Seminars and Colloquia by Series

Asymptotic Properties of Muntz Orthogonal Polynomials

Series
Dissertation Defense
Time
Tuesday, April 27, 2010 - 13:00 for 2 hours
Location
Skiles 269
Speaker
Ulfar Stefansson – School of Mathematics, Georgia Tech
Müntz polynomials arise from consideration of Müntz's Theorem, which is a beautiful generalization of Weierstrass's Theorem. We prove a new surprisingly simple representation for the Müntz orthogonal polynomials on the interval of orthogonality, and in particular obtain new formulas for some of the classical orthogonal polynomials (e.g. Legendre, Jacobi, Laguerre). This allows us to determine the strong asymptotics and endpoint limit asymptotics on the interval. The zero spacing behavior follows, as well as estimates for the smallest and largest zeros. This is the first time that such asymptotics have been obtained for general Müntz exponents. We also look at the asymptotic behavior outside the interval and the asymptotic properties of the associated Christoffel functions.

Quasi-isometries of groups and spaces

Series
Research Horizons Seminar
Time
Tuesday, April 27, 2010 - 12:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Igor Belegradek – Professor, School of Mathematics

Please Note: Hosted by: Huy Huynh and Yao Li

A starting point of geometric group theory is thinking of a group as a geometric object, by giving it a metric induced from the Cayley graph of the group. Gromov initiated a program of studying groups up to quasi-isometries, which are ``bilipschitz maps up to bounded additive error". Quasi-isometries ignore local structure and preserve asymptotic properties of a metric space. In the talk I will give a sample of results, examples, and open questions in this area.

Noncommutative geometry and the field with one element

Series
School of Mathematics Colloquium
Time
Tuesday, April 27, 2010 - 11:05 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Matilde Marcolli – Caltech –
There are presently different approaches to definealgebraic geometry over the mysterious "field with one element".I will focus on two versions, one by Soule' and one by Borger,that appear to have a direct connection to NoncommutativeGeometry via the quantum statistical mechanics of Q-latticesand the theory of endomotives. I will also relate to endomotivesand Noncommutative Geometry the analytic geometry over F1,as defined by Manin in terms of the Habiro ring.

Estimates for Discrepancy and Calderon-Zygmund Operations

Series
Dissertation Defense
Time
Monday, April 26, 2010 - 15:00 for 2 hours
Location
Skiles 255
Speaker
Armen Vagharshakyan – School of Mathematics, Georgia Tech
We improve the lower bound for the L_\infty norm of the discrepancy function. This result makes a partial step towards resolving the Discrepancy Conjecture. Being a theorem in the theory of irregularities of distributions, it also relates to corresponding results in approximation theory (namely, the Kolmogorov entropy of spaces of functions with bounded mixed derivatives) and in probability theory (namely, Small Ball Inequality - small deviation inequality for the Brownian sheet). We also provide sharp bounds for the exponential Orlicz norm and the BMO norm of the discrepancy function in two dimensions. In the second part of the thesis we prove that any sufficiently smooth one-dimensional Calderon-Zygmund convolution operator can be recovered through averaging of Haar shift operators. This allows to generalize the estimates, which had been previously known for Haar shift operators, to Calderon-Zygmund operators. As a result, the A_2 conjecture is settled for this particular type of Calederon-Zygmund operators.

On the categorification of the quantum Casimir

Series
Geometry Topology Seminar
Time
Monday, April 26, 2010 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
A. Beliakova – University of Zurich
In the talk, I will gently introduce the Lauda-Khovanov 2-category, categorifying the idempotent form of the quantum sl(2). Then I will define a complex, whose Euler characteristic is the quantum Casimir. Finally, I will show that this complex naturally belongs to the center of the 2-category. The talk is based on the joint work with Aaron Lauda and Mikhail Khovanov.

Chern classes identities from weak coupling limits

Series
Algebra Seminar
Time
Monday, April 26, 2010 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 171
Speaker
Paolo Aluffi – Florida State University –
We generalize a construction of Ashoke Sen of `weak couplinglimits' for certain types of elliptic fibrations. Physics argumentsinvolving tadpole anomaly cancellations lead to conjectural identitiesof Euler characteristics. We generalize these identities to identitiesof Chern classes, which we are able to verify mathematically inseveral instances. For this purpose we propose a generalization of theso-called `Sethi-Vafa-Witten identity'. We also obtain a typeclassification of configurations of smooth branes satisfying thetadpole condition. This is joint work with Mboyo Esole (Harvard).

CANCELLED - Nonlinear resonance analysis as a base for novel numerical models

Series
Applied and Computational Mathematics Seminar
Time
Monday, April 26, 2010 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Elena Kartaschova – Johannes Kepler University
Nonlinear Resonance Analysis (NRA) is a natural next step after Fourieranalysis developed for linear PDEs. The main subject of NRA isevolutionary nonlinear PDEs, possessing resonant solutions. Importance ofNRA is due to its wide application area -- from climatepredictability to cancer diagnostic to breaking of the wing of an aircraft.In my talk I plan to give a brief overview of the methods and resultsavailable in NRA, and illustrate it with some examples from fluid mechanics.In particular, it will be shown how1) to use a general method of q-class decomposition for computing resonantmodes for a variety of physically relevant dispersion functions;2) to construct NR-reduced models for numerical simulations basing on theresonance clustering; theoretical comparision with Galerkin-like models willbe made and illustrated by the results of some numerical simulations withnonlinear PDE.3) to employ NR-reduced models for interpreting of real-life phenomena (inthe Earth`s atmosphere) and results of laboratory experiments with watertanks.A short presentation of the software available in this area will be given.

Sliding Modes and Fundamental Matrix Solutions of Piecewise Smooth Differential Systems

Series
Applied and Computational Mathematics Seminar
Time
Monday, April 26, 2010 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Luca Dieci – School of Mathematics, Georgia Tech
In this seminar we consider piecewise smooth differential systems of Filippov type, in which the vector field varies discontinuously as solution trajectories reach one or more surfaces. Emphasis is on the fundamental matrix solution associated to these systems. We consider the cases of transversal intersection and of sliding motion on a co-dimension one surface and when sliding motion takes place on a co-dimension two surface (the intersection of two co-dimension one surfaces). [Joint work with L.Lopez, Univ. of Bari]

Asymptotic entropy drops and escape rates for Gibbs measures

Series
CDSNS Colloquium
Time
Monday, April 26, 2010 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Mark Pollicott – University of Warwick
We consider a shift transformation and a Gibbs measure and estimate the drop in entropy caused by deleting an arbitrarily small (cylinder) set. This extends a result of Lind. We also estimate the speed at which the Gibbs measure escapes into the set, which relates to recent work of Bunimovich-Yurchenko and Keller-Liverani. This is joint with Andrew Ferguson.

Giant components in random subgraphs of general graphs

Series
Combinatorics Seminar
Time
Friday, April 23, 2010 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Paul Horn – Emory University
Erd\H{o}s and R\'enyi observed that a curious phase transition in the size of the largest component in arandom graph G(n,p): If pn < 1, then all components have size O(\log n), while if pn > 1 there exists a uniquecomponent of size \Theta(n). Similar transitions can be seen to exist when taking random subgraphs of socalled (n,d,\lambda) graphs (Frieze, Krivelevich and Martin), dense graphs (Bollobas et. al) and several otherspecial classes of graphs. Here we consider the story for graphs which are sparser and irregular. In thisregime, the answer will depend on our definition of a 'giant component'; but we will show a phase transitionfor graphs satisfying a mild spectral condition. In particular, we present some results which supersede ourearlier results in that they have weaker hypotheses and (in some sense) prove stronger results. Additionally,we construct some examples showing the necessity of our new hypothesis.

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