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.

Title:  The combinatorial structure behind the free Lie algebra

Abstract:  We explore a beautiful interaction between algebra and combinatorics in the heart of the free Lie algebra on n generators: The multilinear component of the free Lie algebra Lie(n) is isomorphic as a representation of the symmetric group to the top cohomology of the poset of partitions of an n-set tensored with the sign representation. Then we can understand the algebraic object Lie(n) by applying poset theoretic techniques to the poset of partitions whose description is purely combinatorial. We will show how this relation generalizes further in order to study  free Lie algebras with multiple compatible brackets.

Title:  Cyclotomic Factors of the Descent Set Polynomial

Abstract:  The descent set polynomial is defined in terms of the descent set statistics of a permutation and was first introduced by Chebikin, Ehrenborg, Pylyavskyy, and Readdy. This polynomial was found to have many factors which are cyclotomic polynomials. In this talk, we will continue to explore why these cyclotomic factors exist, focusing on instances of the 2pth cyclotomic polynomial for a prime p.