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
Amit Einav, University of Vienna
"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
Amit Einav, University of Vienna
"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
Jozsef Beck, Rutgers University
"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
Anna Vershynina, BCAM-Basque Center for Applied Mathematics, Spain
"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.
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