Seminars and Colloquia by Series

Surgery Formulas and Heegaard Floer Homology of Mapping Tori

Series
Geometry Topology Seminar
Time
Monday, September 27, 2010 - 17:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Evan Fink – University of Georgia

Please Note: This is the second talk in the Emory-Ga Tech-UGA joint seminar. The first talk will begin at 3:45.

There are many conjectured connections between Heegaard Floer homology and the various homologies appearing in low dimensional topology and symplectic geometry. One of these conjectures states, roughly, that if \phi is a diffeomorphism of a closed Riemann surface, a certain portion of the Heegaard Floer homology of the mapping torus of \phi should be equal to the Symplectic Floer homology of \phi. I will discuss how this can be confirmed when \phi is periodic (i.e., when some iterate of \phi is the identity map). I will recall how a mapping torus can be realized via Dehn surgery; then, I will sketch how the surgery long exact triangles of Heegaard Floer homology can be distilled into more direct surgery formulas involving knot Floer homology. Finally, I'll say a few words about what actually happens when you use these formulas for the aforementioned Dehn surgeries: a "really big game of tic-tac-toe".

HOMFLY-PT polynomial and Legendrian links in the solid torus

Series
Geometry Topology Seminar
Time
Monday, September 27, 2010 - 15:45 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Dan Rutherford – Duke University

Please Note: This is the first talk in the Emory-Ga Tech-UGA joint seminar. The second talk will follow at 5.

A smooth knot in a contact 3-manifold is called Legendrian if it is always tangent to the contact planes. In this talk, I will discuss Legendrian knots in R^3 and the solid torus where knots can be conveniently viewed using their `front projections'. In particular, I will describe how certain decompositions of front projections known as `normal rulings' (introduced by Fuchs and Chekanov-Pushkar) can be used to give combinatorial descriptions for parts of the HOMFLY-PT and Kauffman polynomials. I will conclude by discussing recent generalizations to Legendrian solid torus links. It is usual to identify the `HOMFLY-PT skein module' of the solid torus with the ring of symmetric functions. In this context, normal rulings can be used to give a knot theory description of the standard scalar product determined by taking the Schur functions to form an orthonormal basis.

Theory and applications of fractal transformations

Series
Analysis Seminar
Time
Monday, September 27, 2010 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Michael Barnsley – Department of Mathematics, Australian National University
Let A and B be attractors of two point-fibred iterated function systems with coding maps f and g. A transformations from A into B can be constructed by composing a branch of the inverse of f with g. I will outline the shape of the theory of such transformations, which are termed "fractal" because their graphs are typically of non-integer dimension. I will also describe the remarkable geometry of these transformations when the generating iterated functions systems are projective. Finally, I will show how they can be used to provide new insights into dynamical systems and also how they can be used to manipulate, filter, process and efficiently store digital images, and how they can be used in image synthesis, leading to applications in the visual arts.

Crossings and nestings of two edges in set partitions

Series
Combinatorics Seminar
Time
Friday, September 24, 2010 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Svetlana Poznanovikj – SoM, Georgia Tech
A set partition of [n] can be represented graphically by drawing n dots on a horizontal line and connecting the points in a same block by arcs. Crossings and nestings are then pairs of arcs that cross or nest. Let G be an abelian group, and \alpha, \beta \in G. In this talk I will look at the distribution of the statistic s_{\alpha, \beta} = \alpha * cr + \beta * ne on subtrees of the tree of all set partitions and present a result which says that the distribution of s_{\alpha, \beta} on a subtree is determined by its distribution on the first two levels.

Small-time statistical behavior of Levy processes and its application to the estimation and pricing of Levy-based financial models

Series
Mathematical Finance/Financial Engineering Seminar
Time
Friday, September 24, 2010 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 002
Speaker
J.E. Figueroa-Lopez – Purdue University
The first order small-time approximation of the marginal distribution of a L\'evy process has been known for long-time. In this talk, I present higher order expansions polynomial in time for the distributions of a L\'evy process. As a secondary objective, I illustrate the application of our expansions in the estimation of financial models with jumps as well as in the study of the small-term asymptotic behavior of the implied volatility for this class of financial models. This talk presents joint work with C. Houdr\'e and M. Forde. Associated reading (available in the web site of the speaker): (1) Small-time expansions for the transition distribution of Levy processes. J.E. Figueroa-L\'opez and C. Houdré. Stochastic Processes and their Applications 119 pp. 3862-3889, 2009. (2) Nonparametric estimation of time-changed Levy models under high-frequency data. J.E. Figueroa-L\'opez. Advances in Applied Probability vol. 41, number 4, pp. 1161-1188, 2009. (3) The small-maturity smile for exponential Levy model. J.E. Figueroa-L\'opez and M. Forde. Preprint.

Introduction to (some versions of) Heegaard-Floer Homology

Series
Geometry Topology Working Seminar
Time
Friday, September 24, 2010 - 14:00 for 2 hours
Location
Skiles 171
Speaker
Amey Kaloti – Ga Tech
This will be an introduction to the basic aspects of Heegaard-Floer homology and knot Heegaard-Floer homology. After this talk (talks) we will be organizing a working group to go through various computations and results in knot Heegaard-Floer theory and invariants of Legendrian knots.

Small Noise: Dynamical Systems and Probability put together

Series
SIAM Student Seminar
Time
Friday, September 24, 2010 - 13:05 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Serjio Almanda – School of Mathematics, Georgia Tech
In this talk I will outline a topic that has been of interest due to its applicability in physics and engineering. The so called small noise model is a very technical subject that lies in the center of probability theory and usually study thorough a large deviations approach. I will explain this terminology and why is the correlation with dynamical systems so strong. Recent developments will be given at the end if time allows.

Choosability of planar graphs

Series
Graph Theory Seminar
Time
Thursday, September 23, 2010 - 12:05 for 1 hour (actually 50 minutes)
Location
Skiles 114
Speaker
Zdenek Dvorak – Charles University, Prague, Czech Republic
A graph is k-choosable if it can be properly colored from any assignment of lists of colors of length at least k to its vertices. A well-known results of Thomassen state that every planar graph is 5-choosable and every planar graph of girth 5 is 3-choosable. These results are tight, as shown by constructions of Voigt. We review some new results in this area, concerning 3-choosability of planar graphs with constraints on triangles and 4-cycles.

A non-commutative Wiener Inversion Theorem and Schroedinger dispersive estimates

Series
Analysis Seminar
Time
Wednesday, September 22, 2010 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Michael Goldberg – University of Cincinnati –
We prove an extension of the Wiener inversion theorem for convolution of summable series, allowing the terms to take values in a space of bounded linear operators. The resulting algebra is no longer commutative due to the composition of operators. Inversion theorems arise naturally in the context of proving dispersive estimates for the Schr\"odinger and wave equation and lead to scale-invariant conditions for the class of admissible potentials. All results are joint work with Marius Beceanu.

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