1/1/22 through 1/16/22

Special Colloquium/Candidate Presentation: Cluster algebras, their bases, and categorification

Date Jan 10, 2022
Time 10:00 am
Location Zoom
Speaker Fan Qin (Shanghai Jiao Tong University)
Contact Rinat Kedem
Email rinat@illinois.edu
Sponsor n/a

Abstract: Cluster algebras are certain commutative algebras whose generators are defined recursively. Fomin and Zelevinsky invented these algebras in 2002 in order to establish a combinatorial framework to understand the theory of total positivity due to Lusztig and the dual canonical bases of quantum groups due to Lusztig/Kashiwara. In particular, Fomin-Zelevinsky conjectured that the cluster monomials (certain monomials in the generators) belong to the dual canonical basis. In this talk, we briefly introduce cluster algebras by using concrete examples. We introduce the (common) triangular bases, which are Kazhdan-Lusztig type bases naturally parametrized by the tropical points of the corresponding varieties. They allow us to verify the above motivational conjecture in full generality. We also discuss the relationship between such bases and monoidal categorification.

Fan Qin is an algebraist, working in representation theory and cluster algebras, with interests in quantum groups, geometric representation theory, categorification, algebraic geometry, tropical geometry and higher Teichmuller theory. He is a 2012 graduate of Université Paris VII (Advisor: Bernhard Keller) and is currently at Shanghai Jiao Tong University. Faculty have access to a complete background via mathjobs.org.

Special Colloquium/Candidate Presentation: Thresholds

Date Jan 13, 2022
Time 4:00 pm
Location Zoom
Speaker Jinyoung Park (Stanford University)
Contact Alexandr Kostochka
Email kostochk@illinois.edu
Sponsor n/a

Abstract: Thresholds for increasing properties of random structures are a central concern in probabilistic combinatorics and related areas. In 2006, Kahn and Kalai conjectured that for any nontrivial increasing property on a finite set, its threshold is never far from its "expectation-threshold," which is a natural (and often easy to calculate) lower bound on the threshold. In this talk, I will first introduce the Kahn-Kalai Conjecture with some motivating examples and then talk about the recent resolution of a fractional version of the Kahn-Kalai Conjecture due to Frankston, Kahn, Narayanan, and myself. Some follow-up work, along with open questions, will also be discussed.

Special Colloquium/Candidate Presentation: Beyond the distributions of class groups

Date Jan 14, 2022
Time 3:00 pm
Location 245 Altgeld Hall
Speaker Yuan Liu (University of Michigan)
Contact Iwan Duursma
Email duursma@illinois.edu
Sponsor n/a

Abstract: We will first review several heuristics on the distribution of Galois groups of unramified extensions of global fields, which includes the Cohen—Lenstra Heuristics regarding the class groups of quadratic number fields and the Friedman—Washington Heuristics regarding the Jacobians of hyperelliptic curves. We will then discuss how these heuristics relate to reasonable random group models, and discuss how the function field case is less difficult. Finally, we will explain new conjectures on the distributions of Galois groups of the maximal unramified extensions of Galois number fields or function fields for a fixed Galois group.