Seminars and Colloquia by Series

A functional approach to computer assisted proofs in dynamics

Series
Dynamical Systems Working Seminar
Time
Thursday, November 8, 2012 - 16:30 for 1.5 hours (actually 80 minutes)
Location
Skiles 06
Speaker
Rafael de la Llave – Georgia Tech
The existence of several objects in dynamics can be reduced to the existence of solutions of several functional equations, which then, are dealt with using fixed point theorems (e.g. the contraction mapping principle). This opens the possibility to take numerical approximations and validate them. This requires to take into account truncation, roundoff and other sources of error. I will try to present the principles involved as well as some practical implementations of a basic library. Much of this is work with others including D. Rana, R. Calleja, J. L. Figueras.

A Wong-Zakai Approximation Scheme for Reflected Stochastic Differential Equations

Series
Stochastics Seminar
Time
Thursday, November 8, 2012 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Chris Evans – University of Missouri
In a series of famous papers E. Wong and M. Zakai showed that the solution to a Stratonovich SDE is the limit of the solutions to a corresponding ODE driven by the piecewise-linear interpolation of the driving Brownian motion. In particular, this implies that solutions to Stratonovich SDE "behave as we would expect from ODE theory". Working with my PhD adviser, Daniel Stroock, we have shown that a similar approximation result holds, in the sense of weak convergence of distributions, for reflected Stratonovich SDE.

Colin de Verdiere-type invariants for signed graphs and odd-K_4- and odd-K^2_3-free signed graphs

Series
Graph Theory Seminar
Time
Thursday, November 8, 2012 - 12:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Hein van der Holst – Georgia State University
A signed graph is a pair $(G,\Sigma)$ where $G$ is an undirected graph (in which parallel edges are permitted, but loops are not) and $\Sigma \subseteq E(G)$. The edges in $\Sigma$ are called odd and the other edges are called even. A cycle of $G$ is called odd if it has an odd number of odd edges. If $U\subseteq V(G)$, then re-signing $(G,\Sigma)$ on $U$ gives the signed graph $(G,\Sigma\Delta \delta(U))$. A signed graph is a minor of $(G,\Sigma)$ if it comes from $(G,\Sigma)$ by a series of re-signing, deletions of edges and isolated vertices, and contractions of even edges. If $(G,\Sigma)$ is a signed graph with $n$ vertices, $S(G,\Sigma)$ is the set of all symmetric $n\times n$ matrices $A=[a_{i,j}]$ with $a_{i,j} > 0$ if $i$ and $j$ are connected by only odd edges, $a_{i,j} < 0$ if $i$ and $j$ are connected by only even edges, $a_{i,j}\in \mathbb{R}$ if $i$ and $j$ are connected by both even and odd edges, $a_{i,j}=0$ if $i$ and $j$ are not connected by any edges, and $a_{i,i} \in \mathbb{R}$ for all vertices $i$. The stable inertia set, $I_s(G,\Sigma)$, of a signed graph $(G,\Sigma)$ is the set of all pairs $(p,q)$ such that there exists a matrix $A\in S(G,\Sigma)$ that has the Strong Arnold Hypothesis, and $p$ positive and $q$ negative eigenvalues. The stable inertia set of a signed graph forms a generalization of $\mu(G)$, $\nu(G)$ (introduced by Colin de Verdi\`ere), and $\xi(G)$ (introduced by Barioli, Fallat, and Hogben). A specialization of $I_s(G,\Sigma)$ is $\nu(G,\Sigma)$, which is defined as the maximum of the nullities of positive definite matrices $A\in S(G,\Sigma)$ that have the Strong Arnold Hypothesis. This invariant is closed under taking minors, and characterizes signed graphs with no odd cycles as those signed graphs $(G,\Sigma)$ with $\nu(G,\Sigma)\leq 1$, and signed graphs with no odd-$K_4$- and no odd-$K^2_3$-minor as those signed graphs $(G,\Sigma)$ with $\nu(G,\Sigma)\leq 2$. In this talk we will discuss $I_s(G,\Sigma)$, $\nu(G,\Sigma)$ and these characterizations. Joint work with Marina Arav, Frank Hall, and Zhongshan Li.

Population persistence in the face of demographic and environmental uncertainty

Series
School of Mathematics Colloquium
Time
Thursday, November 8, 2012 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Sebastain Schreiber – UC Davis –
Populations, whether they be viral particles, bio-chemicals, plants or animals, are subject to intrinsic and extrinsic sources of stochasticity. This stochasticity in conjunction with nonlinear interactions between individuals determines to what extinct populations are able to persist in the long-term. Understanding the precise nature of these interactive effects is a central issue in population biology from theoretical, empirical, and applied perspectives. For the first part of this talk, I will discuss, briefly, the relationship between attractors of deterministic models and quasi-stationary distributions of their stochastic, finite population counterpoints i.e. models accounting for demographic stochasticity. These results shed some insight into when persistence should be observed over long time frames despite extinction being inevitable. For the second part of the talk, I will discuss results on stochastic persistence and boundedness for stochastic models accounting for environmental (but not demographic) noise. Stochastic boundedness asserts that asymptotically the population process tends to remain in compact sets. In contrast, stochastic persistence requires that the population process tends to be "repelled" by some "extinction set." Using these results, I will illustrate how environmental noise can facilitate coexistence of competing species and how dispersal in stochastic environments can rescue locally extinction prone populations. Empirical work on Kansas prairies, acorn woodpecker populations, and microcosm experiments demonstrating these phenomena will be discussed.

Horn inequalities for submodules

Series
Analysis Seminar
Time
Wednesday, November 7, 2012 - 14:00 for 1 hour (actually 50 minutes)
Location
005
Speaker
Wing Suet Li – Mathematics, Georgia Tech –
Consider three partitions of integers a=(a_1\ge a_2\ge ... \ge a_n\ge 0), b=(b_1\ge b_2\ge ... \ge b_n \ge 0), and c=(c_1\ge \ge c_2\ge ... \ge c_n\ge 0). It is well-known that a triple of partitions (a,b,c) that satisfies the so-call Littlewood-Richardson rule describes the eigenvalues of the sum of nXn Hermitian matricies, i.e., Hermitian matrices A, B, and C such that A+B=C with a (b and c respectively) as the set of eigenvalues of A (B and C respectively). At the same time such triple also describes the Jordan decompositions of a nilpotent matrix T, T resticts to an invarint subspace M, and T_{M^{\perp}} the compression of T onto the M^{\perp}. More precisely, T is similar to J(c):=J_(c_1)\oplus J_(c_2)\oplus ... J_(c_n)$, and T|M is similar to J(a) and T_{M^{\perp}} is similar to J(b). (Here J(k) denotes the Jordan cell of size k with 0 on the diagonal.) In addition, these partitions must also satisfy the Horn inequalities. In this talk I will explain the connections between these two seemily unrelated objects in matrix theory and why the same combinatorics works for both. This talk is based on the joint work with H. Bercovici and K. Dykema.

An Approach to the Hyperplane Conjecture

Series
Research Horizons Seminar
Time
Wednesday, November 7, 2012 - 12:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Santosh Vempala – Georgia Tech, College of Computing
The hyperplane conjecture of Kannan, Lovasz and Simonovits asserts that the isoperimetric constant of a logconcave measure (minimum surface to volume ratio over all subsets of measure at most half) is approximated by a halfspace to within an absolute constant factor. I will describe the motivation, implications and some developments around the conjecture and an approach to resolving it (which does not seem entirely ridiculous).

Diagonal Actions on Homogeneous Spaces I:

Series
Dynamical Systems Working Seminar
Time
Tuesday, November 6, 2012 - 16:35 for 1.5 hours (actually 80 minutes)
Location
Skiles 006
Speaker
Mikel J. de Viana – Georgia Tech
The study of actions of subgroups of SL(k,\R) on the space of unimodular lattices in \R^k has received considerable attention since at least the 1970s. The dynamical properties of these systems often have important consequences, such as for equidistribution results in number theory. In particular, in 1984, Margulis proved the Oppenheim conjecture on values of indefinite, irrational quadratic forms by studying one dimensional orbits of unipotent flows. A more complicated problem has been the study of the action by left multiplication by positive diagonal matrices, A. We will discuss the main ideas in the work of Einsiedler, Katok and Lindenstrauss where a measure classification is obtained, assuming that there is a one parameter subgroup of A which acts with positive entropy. The first talk is devoted to completing our understanding of the unipotent actions in SL(2,\Z)\ SL(2,\R), a la Ratner, because it is essential to understanding the "low entropy method" of Lindenstrauss. We will then introduce the necessary tools and assumptions, and next week we will complete the classification by application of two complementary methods.

Compressible Navier-Stokes equations with temperature dependent dissipation.

Series
PDE Seminar
Time
Tuesday, November 6, 2012 - 15:01 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Professor Ronghua Pan – Georgia Tech –
From its physical origin, the viscosity and heat conductivity in compressible fluids depend on absolute temperature through power laws. The mathematical theory on the well-posedness and regularity on this setting is widely open. I will report some recent progress made on this direction, with emphasis on the lower bound of temperature, and global existence of solutions in one or multiple dimensions. The relation between thermodynamics laws and Navier-Stokes equations will also be discussed. This talk is based on joint works with Weizhe Zhang.

Discrete Mathematical Biology Working Seminar

Series
Other Talks
Time
Tuesday, November 6, 2012 - 10:00 for 1 hour (actually 50 minutes)
Location
Skiles 114
Speaker
Emily Rogers – Georgia Tech
A discussion of the paper "Genetic network inference: from co-expression clustering to reverse engineering" by P. D'haeseleer, S. Liang, and R. Somogyi (Bioinformatics, 2000).

A STOCHASTIC EXPANSION-BASED APPROACH FOR DESIGN UNDER UNCERTAINTY

Series
CDSNS Colloquium
Time
Monday, November 5, 2012 - 16:00 for 1 hour (actually 50 minutes)
Location
Skiles 06
Speaker
Miguel Walter – Georgia Tech (Aerospace Eng.)
A common practice in aerospace engineering has been to carry out deterministicanalysis in the design process. However, due to variations in design condition suchas material properties, physical dimensions and operating conditions; uncertainty isubiquitous to any real engineering system. Even though the use of deterministicapproaches greatly simplifies the design process since any uncertain parameter is setto a nominal value, the final design can have degraded performance if the actualparameter values are slightly different from the nominal ones.Uncertainty is important because designers are concerned about performance risk.One of the major challenges in design under uncertainty is computational efficiency,especially for expensive numerical simulations. Design under uncertainty is composedof two major parts. The first one is the propagation of uncertainties, and the otherone is the optimization method. An efficient approach for design under uncertaintyshould consider improvement in both parts.An approach for robust design based on stochastic expansions is investigated. Theresearch consists of two parts : 1) stochastic expansions for uncertainty propagationand 2) adaptive sampling for Pareto front approximation. For the first part, a strategybased on the generalized polynomial chaos (gPC) expansion method is developed. Acommon limitation in previous gPC-based approaches for robust design is the growthof the computational cost with number of uncertain parameters. In this research,the high computational cost is addressed by using sparse grids as a mean to alleviatethe curse of dimensionality. Second, in order to alleviate the computational cost ofapproximating the Pareto front, two strategies based on adaptive sampling for multi-objective problems are presented. The first one is based on the two aforementionedmethods, whereas the second one considers, in addition, two levels of fidelity of theuncertainty propagation method.The proposed approaches were tested successfully in a low Reynolds number airfoilrobust optimization with uncertain operating conditions, and the robust design of atransonic wing. The gPC based method is able to find the actual Pareto front asa Monte Carlo-based strategy, and the bi-level strategy shows further computationalefficiency.

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