In the 1980's Gromov and Piatetsky-Shapiro came up with a technique called "hybridization" to produce hyperbolic lattices (and in particular, nonarithmetic lattices). It has been asked whether there is an analogous technique for complex hyperbolic lattices. In this talk I'll present some recent results relating to this question and the construction of complex hyperbolic hybrids in the Picard modular groups. This is joint work with Julien Paupert.

We're going to chat about the Grassmann Manifold $G_{K,N}$?? of all $K$??-dimensional subspaces in $\mathbb{C}^N?$?. A friendly back-pocket example of the Grassmann manifold is $G_{1,N}$??, which is nothing but the very familiar and very well loved projective space on $\mathbb{C}^N$.

With a tiny bit of background in statistical signal processing, we'll discover why we might care about the Grassmann manifold in practice, and why it may be beneficial to regard $G_{K,N}$ as rank-$K$ (orthogonal) projection operators on $\mathbb{C}^N$. We'll use the correspondence between points on $G_{K,N}$ and rank-$K$ projection operators to to develop a local coordinate system on $G_{K,N}$, so we can we can exploit probability measures on the manifold itself.

In multi-channel detection, sufficient statistics for Generalized Likelihood Ratio and Bayesian tests are often functions of the eigenvalues of the Gram matrix formed from data vectors collected at the sensors. When the null hypothesis is that the channels contain only independent complex white Gaussian noise, the distributions of these statistics arise from the joint distribution of the eigenvalues of a complex Wishart matrix. This talk will consider the particular case of the largest eigenvalue of a complex Wishart matrix, which arises in detection of a rank-one signal. Although the distribution of the largest eigenvalue is known, calculating its values numerically has been observed to present formidable difficulties when the dimension of the data is large. This talk will set up the general two hypothesis detection problem, examine the numerical difficulties of classical statistical results arising in modern sensing applications, demonstrate a method for allowing computation of the distribution under the null hypothesis, and propose similar methods for the alternative hypothesis.

Equipped with a certain type of seminorm analogous to the Lipschitz seminorm for continuous functions, a unital $C^*$-algebra becomes a quantum compact metric space, an object arising in the work of M.A. Rieffel and motivated by ideas from high-energy physics. Latremoliere's quantum Gromov-Hausdorff propinquity is a distance between quasi-Leibniz compact quantum metric spaces. This talk focuses on joint work with K. Aguilar, in which we demonstrate that two quantum metric spaces can be built from the same underlying $C^*$-algebra but have positive distance in the quantum propinquity. In particular, we consider the straightforward example of $M_n(\mathbb{C})$, with the quantum metrics from Aguilar and Latremoliere's work on approximately finite-dimensional (AF) algebras.

Many of us are familiar with taking geometric and topological objects and encoding their information by means of an algebraic object. For example, we can encode information about topological spaces using the Fundamental group or Homology and Cohomology groups. Geometric Group Theory exploits the other direction, that is, encoding the information about algebraic objects using geometric objects. In this talk, I will discuss some topics in Geometric Group theory, and the Svarc Milnor lemma and some of its applications.

In this talk, I will provide some background on Ricci flow and discuss recent existence and uniqueness results for instantaneously complete Ricci flow on surfaces.

*Seminar temporarily moved to Room WXLR A206.