Methods of Combinatorial Research

Jesus De Loera and Steve Klee

This RFG will focus on how other fields helped combinatorics evolved and how combinatorics later became useful in return. It aims to broaden the mathematical knowledge and interests of students. Historically there have been several success stories in which the solution of a difficult problem required the intervention of a novel point of view and connecting together fields that appeared to be wide apart. Examples include commutative algebra methods in combinatorics, which were introduced by Richard Stanley was to solve various problems about f-vectors of simplicial complexes using rings and algebraic geometry. Another famous example is the use of probabilistic techniques to solve problems of existence in combinatorics, a method pioneered by Erdos and Turan in the 1940's. A third example is how computer tools have come to affect research and proofs in combinatorial research; this is exemplified by the proof of the 4-color theorem by Appel and Haken in 1977 and by the Wilf-Zeilberger algorithm used to deduced combinatorial identities.

Ping Pong Lemmas in Group Theory

James Forehand

In group theory, a ping pong lemma is a criterion for determining decomposition properties of subgroups. Klein used a ping pong lemma to show that many subgroups generated by two cyclic subgroups are free. A similar ping pong lemma played a crucial role in the proof of Tits alternative, a fundamental theorem in the theory of linear groups. Much of the modern research into ping pong lemmas explores ways to determining what kind of right angled artin groups can be embedded into a given family of groups or ways to generalize and extend Tits' alternative. This talk aims to give a brief survey of ping pong lemmas in group theory.

Some applications of geometry and topology to problems in biology

Joel Hass

I will discuss three biological problems

• Determining the evolutionary connections between primates
• Measuring the progression of Alzheimer's disease
• Understanding the function of a protein

Each of these problems can be viewed as an instance of a geometric problem, namely the problem of finding optimal diffeomorphisms. I'll discuss this problem and some approaches using conformal and hyperbolic geometry.

Cutting Planes for Mixed Integer Optimization

Robert Hildebrand

In the last decade, cuttings planes have proven to be a very useful method for mixed integer optimization, which falls into the class NP-Hard. The technique is summarized as follows: solve the continuous linear optimization problem (with all continuous variables), if the solution is feasible for the mixed integer problem, then we are done, otherwise we generate a new inequality that cuts off the the continuous optimal solution and start again. The strength of these new inequality, or cutting plane, can drastically effect the time it takes to solve the problem, and therefore, we are interested in developing strong cutting planes. Very beautiful geometry results from studying these cutting planes. We will discuss recent research on stronger cutting planes arising from maximal lattice free convex bodies.

Flows on Simplicial Complexes

Yvonne Kemper

Questions about networks helped inspire the study of flows on graphs, and the resulting research has involved a variety fields, including Ehrhart theory, matroids, and optimization, and encompassed many beautiful properties. In this talk, we introduce the idea of flows on graphs, and consider possible generalizations of flows and associated results to higher dimensional objects.

A connection between the $\xxz$ Hamiltonian model and the Asymmetric Simple Exclusion Process

Ricky Kwok

The Heisenberg spin chain and the asymmetric simple exclusion process (ASEP) are two well-studied models in spin-1/2 one-dimensional magnetism and interacting particle system, respectively. Heisenberg hypothesized the $XYZ$ Hamiltonian to govern the dynamics of an infinite number of particles with spin interactions. Up to an overall factor, there are three parameters, $J_x, J_y$ and $J_z$ in the $XYZ$ Hamiltonian. Bethe proposed an ansatz to find eigenvectors and eigenvalues of the special case when all three are identical, now known as the Bethe Ansatz. The ASEP is a continuous-time Markov process on $\Z$ determined by the parameter $0 < p < 1$ and $p\neq 1/2$. Particles hop to the right with this probability and to the left otherwise. Tracy and Widom used ideas of the Bethe Ansatz to show integrability of this model by writing down formulas for the transition probability of the ASEP for the step initial condition. I will discuss the relationship of these two models. Specifically, there is a similarity transformation of the Markov matrix generating the ASEP to the $XXZ$ Hamiltonian, a special case when $J_x = J_y\neq J_z$.

Quantum Phase Transitions

Bruno Nachtergaele

Quantum Phase Transitions and the characterization of Gapped Quantum Phases are once again a hot topic. Solving the classification problem of gapped ground states is important to better understand the possibilities and challenges of implementing quantum computation and information processing with topological quantum ground state phases. The critical points separating different gapped ground state phases (quantum phase transitions) have fascinating mathematical properties. A case in point is the conjectured $E_8$ symmetry of the critical quantum Ising chain, experimental evidence for which was recently reported in the literature (Science, 8 January 2010, 327 (no. 5962) pp. 177-180; see also arXiv:1012.5407). In this RFG, participants who are new to the topic will become familiar with the main techniques to study these topics, with an emphasis on those that have been or might be amenable to mathematical analysis. As needed, more advanced participants will organize tutorials on relevant topics of statistical mechanics, functional analysis, representation theory, Feynman-Kac-type formulas, etc.. For context, there will be a reading course on Quantum Information Theory. In Spring 2012 the RFG will concentrate on topics of current research and there will be a special topics course (MAT280) on quantum phase transitions.

Fluctuations of Matrix Entries of Regular Functions of Random Matrices

David Renfrew

In this talk, I will discuss the fluctuations of matrix entries of regular functions of several classes of random matrices. The Gaussian case (GUE/GOE) was studied by A. Lytova and L. Pastur in 2009. Their approach significantly relies on the the unitary/ orthogonal invariance of the GUE/GOE ensemble. Using a different approach, I will explain how one can extend their results to random matrices with four finite moments on the matrix entries. I will discuss the non-universality of fluctuations for functions of Wigner matrices.

Complex Fluids with Applications to Biology

Becca Thomases

The goal of this Research Focus Group (RFG) is to introduce undergraduate and graduate students to the research areas in Complex Fluids pursued at our campus, with particular emphasis on applications to biology. This RFG will focus on both theoretical, modeling, and computational aspects of the study of complex fluids. Examples of current ongoing work in this area includes the study of biofluid dynamics, viscoelastic instabilities, and multi-scale mixtures in biology. This RFG will be highly interdisciplinary emphasizing mathematical modeling and simulation of macro and multi-scale models of biofluids, and biological phenomena. The group will pursue two goals: (i) prepare postdocs and students to become active part of the ongoing research in Complex Fluids, and (ii) give formal structure to UC Davis faculty doing research in Complex Fluids.

Lieb-Robinson bounds for a class of irreversible quantum dynamics

Anna Vershynina

Lieb-Robinson bounds provide an estimate for the speed of propagation of signals in a spatially extended system. I will consider a general situation, when the dynamics of the system is described by a Markovian dynamical semigroup. The dynamics is generated by both Hamiltonian and dissipative interactions with suitable fast decay in space and that may depend on time. The dissipative interactions arises in every experimental setup due to the coupling of a system with its environment. The bound I will talk about is a generalization of the Lieb-Robinson bounds for the reversible dynamics, which first was proved by Lieb and Robinson in 1972 and was improved by B.Nachtergaele, B. Sims, M.Hastings, and others in the last ten years.