DISCRETE CATS SEMINAR

Emma Pickard, UK

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. 

 

Date:
Location:
745 POT

Emma Pickard, UK

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. 

 

Date:
Location:
745 POT

Discrete CATS Seminar

Speaker:  Jonah Berggren, UK

Title: Framing triangulations and posets from nontrivial netflow vectors

Abstract:

Danilov, Karzanov and Koshevoy introduced framings on a directed acyclic graph and used them to induce regular unimodular “framing triangulations” on the unit flow polytope. The dual graphs of these triangulations have been shown to have the structure of the Hasse diagram of a lattice, generalizing many classical and modern families of lattices in combinatorics. Through a recent/ongoing work of González D’León, Hanusa and Yip, this theory has led to a deeper understanding of h*-polynomials of certain flow polytopes.
 
Thus far, this study of framing triangulations and framing lattices has largely been limited to unit flow polytopes — i.e., from DAGs with one source, one sink and netflow vector (1,0,…,0,-1). I will talk about my recent efforts to generalize beyond the unit case. First, I will give a notion of framed DAGs inducing unimodular framing triangulations in the full generality of arbitrary integer flow polytopes. I will conclude by reducing to a special case of framed DAGs, containing all theories of framing triangulations existing in the literature, to which we can give a theory of framing posets generalizing framing lattices in the unit case.
 

 

Date:
Location:
745 POT
Event Series:

Discrete CATS Seminar

Note:  Rescheduled talk due to technical difficulties last week

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Speaker:  Martha Yip, University of Kentucky

Title:  Permuflows (for the working mathematician)

Abstract:  

Danilov, Karzanov and Koshevoy described a combinatorial method for constructing a family of regular unimodular triangulations of flow polytopes. In recent years it has been shown that the dual graph of a DKK triangulation has the structure of lattice, and many interesting families of lattices arise from these triangulations.
 
We introduce a family of combinatorial objects called permutation flows. Permuflows capture the combinatorics of DKK triangulations in a compact way and highlights the connections to permutations. I will explain how these objects:
 
  1. Correspond with cliques of routes.
  2. Give the face poset of a DKK triangulation of the flow polytope.
  3. Simplify the computation of the lattice of a DKK triangulation.
  4. Lead to a formula for the h* polynomial of the flow polytope.
     
This is joint work with González D'León and Hanusa.
Date:
Location:
745 POT
Event Series:

Discrete CATS seminar

Speaker:  Martha Yip, University of Kentucky

Title:  Permuflows (for the working mathematician)

Abstract:  

Danilov, Karzanov and Koshevoy described a combinatorial method for constructing a family of regular unimodular triangulations of flow polytopes. In recent years it has been shown that the dual graph of a DKK triangulation has the structure of lattice, and many interesting families of lattices arise from these triangulations.
 
We introduce a family of combinatorial objects called permutation flows. Permuflows capture the combinatorics of DKK triangulations in a compact way and highlights the connections to permutations. I will explain how these objects:
  1. Correspond with cliques of routes.
  2. Give the face poset of a DKK triangulation of the flow polytope.
  3. Simplify the computation of the lattice of a DKK triangulation.
  4. Lead to a formula for the h* polynomial of the flow polytope.
This is joint work with González D'León and Hanusa.
 
Date:
Location:
745 POT
Event Series:

Discrete CATS Seminar

Speaker:  Ben Braun

Title:  Flow polytope volumes and Ehrhart theory

Abstract:

The first half of this talk will be a survey regarding flow polytopes and their volumes. The Ehrhart h*-polynomial is a refinement of the volume of a lattice polytope. In the second half of this talk, I will discuss recent joint work with K. Bruegge, R. Davis and D. Hanely regarding Ehrhart h*-polynomials of a class of acyclic directed graphs that we call extensions of bipartite graphs

 

Date:
Location:
745 POT
Event Series:

Emma Pickard - Qualifying Examination

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. 

Date:
-
Location:
POT 745

Emma Pickard - Qualifying Examination

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. 

Date:
-
Location:
POT 745

Colloquium: Pipe dreams, Schubert varieties and the commuting scheme

Colloquium Speaker: Allen Knutson, Cornell University

"Pipe dreams, Schubert varieties and the commuting scheme"

Schubert considered the space of kxn matrices whose Gaussian elimination has fixed pivot columns. The "volume" of this space, in some sense, is a Schur polynomial, with many combinatorial interpretations. Pipe dreams were introduced in 1993 in [Bergeron-Billey] to give a pictorial calculus for "Schubert polynomials," the corresponding volumes of a more general class of Schubert varieties.

In 2005, Miller and I gave a geometric retrodiction of pipe dreams based on Gröbner degeneration. In the same year, I introduced the "lower-upper scheme'' {(X,Y): XY lower triangular, YX upper} to study the scheme of pairs of commuting matrices. I'll explain a (much more natural) pipe dream theory for the lower-upper scheme, use it to rederive the old one (also Lam-Lee-Shimozono's "bumpless pipe dreams'') and give a formula for the degree of the commuting scheme. This is joint with Paul Zinn-Justin.

 

Colloquium tea is at 3:14 pm (= π time) in 745 POT.

 

This colloquium talk is part of the KOI Combinatorics Lectures.

The KOI Combinatorics Lectures are funded by a grant from the National Science Foundation. Partial support provided by the UK Department of Mathematics and the UK College of Arts and Sciences.

 

Date:
Location:
287 Chem-Physics

Colloquium: Pipe dreams, Schubert varieties and the commuting scheme

Colloquium Speaker: Allen Knutson, Cornell University

"Pipe dreams, Schubert varieties and the commuting scheme"

Schubert considered the space of kxn matrices whose Gaussian elimination has fixed pivot columns. The "volume" of this space, in some sense, is a Schur polynomial, with many combinatorial interpretations. Pipe dreams were introduced in 1993 in [Bergeron-Billey] to give a pictorial calculus for "Schubert polynomials," the corresponding volumes of a more general class of Schubert varieties.

In 2005, Miller and I gave a geometric retrodiction of pipe dreams based on Gröbner degeneration. In the same year, I introduced the "lower-upper scheme'' {(X,Y): XY lower triangular, YX upper} to study the scheme of pairs of commuting matrices. I'll explain a (much more natural) pipe dream theory for the lower-upper scheme, use it to rederive the old one (also Lam-Lee-Shimozono's "bumpless pipe dreams'') and give a formula for the degree of the commuting scheme. This is joint with Paul Zinn-Justin.

 

Colloquium tea is at 3:14 pm (= π time) in 745 POT.

 

This colloquium talk is part of the KOI Combinatorics Lectures.

The KOI Combinatorics Lectures are funded by a grant from the National Science Foundation. Partial support provided by the UK Department of Mathematics and the UK College of Arts and Sciences.

 

Date:
Location:
287 Chem-Physics