UCR Differential Geometry Sem.
The Schedule for the Fall of 2020, Fridays 11--11:50AM, starting on Oct. 3.
Email fred "at" math.ucr.edu for the zoom link,
Unless annouced otherwise these talks will be recorded. and posted here.
October 2---Liz Stanhope, Lewis and Clark College, "Eigenvalues and Orbifolds",
Abstract:
The best-known question in spectral geometry is “can you hear the shape of a drum?” Here we take the drum to be a Riemannian manifold, and the drum’s sound to be the eigenvalue spectrum of the associated Laplace operator. The challenge is to deduce properties of the manifold from its Laplace spectrum. After an introduction to this problem, and a few others that are also motivated by physical situations, we will turn our attention to Riemannian orbifolds. An orbifold is a mildly singular generalization of a manifold. We will discuss progress in understanding the extent to which we can ‘hear’ the singularities of an orbifold.
October 9--Anusha Krishnan, Syracuse University, "The prescribed Ricci curvature problem on manifolds with symmetry"
Abstract:
Given a symmetric 2-tensor T on a manifold M, can T be realized as the Ricci curvature of some Riemannian metric on M? We will discuss some classical results in the field, as well as more recent work under certain symmetry assumptions.
October--16--Guofang Wei, UCSB,
Fundamental Gap Estimate in Hyperbolic Space
Abstract:
In their celebrated work, B. Andrews and J. Clutterbuck proved the fundamental gap conjecture that difference of first two eigenvalues of the Laplacian with Dirichlet boundary condition on convex domain with diameter D in the Euclidean space is greater than or equal to $3\pi^2/D^2$. In several joint works with X. Dai, Z. He, S. Seto, L. Wang (in various subsets) the estimate is generalized, showing the same lower bound holds for convex domains in the unit sphere. In sharp contrast, in recent joint work with T. Bourni, J. Clutterbuck, A. Stancu, X. Nguyen and V. Wheeler, we prove that the product of the fundamental gap with the square of the diameter can be arbitrarily small for convex domains of any diameter in hyperbolic space.
October23---Orsola Capovilla-Searle, Duke,
Weinstein handle decompositions of complements of toric divisors in toric 4 manifolds
Abstract:
We consider toric 4 manifolds with certain toric divisors that have normal crossing singularities. The normal crossing singularities can be smoothed, changing the topology of the complement. In specific cases this complement has a Weinstein structure, and we develop an algorithm to construct a Weinstein handlebody diagram of the complement of the smoothed toric divisor.
Weinstein manifolds have a handle decomposition compatible with the symplectic structure. Examples of manifolds with Weinstein structures are Stein manifolds, and cotangent bundles of closed smooth manifolds. The algorithm we construct more generally gives a Weinstein handlebody diagram for Weinstein 4-manifolds constructed by attaching 2 handles to T*S for any surface S, where the 2 handles are attached along the conormal lift of curves on S.
Joint work with Bahar Acu, Agnes Gadbled, Aleksandra Marinkovic, Emmy Murphy, Laura Starkston and Angela Wu.
October 30--Greg Chambers, Rice University
Title: Geometric stability of the Coulomb energy
Abstract: Suppose that $A$ is a measurable subset of $\mathbb{R}^3$ of
finite measure. The Coulomb energy of $A$ is the double integral over
$A$ of $1/|x-y|$, and is maximized when $A$ is a ball. If the Coulomb
energy is close to maximal, then is $A$ geometrically close to a ball?
We will answer this question, and will compare it to the quantitative
isoperimetric inequality. We will also discuss the analogous situation
in higher dimensions. This is joint work with Almut Burchard.
November 6-David Wraith, Maynooth University
Title: Highly connected manifolds and intermediate curvatures
Abstract: It is known that up to connected sum with a homotopy sphere, essentially all highly connected manifolds in dimensions 4k+3 admit a positive Ricci curvature metric. In this talk we consider the curvature of highly connected manifolds in dimensions 4k+1. It turns out that proving an analogous positive Ricci curvature result is out of range at present. However the problem becomes tractable if we consider curvatures which are intermediate between positive scalar and positive Ricci curvature. This is joint work with Diarmuid Crowley.
November 13---Raquel Perales, Inst. de Mat., UAM, Oaxaca
Title: Convergence of manifolds under volume convergence, a tensor and a diameter bound
Abstract: [Based on joint work with Allen, Allen-Sormani and Cabrera Pacheco-Ketterer]
Given a Riemannian manifold $M$ and a pair of Riemannian tensors $g_0 \leq g_j$ on $M$
we have $\vol_0(M) \leq \vol_j(M)$ and the volumes are equal if and only if $g_0=g_j$.
In this talk I will show that if we have a sequence of Riemmanian tensors $g_j$ such that
$g_0\leq g_j$ and $\vol_j(M)\to \vol_0(M)$ then $(M,g_j)$ converges to $(M,g_0)$ in the
volume preserving intrinsic flat sense. I will present examples demonstrating that under these
conditions we do not necessarily obtain smooth, $C^0$ or Gromov-Hausdorff convergence.
Furthermore, this result can be applied to show stability of graphical tori.
November 20--Curtis Pro, CSU Stanislaus
Title: Extending a diffeomorphism finiteness theorem to dimension 4.
Abstract: Cheeger's Finiteness Theorem says: Given numbers $k<K$ in $\mathbb{R}$ and $v, D>0$, there are at most finitely many differentiable structures on the class of $n$-manifolds $M$ that support metrics with $k\leq\sec M\leq K, \mathrm{vol}\,M\geq v,$ and $\mathrm{diam}\,M\leq D.$ In the early 90s, Grove, Petersen, Wu, and (independently) Perelman showed in all dimensions, except possibly $n=4$, this conclusion still holds for the larger class that has no upper bound on sectional curvature. In this talk, I'll present recent work with Fred Wilhelm that shows this conclusion is also true in dimension 4.
December 4---Rina Rotman, Univ. of Toronto
Regina Rotman, University of Toronto
Title: Ricci curvature, the length of a shortest periodic geodesic and quantitative Morse theory on loop spaces
Abstract: I am planning to present the following result of
mine: Let Mn be a closed Riemannian manifold of dimension n and Ric ≥(n-1).
Then the length of a shortest periodic geodesic can be at most 8πn.
The technique involves quantitative Morse theory on loop spaces. We will
discuss some related results in geometry of loop spaces on Riemannian
manifolds.
December 11---Christine Escher, Oregon State Univ.
Title: Non-negative curvature and torus actions
The classification of Riemannian manifolds with positive and non-negative sectional curvature is a long-standing problem in Riemannian
geometry. In this talk I will give an overview of known results and summarize joint work with Catherine Searle on the classification of
closed, simply-connected, non-negatively curved Riemannian manifolds admitting an isometric, effective, isotropy-maximal torus action.
This classification has many applications, in particular the Maximal Symmetry Rank conjecture holds for this class of manifolds.