# Events for 03/31/2023 from all calendars

## Mathematical Physics and Harmonic Analysis Seminar

**Time: ** 1:50PM - 2:50PM

**Location: ** BLOC 302

**Speaker: **Terry Harris, Cornell University

**Title: ***Projections and intersections in the first Heisenberg group.*

**Abstract: **In this talk, I will discuss some recent work on the Hausdorff dimension of projections and intersections in the first Heisenberg group. In Euclidean space, it is known that projections of sets onto k-dimensional subspaces almost surely do not decrease Hausdorff dimension, and that projections of sets of dimension greater than k have projections almost surely of positive k-dimensional area. It has been conjectured that these theorems extend to "vertical projections" in the Heisenberg group. This conjecture is still open, but was recently solved in a significant part of the range by Fassler and Orponen, using a "point-plate incidence" method. I will outline some of my recent work, which also uses the point-plate incidence method, and which proves the "positive area" part of the conjecture. One connection of this talk to harmonic analysis is that it uses the (endpoint) trilinear Kakeya inequality, which grew out of multilinear Fourier analysis inspired by the Fourier restriction and Kakeya conjectures.

## Algebra and Combinatorics Seminar

**Time: ** 3:00PM - 4:00PM

**Location: ** BLOC 302

**Speaker: **Hongdi Huang, Rice University

**Title: ***Twisting Manin's universal quantum groups and comodule algebras*

**Abstract: **Symmetry is an important concept that appears in mathematics and theoretical physics. While classical symmetries arise from group actions on polynomial rings, quantum symmetries are introduced to understand certain quantum objects (e.g., quantum groups) which appear in the theory of quantum mechanics and quantum field theory. In this talk, we will define Manin's universal quantum groups and its 2-cocycle twist. Moreover, we will talk about the invariants under the tensor equivalence of quantum symmetries.

## Free Probability and Operators

**Time: ** 4:00PM - 5:00PM

**Location: ** BLOC 306

**Speaker: **Sheng Yin, Baylor University

**Title: ***Non-commutative rational functions in random matrices and operators.*

**Abstract: **It is well-known that many random matrices have an asymptotical limit which is described by free probability. That is, for any noncommutative polynomial in these d independent random matrices converges to the same polynomial in d freely independent random variables that describe the limit distribution of each sequence of random matrices. In this talk, we will present a natural generalization of this convergence result. Namely, under suitable assumptions, we can enlarge our test function from noncommutative polynomial to noncommutative rational functions. It is based on a joint-work with Benoît Collins, Tobias Mai, Akihiro Miyagawa and Félix Parraud.

## Geometry Seminar

**Time: ** 4:00PM - 5:00PM

**Location: ** BLOC 302

**Speaker: **Máté L. Telek, University of Copenhagen

**Title: ***Reaction networks and a generalization of Descartes’ rule of signs to hypersurfaces*

**Abstract: **The classical Descartes’ rule of signs provides an easily computable upper bound for the number of positive real roots of a univariate polynomial with real coefficients. Descartes' rule of signs is of special importance in applications where positive solutions to polynomial systems are the object of study. This is the case in reaction network theory where variables are concentrations or abundances. Motivated by this setting, we give conditions based on the geometrical configuration of the exponents and the sign of the coefficients of a polynomial that guarantee that the number of connected components of the complement of the hypersurface where the defining polynomial attains a negative value is at most one or two. Furthermore, we discuss how these results can be applied to show that the parameter region of multistationarity of a reaction network is connected.