Below is the Fall 2024 schedule for the Analysis and Probability Seminar.
Time: 9th period (4:05 – 4:55pm)
Room: LIT 423
| Date | Speaker | Title | Abstract |
|---|---|---|---|
| Sep. 6 | Scott McCullough | Krein-Milman, Operator Convexity and Beyond | We discuss analogs of the Krein-Milman Theorem for operator convex sets and see that the results extend to the setting of partially operator convex sets. The work is joint with Igor Klep and Tea Strekelj. It builds on prior work with Igor, Mike Jury, Mark Mancuso, and James Pascoe. |
| Sep.13 | Eric Evert | Carathéodory, matrix convexity and quantum incompatibility. | We discuss positive and negative results for extreme points of matrix convex sets, the finite dimensional restriction of operator convex sets. We will see that for real matrix convex sets defined by a noncommutative polynomial inequality, there is a natural extension of Carathéodory’s theorem on convex hulls. Additionally, we consider an application to incompatibility of quantum measurements. |
| Sep.20 | Mike Jury | Determinants of random unitary pencils | A random unitary pencil (in g variables) is an expression of the form L_A(U) = I - (A_1 U_1 + ... +A_g U_g) where the U's are independent random unitary NxN matrices, and the A's are (deterministic) scalar or matrix coefficients. We state a conjecture about the asymptotic covariances E[ det(L_A(U)) det(L_B(U)*) ] as the size N goes to infinity. We are able to prove the conjecture in the case of scalar A's. I will discuss the conjecture and some of the background from the one-variable (g=1) case, where there result reduces to an instance of the Strong Szego Limit Theorem, which in turn connects it to the theory of Toeplitz determinants and reproducing kernel Hilbert spaces. The Drury-Arveson space will make an appearance. |
| Oct.4 | George Roman | Determinants of random unitary pencils, part 2 | This talk is a continuation of last week's topics. In it, we will outline proofs of some of the theorems discussed last week while exploring connections to the theories of symmetric functions and representations of groups. |
| Oct.25 | Jacob Levenson | Beurling's Theorem | Beurling's Theorem answers the following question: What do the invariant subspaces for a shift operator on a Hilbert space look like? This talk will take a look at Beurling's Theorem as it is applies to invariant subspaces of finite multiplicity. |
| Nov.1 | David Maynoldi | Finite-Dimensional Dilation Theory and Matrix Convexity | We will state a finite-dimensional version of the Sz-Nagy dilation theorem for contractions on a Hilbert space. We will state a more general result which gives conditions under which a u.c.p. map on an operator system in a unital FDI C*-algebra dilates to a finite-dimensional representation. We will give bounds on the dimension when the C*-algebra is r-subhomogeneous. We will discuss other finite-dimensional dilation theorems which can be derived from this result, as well as tools from matrix convexity that were used to prove the general result. |
| Nov.20 | Abdulmajeed Alqasem | Probabilistic approaches in number theory. | We present examples of how probability can be used to tackle problems in number theory including questions related to the prime number theorem, primes in tuples and so on. |