Seminars and Colloquia by Series

Floating-point shadowing for 2D saddle-connection

Series
CDSNS Colloquium
Time
Monday, October 28, 2013 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 202
Speaker
Dmitry Todorov – Chebyshev laboratory, Saint-Petersburg
There is known a lot of information about classical or standard shadowing. Itis also often called a pseudo-orbit tracing property (POTP). Let M be a closedRiemannian manifold. Diffeomorphism f : M → M is said to have POTPif for a given accuracy any pseudotrajectory with errors small enough can beapproximated (shadowed) by an exact trajectory. Therefore, if one wants to dosome numerical investiagion of the system one would definitely prefer it to haveshadowing property.However, now it is widely accepted that good (qualitatively strong) shad-owing is present only in hyperbolic situations. However it seems that manynonhyperbolic systems still could be well analysed numerically.As a step to resolve this contradiction I introduce some sort of weaker shad-owing. The idea is to restrict a set of pseudotrajectories to be shadowed. Onecan consider only pseudotrajectories that resemble sequences of points generatedby a computer with floating-point arithmetic.I will tell what happens in the (simplified) case of “linear” two-dimensionalsaddle connection. In this case even stochastic versions of classical shadowing(when one tries to ask only for most pseudotrajectories to be shadowed) do notwork. Nevertheless, for “floating-point” pseudotrajectories one can prove somepositive results.There is a dichotomy: either every pseudotrajectory stays close to the un-perturbed trajectory forever if one carefully chooses the dependence betweenthe size of errors and requested accuracy of shadowing, or there is always apseudotrajectory that can not be shadowed.

On filtrations of scissors congruence spectra

Series
Algebra Seminar
Time
Monday, October 28, 2013 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Inna Zakharevich – IAS/University of Chicago
The scissors congruence group of polytopes in $\mathbb{R}^n$ is defined tobe the free abelian group on polytopes in $\mathbb{R}^n$ modulo tworelations: $[P] = [Q]$ if $P\cong Q$, and $[P \cup P'] = [P] + [P']$ if$P\cap P'$ has measure $0$. This group, and various generalizations of it,has been studied extensively through the lens of homology of groups byDupont and Sah. However, this approach has many limitations, the chief ofwhich is that the computations of the group quickly become so complicatedthat they obfuscate the geometry and intuition of the original problementirely. We present an alternate approach which keeps the geometry of theproblem central by rephrasing the problem using the tools of algebraic$K$-theory. Although this approach does not yield any new computations asyet (algebraic $K$-theory being notoriously difficult to compute) it hasseveral advantages. Firstly, it presents a spectrum, rather than just agroup, invariant of the problem. Secondly, it allows us to construct suchspectra for all scissors congruence problems of a particular flavor, thusgiving spectrum analogs of groups such as the Grothendieck ring ofvarieties and scissors congruence groups of definable sets. And lastly, itallows us to construct filtrations by filtering the set of generators ofthe groups, rather than the group itself. This last observation allows usto construct a filtration on the Grothendieck spectrum of varieties that does not (necessarily) exist on the ring.

Rogue waves: fantascience or reality?

Series
Applied and Computational Mathematics Seminar
Time
Monday, October 28, 2013 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Francesco G. Fedele – GT Civil Eng and ECE –
Rogue waves are unusually large waves that appear from nowhere at the ocean. In the last 10 years or so, they have been the subject of numerous studies that propose homoclinic orbits of the NLS equation, the so-called breathers, to model such extreme events. Clearly, the NLS equation is an asymptotic approximation of the Euler equations in the spectral narrowband limit and it does not capture strong nonlinear features of the full Euler model. Motivated by the preceding studies, I will present recent results on deep-water modulated wavetrains and breathers of the Hamiltonian Zakharov equation, higher-order asymptotic model of the Euler equations for water waves. They provide new insights into the occurrence and existence of rogue waves and their breaking. Web info: http://arxiv.org/abs/1309.0668

Tight small Seifert fibered manifolds

Series
Geometry Topology Seminar
Time
Monday, October 28, 2013 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Bulent Tosun – University of Virginia
Contact geometry in three dimensions is a land of two disjoint classes ofcontact structures; overtwisted vs. tight. The former ones are flexible,means their geometry is determined by algebraic topology of underlying twoplane fields. In particular their existence and classification areunderstood completely. Tight contact structure, on the other hand, arerigid. The existence problem of a tight contact structure on a fixed threemanifold is hard and still widely open. The classification problem is evenharder. In this talk, we will focus on the classification of tight contactstructures on Seifert fibered manifolds on which the existence problem oftight contact structures was settled recently by Lisca and Stipsicz.

Progressions with a pseudorandom step

Series
Combinatorics Seminar
Time
Friday, October 25, 2013 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Elad Horev – University of Hamburg
An open problem of interest in combinatorial number theory is that of providing a non-ergodic proof to the so called polynomial Szemeredi theorem. So far, the landmark result in this venue is that of Green who considered the emergence of 3-term arithmetic progressions whose gap is a sum of two squares (not both zero) in dense sets of integers. In view of this we consider the following problem. Given two dense subsets A and S of a finite abelian group G, what is the weakest "pseudorandomness assumption" once put on S implies that A contains a 3-term arithmetic progressions whose gap is in S? We answer this question for G=Z_n and G = F_p^n. To quantify pseudorandomness we use Gowers norms.

Markov functions: reflections and musings

Series
ACO Student Seminar
Time
Friday, October 25, 2013 - 13:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Ton Dieker – ISyE, Georgia Tech
This talk evolves around Markov functions, i.e., when a function of a Markov chain results in another Markov chain. We focus on two examples where this concept yields new results and insights: (1) the evolution of reflected stochastic processes in the study of stochastic networks, and (2) spectral analysis for a special high-dimensional Markov chain.

Thresholds for Random Geometric k-SAT

Series
Stochastics Seminar
Time
Thursday, October 24, 2013 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Will Perkins – Georgia Tech, School of Mathematics
Random k-SAT is a distribution over boolean formulas studied widely in both statistical physics and theoretical computer science for its intriguing behavior at its phase transition. I will present results on the satisfiability threshold in a geometric model of random k-SAT: labeled boolean literals are placed uniformly at random in a d-dimensional cube, and for each set of k contained in a ball of radius r, a k-clause is added to the random formula. Unlike standard random k-SAT, this model exhibits dependence between the clauses. For all k we show that the satisfiability threshold is sharp, and for k=2 we find the location of the threshold as well. I will also discuss connections between this model, the random geometric graph, and other probabilistic models. This is based on joint work with Milan Bradonjic.

Breakdown of linear response for smooth families of dynamical systems with bifurcations

Series
School of Mathematics Colloquium
Time
Thursday, October 24, 2013 - 11:00 for 1 hour (actually 50 minutes)
Location
Skyles 006
Speaker
Viviane Baladi – Ecole Normale Superieure, Paris
(Joint with: M. Benedicks and D. Schnellmann) Many interesting dynamical systems possess a unique SRB ("physical")measure, which behaves well with respect to Lebesgue measure. Given a smooth one-parameter family of dynamical systems f_t, is natural to ask whether the SRB measure depends smoothly on the parameter t. If the f_t are smooth hyperbolic diffeomorphisms (which are structurally stable), the SRB measure depends differentiably on the parameter t, and its derivative is given by a "linear response" formula (Ruelle, 1997). When bifurcations are present and structural stability does not hold, linear response may break down. This was first observed for piecewise expanding interval maps, where linear response holds for tangential families, but where a modulus of continuity t log t may be attained for transversal families (Baladi-Smania, 2008). The case of smooth unimodal maps is much more delicate. Ruelle (Misiurewicz case, 2009) and Baladi-Smania (slow recurrence case, 2012) obtained linear response for fully tangential families (confined within a topological class). The talk will be nontechnical and most of it will be devoted to motivation and history. We also aim to present our new results on the transversal smooth unimodal case (including the quadratic family), where we obtain Holder upper and lower bounds (in the sense of Whitney, along suitable classes of parameters).

Riemann's mapping theorem for variable metrics

Series
Geometry Topology Student Seminar
Time
Wednesday, October 23, 2013 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Jing Hu – Georgia Tech
Consider the Beltrami equation f_{\bar z}=\mu *f_{z}. The prime aim is to investigate f in its dependence on \mu. If \mu depends analytically, differentiably, or continuously on real parameters, the same is true for f; in the case of the plane, the results holds also for complex parameters.

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