DISCRETE CATS SEMINAR

Discrete CATS Seminar

Speaker:  Martha Yip, University of Kentucky

Title: The volume of the caracol polytope

Abstract: A number of flow polytopes have volumes that are products of nice combinatorial quantities, but perhaps surprisingly, there are no known combinatorial proofs of these formulas.  In this talk, I will give a combinatorial interpretation of the Lidskii volume formula of Baldoni and Vergne, and use this to prove that the volume of the caracol polytope is the product of a Catalan number and the number of parking functions of length n.  This is joint work with Benedetti, Gonzalez D'Leon, Hanusa, Harris, Khare and Morales.
Date:
Location:
745 Patterson Office Tower
Event Series:

Discrete CATS Seminar

Speaker:  Aida Maraj, University of Kentucky

Title: Hierarchical models and their toric ideals 

Abstract: 
 
This talk will be about Hierarchical Models in Algebraic Statistics. I will present an example where these models apply and introduce their corresponding toric ideals. A formula for calculating Krull dimension will be given. To describe generating sets of these ideals one can use a symmetric group action. Using this tool, we will describe generating sets for some classes of these ideals as decomposable and non-reducible Models.  This is joint work with Uwe Nagel.
Date:
Location:
745 Patterson Office Tower
Event Series:

Qualifying Exam - Julie Vega

Mathematics Qualifying Exam for Julie Vega.

Title: Chromatic Numbers, Hom Complexes, and Topological Obstructions (or a great way to organize your favorite maps)

Abstract: Imagine a pleasant graph, but in color. Now let’s make it a little more exciting by coloring vertices such that no two adjacent vertices have the same color. The smallest number for which you can do this is called the “chromatic number.” In ’78 Lov`asz used the connectivity of the neighborhood complex to find lower bounds for the chromatic number. Later, he generalized this idea to the Hom Complex, Hom(G, H), which encodes graph homomorphisms between G and H and their relationship. In this talk, following an argument by Babson and Kozlov, we will look at Hom complexes and use Stiefel Whitney classes to find lower bounds on the chromatic number of a graph.

Date:
-
Location:
POT 745
Event Series:

Discrete CATS Seminar

Speaker: Karthik Chandrasekhar

Title:  Facing up to metrics

Abstract:
Here we study arrangements of geodesic lines on spaces topologically equivalent to the real 2-plane, but having different metrics. We study the regions of all possible of f-vectors. Mainly we study the hyperbolic plane (the upper half plane with a non-Euclidean metric) which reveals a vastly different region of f-vectors.
 
 
Date:
Location:
745 Patterson Office Tower
Event Series:

Discrete CATS Seminar

Nick Early, University of Minnesota.

Canonical bases for permutohedral plates

Abstract:

There is a natural construction according to which the set of all faces of an arrangement of hyperplanes can be made into a vector space, by taking linear combinations of their characteristic functions.  Our space is equipped with a natural basis of characteristic functions of certain polyhedral cones called permutohedral cones passing through the origin, studied as plates by A. Ocneanu, which are labeled by ordered set partitions; these are in duality with faces of the arrangement of reflection hyperplanes xi=xj.  It is interesting to study the affine case as well: recently we conjectured stating that plates situated compatibly with the lattice of integer points in a generalized hypersimplex provide a new combinatorial interpretation of the h*-vector for that generalized hypersimplex.
 
In this talk, we construct directly a certain canonical basis which is compatible with one or both of two quotients: neglecting characteristic functions of (1) nonpointed cones, and (2) cones of codimension at least 1.  The essential feature here is that subsets of the canonical basis map to bases of the quotients.  As a consequence, we obtain the straightening relations which were originally computed by Ocneanu through the introduction of an auxiliary space of formal linear combinations of layered trees.  Time permitting, we will describe the circumstances which led us to formulate the conjecture for the h*-vector.

For more details, see Discrete CATS website

 

Date:
Location:
745 Patterson Office Tower
Event Series:

Discrete CATS Seminar

McCabe Olsen, University of Kentucky.

Title: Ehrhart theory and ordered set partitions

Abstract: Given a lattice polytope P, one of the central questions in Ehrhart theory is to describe the h*-polynomial (or h*-vector) of P, as this encodes and detects certain algebraic and geometric properties of P. Given that the coefficients of the h*-polynomial are nonnegative integers, it is natural (for a combinatorialist) to wishfully think that these coefficients count something; that is, ideally this polynomial encodes some statistic on some combinatorial object. In this talk, we will discuss some conjectures of Nick Early regarding the h*-polynomial of two well-known polytopes, namely dilated unit simplices and hypersimplices, involving a winding number statistic on certain decorated ordered set partitions. We will provide a proof to one of these conjectures and discuss the other.

For more details, see Discrete CATS website

 

Date:
Location:
745 Patterson Office Tower
Event Series: