Analysis and Probability Seminar

Spring 2026

The Analysis and Probability Seminar meets this semester on Fridays in LIT423 at 3pm.

Date	Talk
January 16	Speaker: Dr. Stanley Snelson Title: Recent results on the relativistic Landau equation Abstract: The Landau equation models the particle density of a diffuse plasma. When particles approach the speed of light with nontrivial probability, relativistic corrections are needed, and the resulting PDE is known as the relativistic Landau equation. Mathematically, this equation features relativistic transport and nonlinear, nonlocal diffusion in the momentum variable. After introducing the model, this talk will discuss some recent results, obtained in collaboration with Henderson, Tarfulea, and Taskovic, on the global regularity of solutions far from equilibrium. Time permitting, we will discuss related follow-up work on the Vlasov-Maxwell-Landau system, which arises from taking into account the electromagnetic field generated by the plasma.
January 23	Speaker: Dr. Eric Evert Title: Tensor decomposition, eigenvalue problems, and existence guarantees. Abstract: Tensors are a natural generalization of matrices to higher-order settings which exhibit several surprising properties. For example, the decomposition of a low-rank tensor is (essentially) unique with very light assumptions. This enables recovery of underlying information in practical signal processing and machine learning problems. Another surprising fact is that the set of low-rank tensors is not closed. This has the consequence that best low-rank tensor approximations can fail to exist, making the problem ill-posed. This talk concerns generalized-eigenvalue-based methods for low-rank tensor decomposition. Intuitively, well-posedness of the underlying eigenvalue decomposition translates to well-posedness of the full tensor decomposition. The former can, for example, be guaranteed with tools such as Gershgorin disks. Building on these ideas, we present methods to guarantee well-posedness of low-rank tensor decomposition. This talk is based on joint work with Lieven De Lathauwer.
January 30	No seminar
February 6	Speaker: Dr. Scott Mccullough Title: Sums of Squares and an update of the convex positivstellensatz. Abstract: Sums of squares (SoS) representations, also known as positivstellensatz, are algebraic certificates of polynomial inequalities. The talk will provide a light historical sampling of the theory before turning to non-commutative (nc) SoS results. It will conclude with an update to the nc convex positivstellensatz recently obtained with Abhay Jindal and Igor Klep.
February 13	Speaker: Lodewyk Jansen van Rensburg Title: Uniform Bounds for Expected Determinants Abstract: We establish bounds on the expected determinants of linear pencils evaluated at independent d×d Haar-distributed unitary random matrices that are uniform in the matrix size d. Our approach combines spectral information for the limiting operator arising in free probability, concentration of measure for the Haar unitary distribution, and strong convergence of Haar unitary ensembles to their free counterparts. This result completes a program developed by Mike Jury and George Roman for computing expected determinants of linear pencils in the most general case. This is joint work with Mike Jury and George Roman.
February 20	Speaker: David Maynoldi Title: Approximate Finite-Dimensional Dilations Abstract: We will discuss known dilations of commuting contractive n x n matrices to commuting unitary matrices. We will explore approximate dilations of a d-tuple of n x n matrices to a d-tuple of m x m matrices in a family N which hold on finite sets of polynomials. One of our results is that if the family N is norming and closed under componentwise finite direct sums, then an approximate dilation involving a pair of isometries holds for a single polynomial. There is a natural generalization of this result. Also, if we strengthen the norming assumption on N and assume it is invariant under componentwise application of any finite-dimensional *-representation, we get the desired approximate dilation using a single isometry. This is joint work with Mike Jury.
February 27 (LIT 368, 4pm)	Speaker: George Roman Title: Determinants of Random Unitary Pencils Abstract: This talk is a continuation of the work discussed in George's oral defense. The oral defense showed how the Szego and Drury-Arveson kernels arose from certain integrals in random matrix theory. These results depended on certain scalar parameters, and the oral defense concluded with a conjecture of a generalization involving matrix parameters. In this talk, we show that the matrix-parameter conjecture is true in some special cases, and go through key steps of the proof.
March 6	Speaker: Jacob Levenson Title: Fejér-Riesz Factorization for Nonegative Commutative and Noncommutative Trigonometric Polynomials Abstract: The classical Fejér-Riesz Theorem says that any nonnegative polynomial on $\mathbb{T}$ is the product of an analytic polynomial on $\mathbb{T}$ and its adjoint. We provide generalizations of this theorem to polynomials indexed by $\mathscr{W} \times \mathfrak{Y}$ where $\mathscr{W}$ is left fractions of the free semigroup on $\tt g$ generators. Results are affected by whether polynomials are positive or nonnegative. In some cases, degree bounds are obtained.
March 13	No seminar
March 20	Spring Break
March 27	Speaker: Dr. Micheal Dritschel Title: Hermitian indices and factorization of selfadjoint operators Abstract: The hermitian indices of a selfadjoint operator C on a Kreǐn (or Hilbert) space H are defined as geometric measures of positivity and negativity of the operator.  A different pair of indices arises in the Bognár-Krámli factorization of C, which writes C as a product AA* where A acts on a Kreǐn space K into H and has zero kernel; the new indices are the positive and negative indices of K.  Such factorizations are far from unique.  When H is separable, it is known that the two notions of indices always coincide.  This has various applications, and is even of interest when the Kreǐn space is a Hilbert space.  A new proof of the equality of indices that does not require separability is discussed. This is joint work with Alejandra Maestripieri and Jim Rovnyak.
April 3	Speaker: Jack Graham Title: Arveson Boundary of Free Quadrilaterals Abstract: A free quadrilateral is a collection of tuples X which have positive semidefinite evaluation on the linear equations defining a classical quadrilateral. Such a set is closed under matrix convex combinations, and it is of interest to determine its Arveson boundary, the set of irreducible free extreme points. In this talk, we present results from Eric Evert on projective maps of free spectrahedra and homogeneous free spectrahedra. Namely, we show that the image of an Arveson extreme point under an invertible projective map is again Arveson extreme. This result will enable us to conclude that any free quadrilateral has an Arveson boundary determined by a noncommutative variety.
April 10	Speaker: Harshika Rathi Title: A Noncommutative Szeg\H{o} type theorem. Abstract: The orthogonal polynomials on the unit circle (OPUC) satisfy recurrence relations and can be used to study the Szeg\H{o} limit theorems. We get a long list of equalities using these OPUC techniques. Motivated by this classical set up, we discuss a Noncommutative Szeg\H{o} type theorem on the row ball. However, the list of equalities now splits. We will also be interested in looking at the location of certain determinantal zeros using the recurrence.
April 17	Speaker: Vangmay Jayant Title: The common range of Co-Analytic Toeplitz operators on the Drury-Arveson space Abstract: In the Hardy space, there exists a precise characterization of the common range of the adjoints of cyclic multiplication operators. A theorem of Helson identifies these functions via the dual space of an inductive topology on the Smirnov class. By considering an alternate topology on the Smirnov class, it can be shown that a function belongs to this common range if and only if its Taylor coefficients satisfy a simple decay condition. In this talk, we present results by Aleman, Hartz, McCarthy, and Richter that extend these results to the Drury-Arveson space. Helson's theorem carries over naturally to this setting, while the growth condition is obtained by introducing a uniform Smirnov class on the ball and giving a description of its dual space.
April 22	Speaker: Dr. Ugur G. Abdulla Title: Kolmogorov Problem and Wiener-type Criteria in Potential Theory Abstract: The central problem in the Analysis of PDEs is understanding the nature of singularities that arise in natural phenomena. This talk will present a full characterization of the fundamental boundary singularity, and equivalently, the unique solvability of the singular Dirichlet problem for the elliptic and parabolic PDEs. The results are threefold. We prove a new Wiener-type criterion for the ”geometric” characterization of the removability of the fundamental singularity for arbitrary open sets in terms of the fine-topological thinness of the complementary set near the singularity point. In the special case when the surface of revolution forms the boundary of the open set near the singularity point, we establish a Kolmogorov-Petrovsky-type test to characterize the removability of the singularity and uniqueness. Finally, in the special case when a continuous graph locally represents the boundary of the open set, the minimal thinness criterion for the removability of the sin gularity is expressed in terms of the minimal regularity of the boundary manifold at the singularity point. From the probabilistic point of view, the criteria present an asymptotic law for conditional Brownian motion. In the topological context, the criteria present a full characterization of the neighborhood base of the boundary singularity point in the minimal fine topology. In the more general framework, the talk will outline a program for the full characterization of the singularities formed by the elliptic and parabolic PDEs.