Sharp bounds for eigenvalues of triangles

Speaker: 

Professor Bartlomiej Siudeja

Institution: 

UIUC

Time: 

Tuesday, October 6, 2009 - 4:00pm

Location: 

RH 306

Eigenvalues of the Laplacian on triangular domains cannot be computed exactly, in general. But the triangles that extremize the first eigenvalue (the fundamental tone of the membrane) often turn out to be equilateral, or degenerate in some way. These special triangles give sharp eigenvalue bounds for the general case.

Among all triangles with fixed diameter, we prove the degenerate acute isosceles triangle minimizes the Neumann fundamental tone. In the other direction, if we fix perimeter (or area) then the equilateral triangle maximizes the Neumann fundamental tone. Our approach involves variational principles and geometric transformations of the domain, and relies on the explicit formulas for eigenfunctions of equilateral triangles and circular sectors. We also prove symmetry/antisymmetry for eigenfunctions of isosceles triangles.

Classical Geometry of curves and completely integrable systems

Speaker: 

Professor Gloria Mari-Beff

Institution: 

University at Wisconsin

Time: 

Tuesday, February 16, 2010 - 4:00pm

Location: 

RH 306

A geometric realization of an integrable system is an evolution of curves on a manifold M, invariant under the action of a group G, and such that it becomes the integrable system when the action of G is mod out. The best known geometric realization is the Vortex filament flow (VF), a flow of curves in Euclidean space which is invariant under the Euclidean group. The VF equation becomes the nonlinear Shrodinger equation when written in terms of the natural curvatures of the flow - Hasimoto proved this way the integrability of VF -. In this talk I will review what is known about the classical geometry of curves in homogeneous spaces and its relation to different types of integrable systems. In particular we will talk about how one can reduce different Hamiltonian structures to the space of curvatures (Euclidean, projective, conformal, etc) and how those reductions indicates the existence of biHamiltonian (integrable) systems. We will also describe how curvatures of Schwarzian type for curves in homogeneous spaces usually describe evolutions of KdV type. I will present background material so the talk should be accessible, at least in part, to different audiences.

Conformal Equivalence of Triangle Meshes

Speaker: 

Professor Ulrich Pinkall

Institution: 

TU Berlin

Time: 

Tuesday, March 10, 2009 - 4:00pm

Location: 

RH 306

We define a notion of conformal equivalence for discrete surfaces (surfaces composed of euclidean triangles). For example, multiplying the lengths of all edges incident with a single vertex by the same factor is considered to be a conformal change of metric. It turns out that finding a conformally equivalent flat metric on a given discrete surface amounts to minimizing a globally convex functional on the space of all metrics. This functional involves the Lobachevski function (known in the context of computing the volume of hyperbolic tetrahedra). This is not an accident, since surprisingly the whole theory is stongly related to hyperbolic geometry. There are important practical applications of our method to Computer Graphics in the context of texture mapping.

The Kahler-Ricci flow on Hirzebruch surfaces

Speaker: 

Professor Benjamin Weinkove

Institution: 

UC San Diego

Time: 

Tuesday, April 28, 2009 - 4:00pm

Location: 

RH 306

I will discuss the metric behavior of the Kahler-Ricci flow on Hirzebruch surfaces assuming that the initial metric is invariant under a maximal compact subgroup of the automorphism group. I will describe how, in the sense of Gromov-Hausdorff, the flow either shrinks to a point, collapses to P^1 or contracts an exceptional divisor. This confirms a conjecture of Feldman-Ilmanen-Knopf. This is a joint work with Jian Song.

Constant mean curvature foliations for isolated systems with general asymptotics

Speaker: 

Dr. Lan-Hsuan Huang

Institution: 

Stanford University

Time: 

Tuesday, May 5, 2009 - 4:00pm

Location: 

RH 306

We will discuss the existence and uniqueness of the foliations by stable spheres with constant mean curvature for asymptotically flat manifolds satisfying the parity condition at infinity. The concept of center of mass in general relativity will also be discussed. This work generalizes the earlier results of Huisken-Yau, R. Ye, and Metzger.

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