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.

Noncommutative Geometry Seminar

Time: 2:00PM - 3:00PM

Location: BLOC 624

Speaker: Weichen Gu, University of New Hampshire

Title: On the zeta-function of some non-commutative semigroups

Abstract: In this talk we introduce a framework on the zeta-functions of some non-commutative semigroups, including the Thompson semigroup and braid semigroups. A generalization $\kappa(n)$ of the Möbius function related to the Thompson group is given, and we will use $\kappa(n)$ to extend the zeta-function of the hompson semigroup to the complex half plane with real part greater than 1/2, and prove that the real pole closest to 1 is a simple pole.

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.