underline manifold. In this talk we study Dirac type operators involving a perturbation term. In particular we think of operators of the form ${cal D} + s{cal A} :Gamma(E) ightarrow Gamma(F)$ over a Riemannian manifold $(X, g)$ for special bundle maps ${cal A} : E ightarrow F$ and study their behavior as $s ightarrow infty$. There are two main aspects of localization being examined: First is the separation of the spectrum of this family of operators into low and high eigenvalues for large $s$. Second is the observation that eigenvectors corresponding to low eigenvalues $L^2$ concentrate near the singular set of the perturbation bundle map ${cal A}$. This gives a new localization formula for the index of $D$ in terms of the singular set of ${cal A}$.

## Friday, February 10th

Special Mathematical Physics Seminar

### "The Almost Cercignani’s Conjecture"

Time: 11:00 AM
Location: Hill 705
Abstract: THIS TALK POSTPONED FROM FEB 9TH DUE TO SNOW STORM.

The validity, and invalidity, of Cercignani’s Conjecture in Kac’s many particle model, is a prominent problem in the field of Kinetic Theory. In its heart, it is an attempt to find a functional inequality, which is independent of the number of particles in the model, that will demonstrate an exponential rate of convergence to equilibrium. Surprisingly enough, this simple conjecture and its underlying functional inequalities contain much of the geometry of the process, and any significant advances in its resolution involves intradisciplinary approach. In this talk I will present recent work with Eric Carlen and Maria Carvahlo, where we have defined new notions of chaoticity on the sphere and managed to give conditions under which an ‘almost’ conjecture is valid. With that in hand, I will show how Kac’s original hope to conclude a rate of decay for his model's limit equation from the model itself, is achieved.

## Thursday, February 9th

### "The Almost Cercignani’s Conjecture"

Time: 12:00 PM
Location: Hill 705
Abstract: The validity, and invalidity, of Cercignani’s Conjecture in Kac’s many particle model, is a prominent problem in the field of Kinetic Theory. In its heart, it is an attempt to find a functional inequality, which is independent of the number of particles in the model, that will demonstrate an exponential rate of convergence to equilibrium. Surprisingly enough, this simple conjecture and its underlying functional inequalities contain much of the geometry of the process, and any significant advances in its resolution involves intradisciplinary approach.

In this talk I will present recent work with Eric Carlen and Maria Carvahlo, where we have defined new notions of chaoticity on the sphere and managed to give conditions under which an ‘almost’ conjecture is valid. With that in hand, I will show how Kac’s original hope to conclude a rate of decay for his model's limit equation from the model itself, is achieved.

## Thursday, February 2nd

### "Dynamical systems: polygon billiards and the geodesic flow on the cube surface"

Time: 12:00 PM
Location: Hill 705
Abstract: One of the simplest dynamical systems is the square billiard, which has a complete theory now. If we change the square to a (say) rhombus, we know much, much less, including even the case of the simplest 60-120 degree rhombus. Similarly, the cube surface consists of 6 squares, but again we know much, much less about the geodesic flow on the cube surface. One good reason is the appearance of singularities, or chaos" (=highly sensitive dependence on the intial condition), which is missing in the square billiard.

Wth my Ph.D. student Michael Donders recently we made some progress in these long-standing open problems. In my lecture I will report on these new results.

## Thursday, January 26th

### "Quantum analogues of geometric inequalities for Information Theory"

Time: 12:00 PM
Location: Hill 705
Abstract: Geometric inequalities, such as entropy power inequality or the isoperimetric inequality, relate geometric quantities, such as volumes and surface areas. Classically, these inequalities have useful applications for obtaining bounds on channel capacities, and deriving log-Sobolev inequalities. In my talk I provide quantum analogues of certain well-known inequalities from classical Information Theory, with the most notable being the isoperimetric inequality for entropies. The latter inequality is useful for the study of convergence of certain semigroups to fixed points. In the talk demonstrate how to apply the isoperimetric inequality for entropies to show exponentially fast convergence of quantum Ornstein-Uhlenbeck (qOU) semigroup to a fixed point of the process. The inequality representing the fast convergence can be viewed as a quantum analogue of a classical Log-Sobolev inequality.

This page was last updated on February 09, 2016 at 10:04 am and is maintained by webmaster@math.rutgers.edu.