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.

Happ Title:  "Generalized Abel-Rothe Polynomials" 

Happ Abstract:  This sequence of polynomials is conjectured to be of a "multi" binomial type, and we will discuss how they count certain trees and generalized parking functions.

Hedmark Title:  TBA

Hedmark Abstract:  TBA

Title:  A categorification of the Stanley symmetric chromatic polynomial

Abstract:  Given a graph G with n vertices, Stanley defined a symmetric polynomial X_G(x_1, x_2, ...) so that for every positive integer k, X_G(1,..,1,0,...) = chi_G(k) is the number of proper k-colourings of G. We build a double chain complex C_*(G) of S_n-modules so that the Frobenius series Frob_G(x;q,t) of the resulting bi-graded homology H_*(G) satisfies Frob_G(x;1,1) = X_G(x_1, x_2, ...).

Title:  "I am not Lou Billera"

Abstract:  I will present the slides from a talk Lou Billera gave at the Stanley birthday conference last summer regarding the history of f-vectors.

Title:  r-Stable Hypersimplices

Abstract:  The n,k-hypersimplices are a well-studied collection of polytopes.  Inside each n,k-hypersimplex we can define a finite nesting of subpolytopes that we call the r-stable n,k-hypersimplices.  In this talk, we will define the r-stable hypersimplices and then see that they share a nice geometric relationship via a well-known regular unimodular triangulation of the n,k-hypersimplex in which they live.  Using this relationship, we will then identify some geometric and combinatorial properties of the r-stable hypersimplices.  In doing so, we will see that a number of the properties of the n,k-hypersimplex also hold for the r-stable hypersimplices within.

Title:  Hopf Lefschetz theorem for posets

Abstract:  The Hopf-Lefschetz theorem is a classical fixed point result from topology relating the Euler characteristic and the traces of certain matrices. In this talk we will prove a generalization of this theorem to order preserving maps on posets due to Baclawski and Björner. Additionally, we will prove a number of sufficient conditions on a poset P guaranteeing that all order preserving maps on P have a fixed point.