# PDE Seminar Fall 2014

### Fridays, 4:10pm, Stevenson Center 1307

Date: **Friday, August 29, 2014**

- Speaker:
**Yuanzhen Shao, Vanderbilt University** - Title: Continuous maximal regularity on manifolds with singularities and applications to geometric flows
- Abstract: In this talk, we study continuous maximal regularity theory for a class of degenerate or singular differential operators on manifolds with singularities. Based on this theory, we show local existence and uniqueness of solutions for several nonlinear geometric flows and diffusion equations on non-compact, or even incomplete, manifolds, including the Yamabe flow and parabolic p-Laplacian equations. In addition, we also establish regularity properties of solutions by means of a technique consisting of continuous maximal regularity theory, a parameter-dependent diffeomorphism and the implicit function theorem.

Date: **Friday, September 5, 2014**

- Speaker:
**Marcelo Disconzi, Vanderbilt University** - Title: Mathematical properties of effective potentials in string theory.
- Abstract: We study effective potentials coming from compactifications of string theory. We show that, under mild assumptions, such potentials are bounded from below in four dimensions, giving an affirmative answer to a conjecture proposed by Michael Douglas. We also derive some sufficient conditions for the existence of critical points, and establish their positivity in the case of slowly varying warp factors. All proofs and mathematical hypotheses are discussed in the context of their relevance to the physics of the problem.

Date: **Friday, September 19, 2014**

- Speaker:
**Vamsi P. Pingali, Johns Hopkins University** - Title: Weighted L^2-Extension of holomorphic functions from singular hypersurfaces.
- Abstract: I shall speak about my results on the following problem: “Given a holomorphic function (which is square-integrable with respect to some weighted measure) on a complex hypersurface of C^n, extend it in a square-integrable manner to C^n.” The techniques used involve the so-called “d-bar” PDE. This is joint work with Dror Varolin.

Date: **Friday, September 26, 2014**

- Speaker:
**Cameron Browne, Vanderbilt University** - Title: Basic Reproduction Number in Population Models with Periodic Forcing
- Abstract: Seasonality and periodic control measures may be important features of certain systems arising in epidemiology and ecology. I will give an overview of the basic reproduction number for structured population models in periodic environments. The definition of the basic reproduction number involves the spectral radius of an integral operator derived from the linearized partial differential equation system which describes the population dynamics. Then I will consider two examples in which calculating the reproduction number can provide insight on optimal timing of periodic interventions. Specifically, I study a within-host HIV model with combination antiviral drug treatment and a multi-patch epidemic model with periodic pulse vaccinations.

Date: ** Thursday, October 2, 2014 **(notice the date; this will be a Colloquium on a PDE topic)

- Speaker:
**Jared Speck, MIT** - Title: Shock Formation in Solutions to 3D Wave Equations
- Abstract: I will provide an overview of the formation of shock waves, developing from small, smooth initial conditions, in solutions to quasilinear wave equations in 3 spatial dimensions. I will first describe prior contributions from many researchers including F. John, S. Alinhac, and especially D. Christodoulou. I will then describe some results from my recent book, in which I show that for two important classes of wave equations, a necessary and sufficient, condition for the phenomenon of small-data shock-formation is the failure of S. Klainerman’s classic null condition. I will highlight some of the main ideas behind the analysis including the critical role played by geometric decompositions based on true characteristic hypersurfaces. Some aspects of this work are joint with G. Holzegel, S. Klainerman, and W. Wong.

Date: **Friday, October 31, 2014**

- Speaker:
**James Benn, Notre Dame University** - Title: L^2 Geometry of the Symplectomorphism Group
- Abstract: The group of Symplectic diffeomorphisms plays a role in Plasma dynamics analogous to the role played by the group of volume preserving diffeomorphisms in Hydrodynamics. I will discuss some features of the global geometry of the Symplectic diffeomorphism group equipped with the L^2 metric. In particular, the exponential map of the L^2 metric is a non-linear Fredholm map of index zero.

Date: **Friday, November 7, 2014**

- Speaker:
**Glenn Webb, Vanderbilt University** - Title: Nonlocal Phenomena in Partial Differential Equations.
- Abstract: Four examples of nonlocal phenomena in partial differential equations will be presented: (1) partial differential equations with time delay (non-local in time – the future depends not only on the present time, but on a history before the present time; (2) age structured population models (nonlocal in the boundary condition – offspring are born at age 0 from a mother with age in a specified age range); (3) cell-cell adhesion models (nonlocal in the transport term – cells have a spatial sensing radius, on the order of several cell diameters, that modulates their adhesion to other cells within their sensing radius); (4) interference phenomena in quantum mechanics (nonlocal in the probability density of spatial position – the detection of a quantum particle is determined only

probabilistically).

Date: **Friday, November 14, 2014**

- Speaker:
**Jeremy LeCrone, Kansas State University** - Title: Velocity of free boundaries for obstacle problems with time dependent data
- Abstract: I will present recent results (joint work with Ivan Blank) on the motion induced on free boundaries of the obstacle problem when either boundary data or the shape of the obstacle are allowed to vary in a time-dependent fashion. Starting with a basic introduction and motivation for the obstacle problem, I will motivate measure-theoretic bounds for the change in contact region under small perturbations and show how this measure-theoretic information leads to precise velocity formulas around regular portions of the free boundary. Application to the Hele-Shaw problem will also be discussed as time allows.

Date: **Friday, November 21, 2014**

- Speaker:
**Xinliang An, Rutgers University** - Title: Formation of Trapped Surfaces in General Relativity
- Abstract: The first is a simplified approach to Christodoulou’s monumental result which showed that trapped surfaces can form dynamically by the focusing of gravitational radiation from past null infinity. We extend the methods of Klainerman-Rodnianski, who gave a simplified proof of this result in a finite region. The second result extends the theorem of Christodoulou by allowing for weaker initial data but still guaranteeing that a trapped surface forms in the causal domain. In particular, we show that a trapped surface can form dynamically from initial data which is merely large in a scale-invariant way. The second result is obtained jointly with Luk

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