Abstracts

Week of October 1, 2023

October 4, 2023
Colloquium Limit measures for topologically and geometrically random surfaces 4:00pm -
KT219

An immersed surface S in a Riemannian manifold M induces a probability measure on the space G_2(M) of two-planes in the tangent bundle of M. If M is a hyperbolic 3-manifold and S_n is a sequence of surfaces with principal curvatures going to zero, then any weak* limit of their induced measures is a convex combination of the Liouville (equidistributed) measure on G_2(M), and measures that come from immersed totally geodesic surfaces in M. We consider two ways of generating a “random” nearly geodesic surface in M, one by bounding the genus, and the other by bounding the area. We show that limits of the measures in the former case must come exclusively from the totally geodesic surfaces (if there are any in M), while limits in the latter case must have some portion that is equidistributed. This is joint work with V. Markovic and I. Smilga.
October 5, 2023
Analysis Nodal count and Morse theory of magnetic operators on graphs. 4:00pm -
KT 219

Our intuition regarding waves suggests that the k-th eigenvector of a graph Laplacian L (or any Discrete Schrodinger operator) should exhibit greater fluctuations as k increases. In this context, the "nodal count" is the number of edges on which the eigenvector changes sign. The works of Fiedler (1975) and Berkolaiko (2007) show that the nodal count is bounded between k-1 and k-1+b, where b is the first Betti number of the graph. We establish that these bounds hold for signed graphs as well when considering sign changes accordingly. The “nodal surplus”, the deviation from k-1, is expected to concentrate around b/2. Numerical observations indicate that the distribution of the nodal surplus, across all eigenvectors and different signings of the graph, resembles a Gaussian distribution centered at b/2, regardless of the graph's characteristics. We prove that it is precisely binomial with mean b/2 in the case of operators on complete graphs with sufficiently high potential.

This outcome, among others, stems from a noteworthy relationship. The magnetic perturbations of L are achieved by multiplying the off-diagonal entries of L by phases (in a Hermitian manner), modulo gauge invariance. The eigenvalues of L extend to piece-wise analytic functions of the phases. At non-degenerate critical points, the Morse index is equal to the associated nodal surplus. If time permits, I will draw the line connecting this work to spectral gaps of periodic operators.
This talk is based on joint works with Mark Goresky and John Urschel
Group Actions, Geometry and Dynamics Patterson--Sullivan theory and equidistribution in Hilbert geometry 4:00pm -
KT801
 In the theory of discrete subgroups of Lie groups, given a length function on the Lie group G, one popular object of study is the asymptotic when R goes to infinity of the number of loxodromic elements with length less R in a discrete subgroup H, and the distribution of the fixed points of these loxodromic elements.
When G is the group of (real) projective transformations and H acts properly discontinuously and cocompactly on a strictly convex domain O of the projective space, Yves Benoist noticed that one can use the Hilbert metric of O, and the associated Hilbert geodesic flow, to estimate the above counting function, for a special length function called the Hilbert length.
An important ingredient in Benoist’s result is the uniform hyperbolicity of the Hilbert geodesic flow, which does not hold when O is not strictly convex.
In this talk we will develop and use instead the theory of Patterson–Sullivan measures for Hilbert geometries O/H that satisfy a mild rank-one assumption, and are not necessarily strictly convex nor compact.
This will yield counting results for H in the cases where the induced Bowen–Margulis measure is finite. (This is joint work with Feng Zhu).
Graduate Learning Seminar Learning seminar on D-modules 4:00pm -
Prospect 204, B-02

This is the fourth lecture in the seminar.
October 6, 2023
Friday Morning Seminar Friday Morning Seminar 10:00am -
809 Commons

A relaxed-pace seminar on impromptu subjects related to the interests of the audience.

Everyone is welcome.

The subjects are geometry, probability, combinatorics, dynamics, and more!
Geometric Analysis and Application Recent Developments in Constant Mean Curvature Hypersurfaces 2:00pm -
KT 906

A constant mean curvature surface is a critical point of the area functional subject to a volume constraint. Min-max theory is a powerful method for finding saddle type critical points of functionals. Recently, Xin Zhou and Jonathan Zhu developed a min-max theory for finding constant mean curvature surfaces in closed manifolds. In this talk, I will discuss some recent results in the min-max theory of constant mean curvature hypersurfaces. In particular, I will discuss an extension of the CMC min-max theory to certain non-compact manifolds. I will also discuss joint work with Xin Zhou on min-max theory with a volume constraint.