The 2023 Workshop in Analysis and Probability at Texas A&M University will be in session from July 5 to July 30. Workshop activities will include a seminars series, SUMIRFAS 2023, and two concentration weeks to be announced later in the Spring.

**Probability and Algebra: New Expressions In Mathematics**

May the odds be ever in your favor!!

**Accessibility Statement**

This webpage is a work in progress. As of right now, it meets the accessibility guidelines of WCAG 2.0AA. External links may or may not meet accessibility guidelines. For any concerns accessing the content on this webpage, please email Jeffrey Kuan.

**Dates and Location**

This program will be held July 10 to July 14 in College Station, Texas, on the campus of Texas A&M University.

**Organizers**

- Jeffrey Kuan
- Local and Scientific Organizer
- Axel Ivan Saenz Rodriguez
- Scientific Organizer
- Daniel Perales Anaya
- Local and Scientific Organizer

**Confirmed Mini-course Speakers**

- Natasha Blitvic
- Leonid Petrov
- Oleg Zaboronski

**Confirmed Speakers**

- Octavio Arizmendi
- Ioana Dumitriu
- James Mingo
- Zhipeng Liu

**Titles and Abstracts**

**Program Description**

This program will bring together researchers, across all career stages and across different research areas. There will be mini-courses aimed at the graduate student level, and also opportunities for short talks for early-career researchers. This workshop will bring together the fields of probability and algebra, in a very general sense. Some examples of active research areas which involve probability and algebra include:

- Integrable Probability:
- Algebraic structures such as Drinfel'd-Jimbo quantum groups yield solutions to the so-called Yang Baxter equation (YBE). With the YBE, the models are integrable in the sense of having commuting transfer matrices, which allow the Hamiltonian to be simultaneously diagonalizable. In the probabilistic side, this results in stochastic models with eigenfunctions which have contour integral expressions, which are amenable to asymptotic analysis.
- Free probability:
- Free probability is a non-commutative analogue of Probability where random variables are no longer required to be commutative. One natural example is when considering random matrices. In this non-commutative world, the notion of independence is not very adequate, instead the more natural concept of freeness is used. The name free comes from the fact that the distribution follows the same behavior of the canonical trace when one considers the free product of group algebras. Random variables can also have q-deformed commutation relations. As such, algebraic structures are also natural sources of free probability.
- Combinatorics:
- One important combinatorial tool in Probability are the cumulants, which are families of functionals that linearize the addition of two independent random variables. Cumulants can be recursively defined in terms of the moments via a precise formula involving set partitions. Analogously, free cumulants linearize the addition of two free variables and can be defined in terms of moments using non-crossing partitions. The interplay between different subsets of set partitions has a rich combinatorial structure. An incidence algebra approach has been typically used to study moment-cumulants and invert them using the MÃ¶bius function. In recent years, a Hopf algebraic approach has been used to do a more systematic study of the transition formulas between cumulants and moments.
- Random Matrix Theory:
- A very natural question regarding eigenvalues of matrices is the following: given the eigenvalues of two dxd Hermitian matrices A and B, what are all the possible sets of eigenvalues of A+B? This apparently simple question, known as Horn's Problem was a long-standing open question until not too long ago (1999). One of the main goals of random matrix theory could be described as solving a randomized Horn's problem: given two random matrices with prescribed eigenvalue distribution, what is the eigenvalue distribution of A+B? One line of study is finite free probability, which concerns with the study of the roots of the expected characteristic polynomial of randomly rotated matrices, where the random rotation refers to conjugation with respect to a matrix uniformly drawn from the set of orthogonal, unitary or signed permutation matrices.