DISCRETE CATS SEMINAR

Discrete Seminar

Title: A biased variant of the ladybug clock problem.

Abstract: Instead of the common assumption of uniform probabilities for steps in a random walk on a graph, we generalize the steps to be biased and compute several statistics related to biased random walks on a cycle graph, more fancifully known as the “ladybug clock problem,” courtesy of Richard Stanley. We obtain new explicit formulas for the distribution of probabilities for a walk to end on a specific node, the expected cover time for the cycle and other similar statistics. We carry out these computations solely with generating functions related to the classic gambler’s ruin problem (and Dyck paths).

 

Date:
-
Location:
POT 745

Discrete Seminar

Title: A biased variant of the ladybug clock problem.

Abstract: Instead of the common assumption of uniform probabilities for steps in a random walk on a graph, we generalize the steps to be biased and compute several statistics related to biased random walks on a cycle graph, more fancifully known as the “ladybug clock problem,” courtesy of Richard Stanley. We obtain new explicit formulas for the distribution of probabilities for a walk to end on a specific node, the expected cover time for the cycle and other similar statistics. We carry out these computations solely with generating functions related to the classic gambler’s ruin problem (and Dyck paths).

 

Date:
-
Location:
POT 745

Discrete Seminar

Title: Region counting on another level
 
Abstract: The number of regions of a hyperplane arrangement is a well-understood invariant, which we can complicate by counting regions of a given level, a statistic that quantifies each region’s "boundedness." Rediscovering a formula of Zaslavsky, we show that the level distribution is a combinatorial invariant and in the process define it for all semi-matroids. The formula allows us to reprove and generalize results on deformations of the braid arrangements and certain specializations beg for alternate interpretations. Joint work with Lani Southern and Su Zhou.
Date:
-
Location:
POT 745

Discrete Seminar

Title: Region counting on another level
 
Abstract: The number of regions of a hyperplane arrangement is a well-understood invariant, which we can complicate by counting regions of a given level, a statistic that quantifies each region’s "boundedness." Rediscovering a formula of Zaslavsky, we show that the level distribution is a combinatorial invariant and in the process define it for all semi-matroids. The formula allows us to reprove and generalize results on deformations of the braid arrangements and certain specializations beg for alternate interpretations. Joint work with Lani Southern and Su Zhou.
Date:
-
Location:
POT 745

Discrete Seminar

Title: T-systems and Grassmannian cluster algebras
 
Abstract: T-system is a certain discrete integrable system that in type A can be visualized as a 3D array of entries satisfying the octahedral recurrence. We will explain the connection to Grassmannians and their associated cluster structure, where Pl\”ucker coordinates become entries of the array. We then study these T-systems from the point of view of additive categorification of cluster algebras.
Date:
-
Location:
POT 745

Discrete Seminar

Title: T-systems and Grassmannian cluster algebras
 
Abstract: T-system is a certain discrete integrable system that in type A can be visualized as a 3D array of entries satisfying the octahedral recurrence. We will explain the connection to Grassmannians and their associated cluster structure, where Pl\”ucker coordinates become entries of the array. We then study these T-systems from the point of view of additive categorification of cluster algebras.
Date:
-
Location:
POT 745

Discrete Seminar

Title: Recent AI-assisted advances in combinatorics

Abstract: Over the past six months, the use of AI tools in mathematics research has expanded rapidly. Analysis of arXiv postings between March and August by Jin, Ke and Sui show that the percentage of math preprints with disclosed AI use increased from 1.39% to 14.09% during this timeframe, and combinatorics has the largest number of such papers. In this talk, I will begin by giving an overview of the major AI tools being used for research in combinatorics. Then, I will survey recent AI-assisted proofs and counterexamples that have been announced, including the types of AI disclosure statements that are being made.

Date:
-
Location:
POT 745

Discrete Seminar

Title: Recent AI-assisted advances in combinatorics

Abstract: Over the past six months, the use of AI tools in mathematics research has expanded rapidly. Analysis of arXiv postings between March and August by Jin, Ke and Sui show that the percentage of math preprints with disclosed AI use increased from 1.39% to 14.09% during this timeframe, and combinatorics has the largest number of such papers. In this talk, I will begin by giving an overview of the major AI tools being used for research in combinatorics. Then, I will survey recent AI-assisted proofs and counterexamples that have been announced, including the types of AI disclosure statements that are being made.

Date:
-
Location:
POT 745

Chloé Napier -- Dissertation Defense

Dissertation Defense

Speaker:  Chloé Napier, University of Kentucky

Extensions between modules defined by lattice paths in the preprojective algebra
 
In 2001, Fomin and Zelevinsky introduced cluster algebras which appear as coordinate rings of many varieties. We study cluster algebras coming from Richardson varieties. Leclerc gives a cluster structure on Richardson varieties using the representation theory of preprojective algebras. While this construction is very algebraic, we take a more combinatorial approach. 
 
The main goal is to find a combinatorial description for when certain cluster variables are compatible or equivalently when modules defined by lattice paths in the preprojective algebra have trivial extensions. We extend the known results from Geiss, Leclerc and Schröer that answer this question in the case of the Grassmannian by using various methods. We introduce the notion of extending a module, describe how add or remove operators applied to pairs of modules affects the extension space between them and provide homological and combinatorial conditions that determine when two arbitrary modules have trivial extension.
Date:
-
Location:
745 POT

Chloé Napier -- Dissertation Defense

Dissertation Defense

Speaker:  Chloé Napier, University of Kentucky

Extensions between modules defined by lattice paths in the preprojective algebra
 
In 2001, Fomin and Zelevinsky introduced cluster algebras which appear as coordinate rings of many varieties. We study cluster algebras coming from Richardson varieties. Leclerc gives a cluster structure on Richardson varieties using the representation theory of preprojective algebras. While this construction is very algebraic, we take a more combinatorial approach. 
 
The main goal is to find a combinatorial description for when certain cluster variables are compatible or equivalently when modules defined by lattice paths in the preprojective algebra have trivial extensions. We extend the known results from Geiss, Leclerc and Schröer that answer this question in the case of the Grassmannian by using various methods. We introduce the notion of extending a module, describe how add or remove operators applied to pairs of modules affects the extension space between them and provide homological and combinatorial conditions that determine when two arbitrary modules have trivial extension.
Date:
-
Location:
745 POT