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.

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.

Title: Quiver representations and syzygies of Gorenstein algebras

Title: Quiver representations and syzygies of Gorenstein algebras

Qualifying Exam

Speaker:  Emma Pickard, University of Kentucky

Title:  Degeneracy loci and permutation groups

Abstract:

Any permutation sigma in S_n determines n^2 rank conditions on any n by n matrix. In doing so, sigma determines a degeneracy locus for a flagged vector bundle on a variety by imposing rank conditions on each fiber. The dimension of such loci and values of the rank can be obtained from diagrams constructed based on only the permutation itself. We will begin exploring these diagrams and loci with our standard symmetric group, and then move to explore other types of permutation groups and their corresponding diagrams. This work is based primarily on the work of William Fulton and David Anderson. 

Qualifying Exam

Speaker:  Emma Pickard, University of Kentucky

Title:  Degeneracy loci and permutation groups

Abstract:

Any permutation sigma in S_n determines n^2 rank conditions on any n by n matrix. In doing so, sigma determines a degeneracy locus for a flagged vector bundle on a variety by imposing rank conditions on each fiber. The dimension of such loci and values of the rank can be obtained from diagrams constructed based on only the permutation itself. We will begin exploring these diagrams and loci with our standard symmetric group, and then move to explore other types of permutation groups and their corresponding diagrams. This work is based primarily on the work of William Fulton and David Anderson. 

Qualifying Exam

Speaker:  Emma Pickard, University of Kentucky

Title:  Degeneracy loci and permutation groups

Abstract:

Any permutation sigma in S_n determines n^2 rank conditions on any n by n matrix. In doing so, sigma determines a degeneracy locus for a flagged vector bundle on a variety by imposing rank conditions on each fiber. The dimension of such loci and values of the rank can be obtained from diagrams constructed based on only the permutation itself. We will begin exploring these diagrams and loci with our standard symmetric group, and then move to explore other types of permutation groups and their corresponding diagrams. This work is based primarily on the work of William Fulton and David Anderson. 

Qualifying Exam

Speaker:  Emma Pickard, University of Kentucky

Title:  Degeneracy loci and permutation groups

Abstract:

Any permutation sigma in S_n determines n^2 rank conditions on any n by n matrix. In doing so, sigma determines a degeneracy locus for a flagged vector bundle on a variety by imposing rank conditions on each fiber. The dimension of such loci and values of the rank can be obtained from diagrams constructed based on only the permutation itself. We will begin exploring these diagrams and loci with our standard symmetric group, and then move to explore other types of permutation groups and their corresponding diagrams. This work is based primarily on the work of William Fulton and David Anderson.