Seminars and Colloquia Schedule

Dynamic Transition Theory and its Application to Gas-Liquid Phase Transitions

Series
CDSNS Colloquium
Time
Monday, October 25, 2010 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 114
Speaker
Shouhong Wang – Indiana University
Gas-liquid transition is one of the most basic problem to study in equilibrium phase transitions. In the pressure-temperature phase diagram, the gas-liquid coexistence curve terminates at a critical point C, also called the Andrews critical point. It is, however, still an open question why the Andrews critical point exists and what is the order of transition going beyond this critical point. To answer this basic question, using the Landau's mean field theory and the Le Chatelier principle, a dynamic model for the gas-liquid phase transitions is established. With this dynamic model, we are able to derive a theory on the Andrews critical point C: 1) the critical point is a switching point where the phase transition changes from the first order with latent heat to the third order, and 2) the liquid-gas phase transition going beyond Andrews point is of the third order. This clearly explains why it is hard to observe the liquid-gas phase transition going beyond the Andrews point. In addition, the study suggest an asymmetry principle of fluctuations, which appears also in phase transitions in ferromagnetic systems. The analysis is based on the dynamic transition theory we have developed recently with the philosophy to search the complete set of transition states. The theory has been applied to a wide range of nonlinear problems. A brief introduction for this theory will be presented as well. This is joint with Tian Ma.

Energy-based fracture evolution

Series
Applied and Computational Mathematics Seminar
Time
Monday, October 25, 2010 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 002 (Ground floor, entrance from Skiles courtyard)
Speaker
Christopher Larsen – WPI
I will describe a sequence of models for predicting crack paths in brittlematerials, with each model based on some type of variational principleconcerning the energy. These models will cover the natural range ofstatics, quasi-statics, and dynamics. Some existence results will bedescribed, but the emphasis will be on deficiencies of the models and openquestions.

A polynomial invariant of pseudo-Anosov maps

Series
Geometry Topology Seminar
Time
Monday, October 25, 2010 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Joan Birman – Barnard College-Columbia University –
Pseudo-Anosov mapping classes on surfaces have a rich structure, uncovered by William Thurston in the 1980's. We will discuss the 1995 Bestvina-Handel algorithmic proof of Thurston's theorem, and in particular the "transition matrix" T that their algorithm computes. We study the Bestvina-Handel proof carefully, and show that the dilatation is the largest real root of a particular polynomial divisor P(x) of the characteristic polynomial C(x) = | xI-T |. While C(x) is in general not an invariant of the mapping class, we prove that P(x) is. The polynomial P(x) contains the minimum polynomial M(x) of the dilatation as a divisor, however it does not in general coincide with M(x).In this talk we will review the background and describe the mathematics that underlies the new invariant. This represents joint work with Peter Brinkmann and Keiko Kawamuro.

Group Dynamics in Phototaxis

Series
School of Mathematics Colloquium
Time
Tuesday, October 26, 2010 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Doron Levy – CSCAMM University of Maryland (College Park)
Microbes live in environments that are often limiting for growth. They have evolved sophisticated mechanisms to sense changes in environmental parameters such as light and nutrients, after which they swim or crawl into optimal conditions. This phenomenon is known as "chemotaxis" or "phototaxis." Using time-lapse video microscopy we have monitored the movement of phototactic bacteria, i.e., bacteria that move towards light. These movies suggest that single cells are able to move directionally but at the same time, the group dynamics is equally important. Following these observations, in this talk we will present a hierarchy of mathematical models for phototaxis: a stochastic model, an interacting particle system, and a system of PDEs. We will discuss the models, their simulations, and our theorems that show how the system of PDEs can be considered as the limit dynamics of the particle system. Time-permitting, we will overview our recent results on particle, kinetic, and fluid models for phototaxis. This is a joint work with Devaki Bhaya (Department of Plant Biology, Carnegie Institute), Tiago Requeijo (Math, Stanford), and Seung-Yeal Ha (Seoul, Korea).

Well-posedness theory for compressible Euler equations in a physical vacuum

Series
PDE Seminar
Time
Tuesday, October 26, 2010 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Prof. Juhi Jang – Department of Mathematics, University of California, Riverside –
An interesting problem in gas and fluid dynamics is to understand the behavior of vacuum states, namely the behavior of the system in the presence of vacuum. A particular interest is so called physical vacuum which naturally arises in physical problems. The main difficulty lies in the fact that the physical systems become degenerate along the boundary. I'll present the well- posedness result of 3D compressible Euler equations for polytropic gases. This is a joint work with Nader Masmoudi.

Tropical Bernstein's theorem

Series
Tropical Geometry Seminar
Time
Wednesday, October 27, 2010 - 10:05 for 1 hour (actually 50 minutes)
Location
Skiles 114
Speaker
Anton Leykin – Georgia Tech
The classical Bernstein's theorem says that the number of roots of a system of sparse polynomials with generic coefficients equals the mixed volume of the Newton polytopes of the polynomials. We shall sketch a constructive proof by describing the solutions in the field of Puiseux series. The tropical Bernstein's theorem says that the number of tropical roots of a system of sparse tropical polynomials with generic coefficients equals the mixed volume of the Newton polytopes. We will prove this using the Huber--Sturmfels method for computing mixed volumes with regular mixed subdivisions of polytopes. Side topics: computation of mixed volumes, polyhedral homotopy continuation (finding complex solutions of a sparse polynomial system).

Some Applications of Nonlinear Dynamics and Statistical Physics in Critical Care

Series
Mathematical Biology Seminar
Time
Wednesday, October 27, 2010 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 169
Speaker
Anton Burykin – Emory University Center for Critical Care
Critical care is a branch of medicine concerned with the provision of life support or organ support systems in patients who are critically ill and require intensive monitoring. Such monitoring allows us to collect massive amounts of data (usually at the level of organ dynamics, such as electrocardiogram, but recently also at the level of genes). In my talk I’ll show several examples of how ideas from nonlinear dynamics and statistical physics can be applied for the analysis of these data in order to understand and eventually predict physiologic status of critically ill patients: (1) Heart beats, respiration and blood pressure variations can be viewed as a dynamics of a system of coupled nonlinear oscillators (heart, lungs, vessels). From this perspective, a live support devise (e.g. mechanical ventilator used to support breathing) acts as an external driving force on one of the oscillators (lungs). I’ll show that mechanical ventilator entrances the dynamics of whole cardiovascular system and leads to phase synchronization between respiration and heart beats. (2) Then I’ll discuss how fluctuation-dissipation theorem can be used in order to predict heart rate relaxation after a stress (e.g. treadmill exercise test) from the heart rate fluctuations during the stress. (3) Finally, I’ll demonstrate that phase space dynamics of leukocyte gene expression during critical illness and recovery has an attractor state, associated with immunological health.

Branched Covers in Contact Geometry

Series
Other Talks
Time
Wednesday, October 27, 2010 - 12:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Meredith Casey – School of Mathematics, Georgia Tech

This talk will be the oral examination for Meredith Casey.

I will first discuss the motivation and background information necessary to study the subjects of branched covers and of contact geometry. In particular we will give some examples and constructions of topological branched covers as well as present the fundamental theorems in this area. But little is understood about the general constructions, and even less about how branched covers behave in the setting of contact geometry, which is the focus of my research. The remainder of the talk will focus on the results I have thus far and current projects.

Sticky particle dynamics with interactions

Series
Research Horizons Seminar
Time
Wednesday, October 27, 2010 - 12:00 for 1 hour (actually 50 minutes)
Location
Skiles 171
Speaker
Michael Westdickenberg – School of Mathematics - Georgia Institute of Technology

Hosts: Yao Li and Ricardo Restrepo

We consider compressible fluid flows in Lagrangian coordinates in one space dimension. We assume that the fluid self-interacts through a force field generated by the fluid. We explain how this flow can be described by a differential inclusion on the space of transport maps, when the sticky particle dynamics is assumed. We prove a stability result for solutions of this system. Global existence then follows from a discrete particle approximation.

Rational Inner Functions in the Schur-Agler Class

Series
Analysis Seminar
Time
Wednesday, October 27, 2010 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Greg Knese – University of Alabama
The Schur-Agler class is a subclass of the bounded analytic functions on the polydisk with close ties to operator theory. We shall describe our recent investigations into the properties of rational inner functions in this class. Non-minimality of transfer function realization, necessary and sufficient conditions for membership (in special cases), and low degree examples are among the topics we will discuss.

Euler's pentagonal numbers theorem - refinements, variations and companions

Series
School of Mathematics Colloquium
Time
Thursday, October 28, 2010 - 11:05 for 1 hour (actually 50 minutes)
Location
Skiles 249
Speaker
Krishnaswami Alladi – University of Florida –
Euler's celebrated pentagonal numbers theorem is one themost fundamental in the theory of partitions and q-hypergeometric series.The recurrence formula that it yields is what MacMahon used to compute atable of values of the partition function to verify the deep Hardy-Ramanujanformula. On seeing this table, Ramanujan wrote down his spectacular partition congruences. The author recently proved two new companions to Euler'stheorem in which the role of the pentagonal numbers is replaced by the squares.These companions are deeper in the sense that lacunarity can be achievedeven with the introduction of a parameter. One of these companions isdeduced from a partial theta identity in Ramanujan's Lost Notebook and theother from a q-hypergeometric identity of George Andrews. We will explainconnections between our companions and various classical results such asthe Jacobi triple product identity for theta functions and the partitiontheorems of Sylvester and Fine. The talk will be accessible to non-experts.

Balanced Vertices in Trees and a Simpler Algorithm to Compute the Genomic Distance

Series
Combinatorics Seminar
Time
Thursday, October 28, 2010 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Peter L.Erdos – Alfred Renyi Inst. of Mathematics, Budapest
In this talk we will report a short and transparent solution for the covering cost of white--grey trees which play a crucial role in the algorithm of Bergeron et al. to compute the rearrangement distance between two multi-chromosomal genomes in linear time (Theor. Comput. Sci., 410:5300-5316, 2009). In the process it introduces a new center notion for trees, which seems to be interesting on its own.

Displaying blocking pairs in signed graphs

Series
ACO Seminar
Time
Thursday, October 28, 2010 - 16:30 for 1.5 hours (actually 80 minutes)
Location
Skiles 255
Speaker
Bertrand Guenin – Dept. of Combinatorics and Optimization, University of Waterloo
A signed graph is a pair (G, \Sigma) where G is a graph and \Sigma is a subset of the edges of G. A cycle C in G is even (resp. odd) if E(C) \cap \Sigma is even (resp. odd). A blocking pair in a signed graph is a pair of vertices {x, y} such that every odd cycle in (G, \Sigma) intersects at least one of the vertices x and y. Blocking pairs arise in a natural way in the study of even cycle matroids on signed graphs as well as signed graphs with no odd K_5 minor. In this article, we characterize when the blocking pairs of a signed graph can be represented by 2-cuts in an auxiliary graph. We discuss the relevance of this result to the problem of characterizing signed graphs with no odd K_5 minor and determing when two signed graphs represent the same even cycle matroid. This is joint work with Irene Pivotto and Paul Wollan.

When do random CSPs become hard?

Series
SIAM Student Seminar
Time
Friday, October 29, 2010 - 13:05 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Ricardo Restrepo – School of Mathematics, Georgia Tech
A constraint satisfaction problem (CSP) is an ensemble of boolean clauses, where satisfaction is obtained by an assignment of the variables if every clause is satisfied by such assignment. We will see that when such CSP is arranged following certain random structure, the Fourier expansion of the corresponding clauses allows us to understand certain properties of the solution space, in particular getting a partial understanding of when the 'usual suspects' of the drastical failure of all known satisfiability algorithms, namely long range correlations and clustering, appear. Based in joint work with Prasad Tetali and Andrea Montanari.

Non-commutative Geometry III: Crossed Product and Orbit Space of Group Actions

Series
Geometry Topology Working Seminar
Time
Friday, October 29, 2010 - 14:00 for 2 hours
Location
Skiles 171
Speaker
Jean Bellissard – Ga Tech

Note this is a 2 hour talk.

An action of the real line on a compact manifold defines a topological dynamical system. The set of orbits might be very singular for the quotient topology. It will be shown that there is, however, a C*-algebra, called the crossed product, which encodes the topology of the orbit space. The construction of this algebra can be done for an group action, if the group is locally compact.

Decimations of l-sequences and permutations of even residues mod p

Series
Combinatorics Seminar
Time
Friday, October 29, 2010 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Todd Cochrane – Math, Kansas State University
\ell-sequences are periodic binary sequences {a_i} that arise from Feedback with Carry Shift Registers and in many other ways. A decimation of {a_i} is a sequence of the form {a_{di}}. Goresky and Klapper conjectured that for any prime p>13 and any \ell-sequence based on p, every pair of allowable decimations of {a_i} is cyclically distinct. If true this would yield large families of binary sequences with ideal arithmetic cross correlations. The conjecture is essentially equivalent to the statement that if p>13 then the mapping x \to Ax^d on \mathbb Z/(p) with (d,p-1)=1, p \nmid A, permutes the even residues only if it is the identity mapping. We will report on the progress towards resolving this conjecture, focussing on our joint work with Bourgain, Paulhus and Pinner.