Seminars and Colloquia by Series

Phantom Jams and Jamitons in Macroscopic Traffic Models

Series
Applied and Computational Mathematics Seminar
Time
Monday, March 31, 2014 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Benjamin Seibold – Temple University
Initially homogeneous vehicular traffic flow can become inhomogeneous even in the absence of obstacles. Such ``phantom traffic jams'' can be explained as instabilities of a wide class of ``second-order'' macroscopic traffic models. In this unstable regime, small perturbations amplify and grow into nonlinear traveling waves. These traffic waves, called ``jamitons'', are observed in reality and have been reproduced experimentally. We show that jamitons are analogs of detonation waves in reacting gas dynamics, thus creating an interesting link between traffic flow, combustion, water roll waves, and black holes. This analogy enables us to employ the Zel'dovich-von Neumann-Doering theory to predict the shape and travel velocity of the jamitons. We furthermore demonstrate that the existence of jamiton solutions can serve as an explanation for multi-valued parts that fundamental diagrams of traffic flow are observed to exhibit.

Algebraic degrees of stretch factors in mapping class groups

Series
Dissertation Defense
Time
Monday, March 31, 2014 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Hyunshik Shin – Georgia Institute of Technology –
Given a closed surface S_g of genus g, a mapping class f is said to be pseudo-Anosov if it preserves a pair of transverse measured foliations such that one is expanding and the other one is contracting by a number $\lambda$. The number $\lambda$ is called a stretch factor (or dilatation) of f. Thurston showed that a stretch factor is an algebraic integer with degree bounded above by 6g-6. However, little is known about which degrees occur. Using train tracks on surfaces, we explicitly construct pseudo-Anosov maps on S_g with orientable foliations whose stretch factor $\lambda$ has algebraic degree 2g. Moreover, the stretch factor $\lambda$ is a special algebraic number, called Salem number. Using this result, we show that there is a pseudo-Anosov map whose stretch factor has algebraic degree d, for each positive even integer d such that d is less than or equal to g. Our examples also give a new approach to a conjecture of Penner.

Southeast Geometry Seminar XXIV

Series
Other Talks
Time
Sunday, March 30, 2014 - 08:30 for 8 hours (full day)
Location
Skiles 005
Speaker
Southeast Geometry Seminar – School of Mathematics, Georgia Tech
The Southeast Geometry Seminar is a series of semiannual one-day events focusing on geometric analysis. These events are hosted in rotation by the following institutions: The University of Alabama at Birmingham, The Georgia Institute of Technology, Emory University, The University of Tennessee Knoxville. The following six speakers will give presentations on topics that include geometric analysis, and related fields, such as partial differential equations, general relativity, and geometric topology: Robert Finn (Stanford University), Bo Guan (Ohio State University), John Harvey (University of Notre Dame), Fernando Schwartz (University of Tennessee), Henry Wente (Toledo, Ohio), Xiangwen Zhang (Columbia University) .

Topics in Ergodic Theory IV: Shannon-McMillan-Breiman Theorem.

Series
Dynamical Systems Working Seminar
Time
Friday, March 28, 2014 - 13:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Lei Zhang – Georgia Tech
We present the proof of the Shannon-McMillan-Breiman Theorem. This is part of a reading seminar geared towards understanding of Smooth Ergodic Theory. (The study of dynamical systems using at the same time tools from measure theory and from differential geometry)It should be accesible to graduate students and the presentation is informal. The first goal will be a proof of the Oseledets multiplicative ergodic theorem for random matrices. Then, we will try to cover the Pesin entropy formula, invariant manifolds, etc.

Average Case Equilibria

Series
ACO Student Seminar
Time
Friday, March 28, 2014 - 12:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Ioannis Panageas – Georgia Tech
Since the 50s and Nash general proof of equilibrium existence in games it is well understood that even simple games may have many, even uncountably infinite, equilibria with different properties. In such cases a natural question arises, which equilibrium is the right one? In this work, we perform average case analysis of evolutionary dynamics in such cases of games. Intuitively, we assign to each equilibrium a probability mass that is proportional to the size of its region of attraction. We develop new techniques to compute these likelihoods for classic games such as the Stag Hunt game (and generalizations) as well as balls-bins games. Our proofs combine techniques from information theory (relative entropy), dynamical systems (center manifold theorem), and algorithmic game theory. Joint work with Georgios Piliouras

Maximal Ideal Spaces

Series
Analysis Working Seminar
Time
Friday, March 28, 2014 - 10:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Ishawari Kunwar – School of Math
Ishwari will cover chapter 5 section 1 of Bounded Analytic Functions.

Heat kernel asymptotics at the cut locus

Series
Stochastics Seminar
Time
Thursday, March 27, 2014 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Robert Neel – Lehigh Univ.
We discuss a technique, going back to work of Molchanov, for determining the small-time asymptotics of the heat kernel (equivalently, the large deviations of Brownian motion) at the cut locus of a (sub-) Riemannian manifold (valid away from any abnormal geodesics). We relate the leading term of the expansion to the structure of the cut locus, especially to conjugacy, and explain how this can be used to find general bounds as well as to compute specific examples. We also show how this approach leads to restrictions on the types of singularities of the exponential map that can occur along minimal geodesics. Further, time permitting, we extend this approach to determine the asymptotics for the gradient and Hessian of the logarithm of the heat kernel on a Riemannian manifold, giving a characterization of the cut locus in terms of the behavior of the log-Hessian, which can be interpreted in terms of large deviations of the Brownian bridge. Parts of this work are joint with Davide Barilari, Ugo Boscain, and Grégoire Charlot.

Generalized Measures of Correlation and Their Implications in GARCH and Heston Models

Series
Mathematical Finance/Financial Engineering Seminar
Time
Thursday, March 27, 2014 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Zhengjun Zhang – University of Wisconsin
Applicability of Pearson's correlation as a measure of explained variance is by now well understood. One of its limitations is that it does not account for asymmetry in explained variance. Aiming to obtain broad applicable correlation measures, we use a pair of r-squares of generalized regression to deal with asymmetries in explained variances, and linear or nonlinear relations between random variables. We call the pair of r-squares of generalized regression generalized measures of correlation (GMC). We present examples under which the paired measures are identical, and they become a symmetric correlation measure which is the same as the squared Pearson's correlation coefficient. As a result, Pearson's correlation is a special case of GMC. Theoretical properties of GMC show that GMC can be applicable in numerous applications and can lead to more meaningful conclusions and decision making. In statistical inferences, the joint asymptotics of the kernel based estimators for GMC are derived and are used to test whether or not two random variables are symmetric in explaining variances. The testing results give important guidance in practical model selection problems. In real data analysis, this talk presents ideas of using GMCs as an indicator of suitability of asset pricing models, and hence new pricing models may be motivated from this indicator.

The reduction of PPAD linear complementarity problems to bimatrix games

Series
ACO Colloquium
Time
Thursday, March 27, 2014 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 268
Speaker
Ilan Adler – University of California, Berkeley
It is well known that many optimization problems, ranging from linear programming to hard combinatorial problems, as well as many engineering and economics problems, can be formulated as linear complementarity problems (LCP). One particular problem, finding a Nash equilibrium of a bimatrix game (2 NASH), which can be formulated as LCP, motivated the elegant Lemke algorithm to solve LCPs. While the algorithm always terminates, it can generates either a solution or a so-called ‘secondary ray’. We say that the algorithm resolves a given LCP if a secondary ray can be used to certify, in polynomial time, that no solution exists. It turned out that in general, Lemke-resolvable LCPs belong to the complexity class PPAD and that, quite surprisingly, 2 NASH is PPAD-complete. Thus, Lemke-resolvable LCPs can be formulated as 2 NASH. However, the known formulation (which is designed for any PPAD problem) is very complicated, difficult to implement, and not readily available for potential insights. In this talk, I’ll present and discuss a simple reduction of Lemke-resolvable LCPs to bimatrix games that is easy to implement and have the potential to gain additional insights to problems (including several models of market equilibrium) for which the reduction is applicable.

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