# PDE's and Applied Mathematics Seminar

## Fall 2013

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September 17, (CAS 130, 2:30-3:30 pm).
Wilfrid Gangbo (Georgia Institute of Technology )
will speak on "Optimal Transport and Large Number of Particles".

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Abstract: We introduce a concept of viscosity solutions of Hamilton-Jacobi equations in metric
spaces and in some cases relate it to viscosity solutions in the sense of differentials in the
Wasserstein space. Our study is motivated physical systems which consist of infinitely many
particles in motion (This is a joint work with Andzrej Swiech).

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September 24, (Leigh Hall 305, 3:00-4:00 pm). Sarah Wingfield (Wolfram Research, Inc.)
will give "An Overview of Mathematica for Education and Research"

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October 8, (CAS 130, 2:30-3:30pm). Artem Zvavitch (Kent State University)
will speak on "Uniqueness questions in Convex Geometry".

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Abstract: Classical theorems in Convex geometry (and Harmonic Analysis) tell us that a symmetric star-shaped body is uniquely determined by the volumes of its hyperplane sections, moreover, if in addition to symmetricity we would also require convexity, then volume of orthogonal projections can be used instead of sections. It is also well know that in both cases symmetricity is required. A very natural question is what information on section/projections we need to require to have uniqueness for non-symmetric case? Volume of projections and Sections? Volume of "maximal" hyperplane section in all fixed directions?
We will start this talk with the discussion on some open problems related to those questions. We will also present a couple, recently constructed, counterexamples: we show that in all dimensions higher then 2, there exists an asymmetric convex body of revolution all of whose maximal hyperplane sections have the same volume. In addition we will show that if we fix even dimension greater or equal then 4, then one can find two essentially different convex bodies such that the volumes of their maximal sections, central sections, and projections coincide for all directions. The talk is combination of joint works with R. Gardner, F. Nazarov, D. Ryabogin and V. Yaskin.

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October 21, (CAS 220D, 1:00-2:00pm). Yaniv Almog (Louisiana State University)
will speak on "Global stability of the normal state of superconductors under the effect of strong electric current."

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Abstract:
Consider a superconducting wire whose temperature is lower than the
critical one. When one flows a sufficiently strong current through the wire,
it is well known from experimental observation that the wire becomes resistive, behaving like a normal metal. We prove that the time-dependent
Ginzburg-Landau model anticipates this behavior. We first prove that, for
sufficiently strong currents, the semi-group associated with the model, be-
comes a contraction semi-group. Then, we obtain an upper bound for the
critical current where the semi-group becomes stable. We relate this current
to the resolvent of the linearized elliptic operator.
Joint work with Bernard Helffer

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October 22, (CAS 130, 2:30-3:30 pm). Benjamin Jaye (Kent State University)
will speak on
"Geometric properties of measures with bounded Riesz transform".

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Abstract:
In this talk, we shall describe an approach to studying the geometric properties of measures for which an associated Calderon-Zygmund operator is bounded. This problem is closely related to classifying the removable sets for Lipschitz continuous solutions of the fractional Laplacian equation. The results described are part of some joint work with Fedor Nazarov.

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October 29, (Student Union Meeting Room 312,2:00-3:00 pm) [Jointly with Research for Lunch Seminar] .
Peter Gordon (University of Akron) will speak on
"Gelfand type problem for two phase porous media".

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Abstract:
We consider a generalization of the Gelfand problem arising in a Frank-Kamenetskii theory of thermal explosion.
This generalization is a natural extension of the Gelfand problem to two phase materials, where, in contrast to classical
Gelfand problem which utilizes single temperature approach,
the state of the system is described by two different temperatures.
We show that similar to classical Gelfand
problem the thermal explosion occurs exclusively due to the absence of
stationary temperature distribution. We also show that the presence of
inter-phase heat exchange delays a thermal explosion.
Moreover, we prove that in the limit of infinite heat exchange between phases
the problem of thermal explosion in two phase porous media reduces to
classical Gelfand problem with renormalized constants.
This is a joint work with Vitaly Moroz ( Swansea University, UK).

### December 12, (CAS 220D, 2:00-3:00pm). Vitaly Moroz (Swansea University, UK)
will speak on "Asymptotic properties of ground states of a semilinear elliptic problem with a vanishing parameter".

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Abstract:
We consider a semilinear elliptic problem with a double-well type nonlinearity on the entire space.
When one of the parameters vanishes the ground state of the problem converges after a suitable rescaling
to the ground state of a limiting problem, however the form of the limiting equation depends on
whether the original problem contains a subcritical, critical or supercritical term.
We establish the precise asymptotic rate of the limiting rescaling and derive
asymptotic behaviour of the ground states at the origin and near infinity.
This is a joint work with Cyrill Muratov (NJIT).

Previous seminars:

Spring 2013

Fall 2012