Title:  Deletion-Induced Triangulations

Abstract:   Let $d > 0$ be  a fixed integer and let $\A \subseteq \mathbb{R}^d$ be a collection of $n \geq d+2$ points which we lift into $\mathbb{R}^{d+1}$. Further let $k$ be an integer satisfying $0 \leq k \leq n-(d+2)$ and assign to each $k$-subset of the points of $\A$ a (regular) triangulation obtained by deleting the specified $k$-subset and projecting down the lower hull of the convex hull of the resulting lifting. Next, for each triangulation we form the characteristic vector outlined by Gelfand, Kapranov, and Zelevinsky by assigning to each vertex the sum of the volumes of all adjacent simplices. We then form a vector for the lifting, which we call the compound GKZ-vector, by summing all the characteristic vectors. Lastly, we construct a polytope $\Sigma_k(\A) \subseteq \mathbb{R}^{| \A |}$ by taking the convex hull of all obtainable compound GKZ-vectors by various liftings of $\A$, and note that $\Sigma_0(\A)$ is the well-studied secondary polytope corresponding to $\A$. We will see that by varying $k$, we obtain a family of polytopes with interesting properties relating to Minkowski sums, Gale transforms, and Lawrence constructions, with the member of the family with maximal $k$ corresponding to a zonotope studied by Billera, Fillamen, and Sturmfels. We will also discuss the case $k=d=1$, in which we can outline a combinatorial description of the vertices allowing us to better understand the graph of the polytope and to obtain formulas for the numbers of vertices and edges present.

Title: Representing discrete Morse functions with polyhedra

Abstract: Discrete Morse theory is a method of reducing a CW complex to a simpler complex with similar topological properties. Well-known approaches to this task are due to Banchoff, whose process involves embedding a polyhedron in Euclidean space and considering the projections of its vertices onto a straight line, and to Forman, whose process involves finding special functions from the face poset of a complex to the real numbers. In this talk, I will discuss a result by Bloch which gives a relationship between these two methods. In particular, given a discrete Morse function on a CW complex, there exists a corresponding polyhedral embedding of the barycentric subdivision of X such that the discrete Morse function and the projection of the vertices of the polyhedron onto a line give the same critical cells.

 

Title: An Introduction to Symmetric Functions, part II

Abstract: In this pair of talks, I will provide an overview of basic results regarding symmetric functions.  My goal will be to create a "road map" for anyone who is interested in reading more about these objects in Chapter 7 of Stanley's Enumerative Combinatorics, Volume 2 (if you have a copy and are interested, it might be helpful to bring it with you).  We will motivate the study of symmetric functions by interpreting them as generalizations of subsets and multisubsets of [n], so these talks should be accessible to anyone who is familiar with the material from the first part of MA 614.

Title: An Introduction to Symmetric Functions, part I

Abstract: In this pair of talks, I will provide an overview of basic results regarding symmetric functions.  My goal will be to create a "road map" for anyone who is interested in reading more about these objects in Chapter 7 of Stanley's Enumerative Combinatorics, Volume 2 (if you have a copy and are interested, it might be helpful to bring it with you).  We will motivate the study of symmetric functions by interpreting them as generalizations of subsets and multisubsets of [n], so these talks should be accessible to anyone who is familiar with the material from the first part of MA 614.

Title: Single Splitter Details

Abstract: Lee defined the winding number w_k in a Gale diagram corresponding to a given simplicial polytope. He showed that w_k equals g_k of the corresponding polytope. We are working on extending Lee's definition of w_k to nonsimplicial polytopes. In this talk, we will discuss our results when the origin in the Gale diagram falls on a single k-splitter, a hyperplane that separates k points from the rest.

Title:  The polytope of Tesler matrices

Abstract:  Tesler matrices are upper triangular matrices with nonnegative integer entries with certain restrictions on the sums of their rows and columns. Glenn Tesler studied these matrices in the 1990s and in 2011 Jim Haglund rediscovered them in his study of diagonal harmonics. We investigate a polytope whose integer points are the Tesler matrices. It turns out that this polytope is a flow polytope of the complete graph thus relating its lattice points to vector partition functions. We study the face structure of this polytope and show that it is a simple polytope. We show its h-vector is given by Mahonian numbers and its volume is a product of consecutive Catalan numbers and the number of Young tableaux of staircase shape.  This is joint work with Brendon Rhoades and Karola Mészàros.

Title:  Algebraic models in systems biology

Abstract:  Progress in systems biology relies on the use of mathematical and statistical models for system level studies of biological processes. Several different modeling frameworks have been used successfully, including traditional differential equations based models, a variety of stochastic models, agent-based models, and Boolean networks, to name some common ones. This talk will focus on discrete models and the challenges they present, in particular model stability and data selection.

Title:  Conditions for the toric homogenous Markov Chain models to have square-free quadratic Groebner basis

Abstract:  Discrete time Markov chains are often used in statistical models to fit the observed data from a random physical process. Sometimes, in order to simplify the model, it is convenient to consider time-homogeneous Markov chains, where the transition probabilities do not depend on the time.  While under the time-homogeneous Markov chain model it is assumed that the row sums of the transition probabilities are equal to one, under the the toric homogeneous Markov chain (THMC) model the parameters are free and the row sums of the transition probabilities are not restricted.

In this talk we consider a Markov basis and a Groebner basis for the toric ideal associate with the design matrix (configuration) defined by THMC model with the state space with $m$ states where $m \geq 2$ and we study when THMC with $m$ states have a square-free quadratic Groebner basis.  One such example is the embedded discrete Markov chain for the Kimura three parameter model. This is joint work with Abraham Martin del Campo and Akimichi Takemura.

Title:  An Algebraic Approach to Systems Biology.

Abstract:  This talk will present an algebraic perspective for modeling gene regulatory networks. Algebraic models can be represented by polynomials over finite fields. In this setting, several problems relevant to biology can be studied. For instance, the algebraic view has been successfully applied for the development of computational tools to determine the attractors of Boolean Networks, for network inference algorithms, and for the development of a theoretical framework for agent based models. In this talk, the algebraic perspective of discrete models will be applied for control problems. No background in mathematical biology will be assumed for this talk.