DISCRETE CATS SEMINAR

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

Pablo Castilla -- Qualifying Exam

Qualifying Exam

Speaker:  Pablo Castilla, University of Kentucky

Title:  Understanding the Túran polytope with cutting planes

Abstract:

For a 3-regular hypergraph with n vertices, what is the most number of edges it can have without having a 4-clique as a subgraph? Túran posed this problem in 1941 and constructed what he conjectured was optimal, but to date the question remains open. 

While it has had great deal of attention combinatorially, recently Raymond has taken a polytopal approach, formulating the problem as an integer linear program. The convex hull of the admissible hypergraphs, known as the Túran polytope, is combinatorially interesting in its own right, having many correspondences with the stable set polytope. 

We propose to further understand the Túran polytope using cutting planes, a technique from integer linear programming. By observing cutting plane algorithms applied by software to solve the integer program, we can discover more of the Túran polytope’s facet structure and make progress on proving Túran’s conjecture.

Date:
Location:
745 POT

Pablo Castilla -- Qualifying Exam

Qualifying Exam

Speaker:  Pablo Castilla, University of Kentucky

Title:  Understanding the Túran polytope with cutting planes

Abstract:

For a 3-regular hypergraph with n vertices, what is the most number of edges it can have without having a 4-clique as a subgraph? Túran posed this problem in 1941 and constructed what he conjectured was optimal, but to date the question remains open. 

While it has had great deal of attention combinatorially, recently Raymond has taken a polytopal approach, formulating the problem as an integer linear program. The convex hull of the admissible hypergraphs, known as the Túran polytope, is combinatorially interesting in its own right, having many correspondences with the stable set polytope. 

We propose to further understand the Túran polytope using cutting planes, a technique from integer linear programming. By observing cutting plane algorithms applied by software to solve the integer program, we can discover more of the Túran polytope’s facet structure and make progress on proving Túran’s conjecture.

Date:
Location:
745 POT

Lok Yam - Master's Examination

Masters Examination

Speaker:  Lok Yam, University of Kentucky

Title:  Facets of symmetric edge polytopes

Abstract:

Symmetric edge polytopes are lattice polytopes arising from graphs. We define the correspondence between simple connected graphs and their associated symmetric edge polytopes. We discuss the central symmetry and reflexivity of symmetric edge polytopes. We state and prove a combinatorial description of their facet structure given by Higashitani, Jochemko and Michalek (2018).
 
Date:
-
Location:
POT 110

Lok Yam - Master's Examination

Masters Examination

Speaker:  Lok Yam, University of Kentucky

Title:  Facets of symmetric edge polytopes

Abstract:

Symmetric edge polytopes are lattice polytopes arising from graphs. We define the correspondence between simple connected graphs and their associated symmetric edge polytopes. We discuss the central symmetry and reflexivity of symmetric edge polytopes. We state and prove a combinatorial description of their facet structure given by Higashitani, Jochemko and Michalek (2018).
 
Date:
-
Location:
POT 110

Williem Rizer - Doctoral Defense

Doctoral Defense (Note time)

Speaker:  Williem Rizer

Title:  Combinatorial models for nonnegativity in flag varieties

Abstract:  

The nonnegative Grassmannian admits a widely studied cell decomposition due to Alexander Postnikov, whose cells are indexed by positroids and modeled by several equivalent combinatorial objects. Subsequent work by authors including Lauren Williams, Suho Oh and Carolina Benedetti has further developed the combinatorics and geometry of these structures. 

In this talk, we will extend some of Postnikov’s combinatorial framework to the nonnegative flag variety. We introduce flag positroid pipe dreams, a diagrammatic model analogous to Le diagrams, together with associated directed graphs and networks that parameterize Richardson cells indexed by flag positroids. Using this framework, we give a constructive proof of a conjecture from Benedetti-Chavez-Tamayo in the case of nonnegatively representable quotients, giving a complete characterization of all positroids of which a fixed positroid is such a quotient in terms of decorated permutations and diagrammatic data. 

Our approach also highlights the connection between flag positroids and intervals in the Bruhat order. Building on work of onathan Boretsky, Christopher Eur and Williams, we show that flag positroid pipe dreams are in bijection with Bruhat intervals, where the number of ones in the diagram encodes the interval length and hence the dimension of the corresponding Richardson cell. Together, these results provide a unified combinatorial perspective on nonnegativity in flag varieties.

 

Date:
-
Location:
POT 745

Williem Rizer - Doctoral Defense

Doctoral Defense (Note time)

Speaker:  Williem Rizer

Title:  Combinatorial models for nonnegativity in flag varieties

Abstract:  

The nonnegative Grassmannian admits a widely studied cell decomposition due to Alexander Postnikov, whose cells are indexed by positroids and modeled by several equivalent combinatorial objects. Subsequent work by authors including Lauren Williams, Suho Oh and Carolina Benedetti has further developed the combinatorics and geometry of these structures. 

In this talk, we will extend some of Postnikov’s combinatorial framework to the nonnegative flag variety. We introduce flag positroid pipe dreams, a diagrammatic model analogous to Le diagrams, together with associated directed graphs and networks that parameterize Richardson cells indexed by flag positroids. Using this framework, we give a constructive proof of a conjecture from Benedetti-Chavez-Tamayo in the case of nonnegatively representable quotients, giving a complete characterization of all positroids of which a fixed positroid is such a quotient in terms of decorated permutations and diagrammatic data. 

Our approach also highlights the connection between flag positroids and intervals in the Bruhat order. Building on work of onathan Boretsky, Christopher Eur and Williams, we show that flag positroid pipe dreams are in bijection with Bruhat intervals, where the number of ones in the diagram encodes the interval length and hence the dimension of the corresponding Richardson cell. Together, these results provide a unified combinatorial perspective on nonnegativity in flag varieties.

 

Date:
-
Location:
POT 745