Seminars and Colloquia by Series

Tolerance Graphs and Orders

Series
Combinatorics Seminar
Time
Wednesday, March 11, 2009 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Ann Trenk – Department of Mathematics, Wellesley College
Tolerance graphs were introduced in 1982 by Golumbic and Monma as a generalization of the class of interval graphs. A graph G= (V, E) is an interval graph if each vertex v \in V can be assigned a real interval I_v so that xy \in E(G) iff I_x \cap I_y \neq \emptyset. Interval graphs are important both because they arise in a variety of applications and also because some well-known recognition problems that are NP-complete for general graphs can be solved in polynomial time when restricted to the class of interval graphs. In certain applications it can be useful to allow a representation of a graph using real intervals in which there can be some overlap between the intervals assigned to non-adjacent vertices. This motivates the following definition: a graph G= (V, E) is a tolerance graph if each vertex v \in V can be assigned a real interval I_v and a positive tolerance t_v \in {\bf R} so that xy \in E(G) iff |I_x \cap I_y| \ge \min\{t_x,t_y\}. These topics can also be studied from the perspective of ordered sets, giving rise to the classes of Interval Orders and Tolerance Orders. In this talk we give an overview of work done in tolerance graphs and orders . We use hierarchy diagrams and geometric arguments as unifying themes.

Classical solutions in Hölder and Sobolev spaces for the thin film equation

Series
PDE Seminar
Time
Tuesday, March 10, 2009 - 15:05 for 1.5 hours (actually 80 minutes)
Location
Skiles 255
Speaker
Hans Knüpfer – Courant Institute, New York
We consider the the following fourth order degenerate parabolic equation h_t + (hh_xxx)_x = 0. The equation arises in the lubrication approximation regime, describing the spreading of a thin film liquid with height profile h >= 0 on a plate. We consider the equation as free boundary problem, defined on its positivity set. We address existence and regularity of classical solutions in weighted Hölder and Sobolev spaces.

Fluctuation Theorems

Series
CDSNS Colloquium
Time
Monday, March 9, 2009 - 16:30 for 2 hours
Location
Skiles 255
Speaker
Mark Pollicott – University of Warwick
The Cohen-Gallavotti Fluctuation theorem is a result describing the behaviour of simple hyperbolic dynamical systems. It was introduced to illustrate, in a somewhat simpler context, anomalies in the second law of thermodynamics. I will describe the mathematical formulation of this Fluctuation Theorem, and some variations on it.

Recent Progresses and Challenges in High-Order Unstructured Grid Methods in CFD

Series
Applied and Computational Mathematics Seminar
Time
Monday, March 9, 2009 - 13:05 for 1.5 hours (actually 80 minutes)
Location
Skiles 255
Speaker
Zhi J. Wang – Aerospace Engineering, Iowa State University
The current breakthrough in computational fluid dynamics (CFD) is the emergence of unstructured grid based high-order (order > 2) methods. The leader is arguably the discontinuous Galerkin method, amongst several other methods including the multi-domain spectral, spectral volume (SV), and spectral difference (SD) methods. All these methods possess the following properties: k-exactness on arbitrary grids, and compactness, which is especially important for parallel computing. In this talk, recent progresses in the DG, SV, SD and a unified formulation called lifting collocation penalty will be presented. Numerical simulations with the SV and the SD methods will be presented. The talk will conclude with several remaining challenges in the research on high-order methods.

Invariants of Legendrian Knots from Open Book Decompositions

Series
Geometry Topology Seminar
Time
Monday, March 9, 2009 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Sinem Onaran – School of Mathematics, Georgia Tech
Given any contact 3-manifold, Etnyre and Ozbagci defined new invariants of the contact structures in terms of open book decompositions supporting the contact structure. One of the invariants is the support genus of the contact structure which is defined as the minimal genus of a page of an open book that supports the contact structure. In a similar fashion, we define the support genus sg(L) of a Legendrian knot L in a contact manifold M as the minimal genus of a page of an open book of M supporting the contact structure such that L sits on a page and the framing given by the contact structure and by the page agree. In this talk, we will discuss the support genus of Legendrian knots in contact 3-manifolds. We will show any null-homologous loose knot has support genus zero. To prove this, we observe an interesting topological property of knots and links on the way. We observe any topological knot or link in a 3-manifold sits on a planar page (genus zero page) of an open book decomposition.

Coupling with respect to initial conditions for deterministic dynamics

Series
Probability Working Seminar
Time
Friday, March 6, 2009 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 268
Speaker
Alex Grigo – School of Mathematics, Georgia Tech
The talk is based on the paper titled "Anosov diffeomorphisms and coupling" by Bressaud and Liverani. Existence and uniqueness of SRB invariant measure for the dynamics is established via a coupling of initial conditions introduced to dynamics by L.-S. Young.

Graph parallel rigidity

Series
Combinatorics Seminar
Time
Friday, March 6, 2009 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Alexey Spiridonov – MIT
This is joint work with Alex Postnikov. Imagine that you are to build a system of space stations (graph vertices), which communicate via laser beams (edges). The edge directions were already chosen, but you must place the stations so that none of the beams miss their targets. In this talk, we let the edge directions be generic and independent, a choice that constrains vertex placement the most. For K_{3} in \mathbb{R}^{2}, the edges specify a unique triangle, but its size is arbitrary --- D_{2}(K_{3})=1 degree of freedom; we say that K_{3} is rigid in \mathbb{R}^{2}. We call D_{n}(G) the degree of parallel rigidity of the graph for generic edge directions. We found an elegant combinatorial characterization of D_{n}(G) --- it is equal to the minimal number of edges in the intersection of n spanning trees of G. In this talk, I will give a linear-algebraic proof of this result, and of some other properties of D_{n}(G). The notion of parallel graph rigidity was previously studied by Whiteley and Develin-Martin-Reiner. The papers worked with the generic parallel rigidity matroid; I will briefly compare our results in terms of D_{n}(G) with the previous work.

An introduction to mathematical learning theory

Series
SIAM Student Seminar
Time
Friday, March 6, 2009 - 12:30 for 2 hours
Location
Skiles 269
Speaker
Kai Ni – School of Mathematics, Georgia Tech
In this talk, I will briefly introduce some basics of mathematical learning theory. Two basic methods named perceptron algorithm and support vector machine will be explained for the separable classification case. Also, the subgaussian random variable and Hoeffding inequality will be mentioned in order to provide the upper bound for the deviation of the empirical risk. If time permits, the Vapnik combinatorics will be involved for shaper bounds of this deviation.

Shot Noise Process

Series
Stochastics Seminar
Time
Thursday, March 5, 2009 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Yuanhui Xiao – Department of Mathematics and Statistics, Georgia State University
A shot noise process is essentially a compound Poisson process whereby the arriving shots are allowed to accumulate or decay after their arrival via some preset shot (impulse response) function. Shot noise models see applications in diverse areas such as insurance, fi- nance, hydrology, textile engineering, and electronics. This talk stud- ies several statistical inference issues for shot noise processes. Under mild conditions, ergodicity is proven in that process sample paths sat- isfy a strong law of large numbers and central limit theorem. These results have application in storage modeling. Shot function parameter estimation from a data history observed on a discrete-time lattice is then explored. Optimal estimating functions are tractable when the shot function satisfies a so-called interval similar condition. Moment methods of estimation are easily applicable if the shot function is com- pactly supported and show good performance. In all cases, asymptotic normality of the proposed estimators is established.

Dimers and random interfaces

Series
School of Mathematics Colloquium
Time
Thursday, March 5, 2009 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Rick Kenyon – Mathematics Department, Brown University
This is joint work with Andrei Okounkov. The ``honeycomb dimer model'' is a natural model of discrete random surfaces in R^3. It is possible to write down a ``Law of Large Numbers" for such surfaces which describes the typical shape of a random surface when the mesh size tends to zero. Surprisingly, one can parameterize these limit shapes in a very simple way using analytic functions, somewhat reminiscent of the Weierstrass parameterization of minimal surfaces. This is even more surprising since the limit shapes tend to be facetted, that is, only piecewise analytic. There is a large family of boundary conditions for which we can obtain exact solutions to the limit shape problem using algebraic geometry techniques. This family includes the (well-known) solution to the limit shape of a ``boxed plane partition'' and has many generalizations.

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