The Analysis Seminar

Fall 2013

The seminar meets Wednesdays in room LN 2205 at 3:30 p.m. There are refreshments and snacks in the Anderson Reading Room at 3:15.

The seminar is partly funded as one of Dean's Speaker Series in Harpur College (College of Arts and Sciences) at Binghamton University.

Organizer: Xiangjin Xu

September 11th: Organizational meeting

September 18th: Paul Loya (Binghamton)
Title: The Casimir energy and its relation to zeta functions of Laplacians and to stretching/breaking manifolds.

Abstract: In 1948, Hendrik Casimir published a surprising formula involving an "energy from nowhere''. In this talk I will explain Casimir's energy and mathematics that can be used to explain the origin and meaning of the mysterious constants that appear in Casimir's formula.

October 9th: Marcelo Disconzi (Vanderbilt)
Title: On the well-posedness of relativistic viscous fluids

Abstract: Using a simple and well-motivated modification of the stress-energy tensor for a viscous fluid proposed by Lichnerowicz, we prove that Einstein's equations coupled to a relativistic version of the Navier-Stokes equations are well-posed in a suitable Gevrey class if the fluid is incompressible and irrotational. These last two conditions are given a suitable relativistic interpretation. We also derive a full set of equations, describing a relativistic fluid that is not necessarily incompressible or irrotational, which is well-suited for comparisons with the system of an inviscid fluid.
October 23rd: Gideon Simpson (Drexel)
Title: Numerical Analysis of Parallel Replica Dynamics and Metastability

Abstract: A figurative, and sometimes literal, bottleneck in the simulation of Molecular Dynamics (MD) models of metals, proteins and other atomistic systems is the presence of metastable regions in configuration space, where a trajectory for the system can be trapped for a relatively long time. A.F. Voter (LANL) introduced Parallel Replica Dynamics as one method for overcoming this obstacle by exploiting parallel computation. The gain in performance is achieved by a suitable coarse graining, in time, of the trajectory. I will present recent work, with M. Luskin, on rigorously justifying the algorithm for stochastic differential equations with favorable spectral properties (i.e., the generator of the associated Fokker-Plank equation is elliptic). Related work with T. L\`elievre and D. Aristoff, extending the algorithm to discrete in time Markov Chains, which may come from problems unrelated to MD, will also be presented. Open problems and ongoing work will also be discussed.

4:00-5:00pm, November 4th (Special Date and Time): Tsuyoshi Yoneda (Hokkaido)
Title: A differential geometric consideration on the Navier-Stokes flow and its numerical computation

Abstract: We constructed a regularity criterion of the Navier-Stokes equation in a supercritical function space with a streamline condition (Chan-Y, MAA, 2012). After the construction of the theorem, we realized two things:
1. The differential geometric consideration along streamlines is a very important idea, but there are not so many works in NS field.
2. We need to analyze the pressure term more and more, but up to now, it has been very slow.

In the above two points of view, we constructed a new type of blowup criteria (a sophistication of Beale-Kato-Majda's blowup criterion) in terms of a symmetric structure near maximum points (the paper is in preparation). By the above blowup criterion, we can say that ``the pressure is a nonlocal operator, but local behavior affects the local behavior itself strongly". In order to progress the local pressure analysis further, analyzing ``separation phenomena" is one of the suitable directions. In the beginning of 20th century, Prandtl proposed Boundary Layer Theory, and it has been developing extensively. In our study, creation of the reverse flow phenomena (in a differential geometric approach) is observed. More precisely, in 2D-NS with circular arc no-slip boundary conditions and diffusing laminar initial data (we can define them rigorously in differential geometry), topologically changing flow, namely, instability is observed.

November 27th: Thanksgiving break

December 11th :Peng Wu(Cornell)
Title: Einstein four-manifolds of three-positive curvature operator

Abstract: In this talk we will show that Einstein four-manifolds of three-positive curvature operator are isometric to $(S^4, g_0)$, $(\mathbb{R}P^4, g_0)$, or $(\mathbb{C}P^2, g_{FS})$. The tools we use are curvature decompositions and Weitzenbock formula.

December 17th-- Mechanical Engineering seminar series

Speaker: Richard Rand (Dept. of Mechanical and Aerospace Engineering, and Dept. of Mathematics, Cornell University)
Time:12:00- 1:00 pm, Tuesday
Location:Mechanical Engineering Conference Room- 1st floor, ES 1337, Engineering and Science Building (Murray Hill Road)
Title: Differential Equations with Fractional Derivatives

Abstract: What are fractional derivatives?

Fractional derivatives, for example the half-derivative of $t$, $D^{1/2}_t$, are a generalization of ordinary integer-order derivatives, in the same way that the factorial function $n!$ can be extended to non-integer values by the gamma function $Γ(n)$. In this seminar, we will introduce the fractional calculus, and will discuss the nonlinear dynamics of differential equations which contain fractional derivatives, for example: $\ddot{x}+D^{1/2}_t x=0$.

Biography

Rand joined the Cornell faculty in 1967 after receiving his doctorate from Columbia University. He was a visiting professor at the University of California at Berkeley in 1981 and at the University of California at Los Angeles in 1989. He is a fellow of the American Society of Mechanical Engineers. Current research work involves using perturbation methods and bifurcation theory to obtain approximate solutions to differential equations arising from nonlinear dynamics problems in engineering and biology. Recent projects involve NEMS (nano electrical mechanical systems); evolutionary dynamics; and differential-delay equations.

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