Seminars and Colloquia by Series

Even K3,3's in Bipartite Graphs

Series
Graph Theory Seminar
Time
Thursday, March 28, 2013 - 12:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Peter Whalen – Georgia Tech
We show that any internally 4-connected non-planar bipartite graph contains a subdivision of K3,3 in which each subdivided path contains an even number of vertices. In addition to being natural, this result has broader applications in matching theory: for example, finding such a subdivision of K3,3 is the first step in an algorithm for determining whether or not a bipartite graph is Pfaffian. This is joint work with Robin Thomas.

Semidefinite method in extremal graph theory

Series
Job Candidate Talk
Time
Thursday, March 28, 2013 - 11:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Sergey Norin – McGill University
Many fundamental theorems in extremal graph theory can be expressed as linear inequalities between homomorphism densities. Lovasz and, in a slightly different formulation, Razborov asked whether it is true that every such inequality follows from a finite number of applications of the Cauchy-Schwarz inequality. In this talk we will show that the answer to this question is negative. Further, we will show that the problem of determining the validity of a linear inequality between homomorphism densities is undecidable. Hence such inequalities are inherently difficult in their full generality. These results are joint work with Hamed Hatami. On the other hand, the Cauchy-Schwarz inequality (a.k.a. the semidefinite method) represents a powerful tool for obtaining _particular_ results in asymptotic extremal graph theory. Razborov's flag algebras provide a formalization of this method and have been used in over twenty papers in the last four years. We will describe an application of flag algebras to Turan’s brickyard problem: the problem of determining the crossing number of the complete bipartite graph K_{m,n}. This result is based joint work with Yori Zwols.

Wolff's Ideal Problem in the Multiplier Algebra on weighted Dirichlet Space

Series
Analysis Seminar
Time
Wednesday, March 27, 2013 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Debendra Banjade – University of Alabama
In 1980, T. M. Wolff has given the following version of the ideal membership for finitely generated ideals in $H^{\infty}(\mathbb{D})$: \[\ensuremath{\mbox{If \,\,}\left\{ f_{j}\right\} _{j=1}^{n}}\subset H^{\infty}(\mathbb{D}),\, h\in H^{\infty}(\mathbb{D})\,\,\mbox{and }\]\[\vert h(z)\vert\leq\left(\underset{j=1}{\overset{n}{\sum}}\vert f_{j}(z)\vert^{2}\right)^{\frac{1}{2}}\,\mbox{for all \ensuremath{z\in\mathbb{D},}}\]then \[h^{3}\in\mathcal{I}\left(\left\{ f_{j}\right\} _{j=1}^{n}\right),\,\,\mbox{the ideal generated by \ensuremath{\left\{ f_{j}\right\} _{j=1}^{n}}in \ensuremath{H^{\infty}}\ensuremath{(\mathbb{D})}. }\]In this talk, we will give an analogue of the Wolff's ideal problem in the multiplier algebra on weighted Dirichlet space. Also, we will give a characterization for radical ideal membership.

Integrable systems as a tool in math-physics problems

Series
Research Horizons Seminar
Time
Wednesday, March 27, 2013 - 12:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Plamen Iliev – Georgia Tech, School of Math
In the last few years many problems of mathematical and physical interest, which may not be Hamiltonian or even dynamical, were solved using techniques from integrable systems. I will review some of these techniques and their connections to some open research problems.

Weak KAM theorem for the most general first-order Nonlinear partial differential equation

Series
Dynamical Systems Working Seminar
Time
Tuesday, March 26, 2013 - 16:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Xifeng Su – Academy of Mathematics and Systems Science, Chinese Academy of Sciences –
We consider the evolutionary first order nonlinear partial differential equations of the most general form \frac{\partial u}{\partial t} + H(x, u, d_x u)=0.By virtue of introducing a new type of solution semigroup, we establish the weak KAM theorem for such partial differential equations, i.e. the existence of weak KAM solutions or viscosity solutions. Indeed, by employing dynamical approach for characteristics, we develop the theory of associated global viscosity solutions in general. Moreover, the solution semigroup acting on any given continuous function will converge to a uniform limit as the time goes to infinity. As an application, we prove that such limit satisfies the the associated stationary first order partial differential equations: H(x, u, d_x u)=0.

Wasserstein distances in the analysis of time series and dynamical systems

Series
CDSNS Colloquium
Time
Tuesday, March 26, 2013 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Sjoerd Verduyn Lunel – Universiteit Utrecht –
A new approach based on Wasserstein distances, which are numerical costs ofan optimal transportation problem, allows to analyze nonlinear phenomena ina robust manner. The long-term behavior is reconstructed from time series, resulting in aprobability distribution over phase space. Each pair of probabilitydistributions is then assigned a numerical distance that quantifies thedifferences in their dynamical properties. From the totality of all these distances a low-dimensional representation ina Euclidean spaceis derived. This representation shows the functional relationships betweenthe dynamical systems under study. It allows to assess synchronizationproperties and also offers a new way of numerical bifurcation analysis.

Short proofs of coloring theorems on planar graphs

Series
Graph Theory Seminar
Time
Tuesday, March 26, 2013 - 12:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Bernard Lidicky – University of Illinois at Urbana-Champaign
A recent lower bound on the number of edges in a k-critical n-vertex graph by Kostochka and Yancey yields a half-page proof of the celebrated Grotzsch Theorem that every planar triangle-free graph is 3-colorable. We use the same bound to give short proofs of other known theorems on 3-coloring of planar graphs, among whose is the Grunbaum-Aksenov Theorem that every planar with at most three triangles is 3-colorable. We also prove the new result that every graph obtained from a triangle-free planar graph by adding a vertex of degree at most four is 3-colorable. Joint work with O. Borodin, A. Kostochka and M. Yancey.

Conditional independence models

Series
Other Talks
Time
Monday, March 25, 2013 - 17:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Pedro Rangel – Georgia Tech
(algebraic statistics reading seminar)

Convergent series and domains of analyticity for response solutions in quasi-periodically forced strongly dissipative systems

Series
CDSNS Colloquium
Time
Monday, March 25, 2013 - 16:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Livia Corsi – University of Naples ``Federico II'' –
We study the ordinary differential equation \varepsilon \ddot x + \dot x + \varepsilon g(x) = \e f(\omega t), with f and g analytic and f quasi-periodic in t with frequency vector \omega\in\mathds{R}^{d}. We show that if there exists c_{0}\in\mathds{R} such that g(c_{0}) equals the average of f and the first non-zero derivative of g at c_{0} is of odd order \mathfrak{n}, then, for \varepsilon small enough and under very mild Diophantine conditions on \omega, there exists a quasi-periodic solution "response solution" close to c_{0}, with the same frequency vector as f. In particular if f is a trigonometric polynomial the Diophantine condition on \omega can be completely removed. Moreover we show that for \mathfrak{n}=1 such a solution depends analytically on \e in a domain of the complex plane tangent more than quadratically to the imaginary axis at the origin. These results have been obtained in collaboration with Roberto Feola (Universit\`a di Roma ``La Sapienza'') and Guido Gentile (Universit\`a di Roma Tre).

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