Dissertation Defense

Speaker:  Chloé Napier, University of Kentucky

Title:  TBA

Abstract

Dissertation Defense

Speaker:  Chloé Napier, University of Kentucky

Title:  TBA

Abstract

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.

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.

Masters Examination

Speaker:  Lok Yam

Title:  TBA

Abstract:

Masters Examination

Speaker:  Lok Yam

Title:  TBA

Abstract:

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.

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.

Speaker:  Maxwell Hosler, University of Kentucky

Title:  Alcolved polytopes

Qualifying Exam

We will discuss work of Lam and Postnikov on alcoved polytopes and some recent progress which extends their work.
 

The type-A affine Coxeter arrangement divides real space into unit simplices, called alcoves. Convex unions of these alcoves are called alcoved polytopes. We examine three additional ways of triangulating a particular family of alcoved polytopes called hypersimplices. It is shown they are all, in fact, identical to the alcoved triangulation, and that the logic behind them generalizes to all alcoved polytopes. This innovation gives us multiple ways to express the structure of alcoved polytopes, as well as drawing connections to commutative algebra.

Speaker:  Maxwell Hosler, University of Kentucky

Title:  Alcolved polytopes

Qualifying Exam

We will discuss work of Lam and Postnikov on alcoved polytopes and some recent progress which extends their work.
 

The type-A affine Coxeter arrangement divides real space into unit simplices, called alcoves. Convex unions of these alcoves are called alcoved polytopes. We examine three additional ways of triangulating a particular family of alcoved polytopes called hypersimplices. It is shown they are all, in fact, identical to the alcoved triangulation, and that the logic behind them generalizes to all alcoved polytopes. This innovation gives us multiple ways to express the structure of alcoved polytopes, as well as drawing connections to commutative algebra.