Seminars and Colloquia by Series

Unknotting knots: Khovanov Homology and a less violent approach to the Gordian problem

Series
Research Horizons Seminar
Time
Wednesday, September 15, 2010 - 12:00 for 1 hour (actually 50 minutes)
Location
Siles 171
Speaker
John Etnyre – School of Mathematics - Georgia Institute of Technology

Please Note: Hosted by Yao Li and Ricardo Restrepo.

 Deciding how to unknot a knotted piece of string (with its ends glued together) is not only a difficult problem in the real world, it is also a difficult and long studied problem in mathematics. (There are several notions of what one might mean by "unknotting" and I will leave the exact meaning a bit vague in this abstract.) In the past mathematicians have used a vast array of techniques --- from geometry to algebra, and even PDEs --- to study this question. I will discuss this question and (partially) recast it in terms of 4 dimensional topology. This new perspective will allow us to use a powerful new knot invariant called Khovanov Homology to study the problem. I will give an overview of Khovanov Homology and indicate how to study our unknotting question using it.

Incremental mutual information: a new method for characterizing the strength and dynamics of connections in neuronal circuits

Series
Mathematical Biology Seminar
Time
Wednesday, September 15, 2010 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 169
Speaker
Abhinav Singh – University College London –
Understanding the computations performed by neuronal circuits requires characterizing the strength and dynamics of the connections between individual neurons. This characterization is typically achieved by measuring the correlation in the activity of two neurons through the computation of a cross-correlogram or one its variants. We have developed a new measure for studying connectivity in neuronal circuits based on information theory, the incremental mutual information (IMI). IMI improves on correlation in several important ways: 1) IMI removes any requirement or assumption that the interactions between neurons is linear, 2) IMI enables interactions that reflect the connection between neurons to be differentiated from statistical dependencies caused by other sources (e.g. shared inputs or intrinsic cellular or network mechanisms), and 3) for the study of early sen- sory systems, IMI does not require that the external stimulus have any specific properties, nor does it require responses to repeated trials of identical stimulation. We describe the theory of IMI and demonstrate its utility on simulated data and experimental recordings from the visual system.

ACO/OR Colloquium - Advances in multistage optimization

Series
Other Talks
Time
Tuesday, September 14, 2010 - 11:00 for 1 hour (actually 50 minutes)
Location
ISyE Executive Classroom
Speaker
Dimitris Bertsimas – Operations Research/Statistics, Sloan School of Management, MIT
In this presentation, we show a significant role that symmetry, a fundamental concept in convex geometry, plays in determining the power of robust and finitely adaptable solutions in multi-stage stochastic and adaptive optimization problems. We consider a fairly general class of multi-stage mixed integer stochastic and adaptive optimization problems and propose a good approximate solution policy with performance guarantees that depend on the geometric properties such as symmetry of the uncertainty sets. In particular, we show that a class of finitely adaptable solutions is a good approximation for both the multi-stage stochastic as well as the adaptive optimization problem. A finitely adaptable solution specifies a small set of solutions for each stage and the solution policy implements the best solution from the given set depending on the realization of the uncertain parameters in the past stages. To the best of our knowledge, these are the first approximation results for the multi-stage problem in such generality. (Joint work with Vineet Goyal, Columbia University and Andy Sun, MIT.)

Applying for Graduate School

Series
Other Talks
Time
Monday, September 13, 2010 - 16:30 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Michael Lacey – GT
The why and how of applying to graduate school, with examples of different opportunities drawn from the past 10 years of undergraduate mathematics majors that have gone on to programs in EE, Physics, Applied Math, Statistics, Math, and even Public Policy. Useful for all undergraduate math majors. This is part of the regular Club Math meetings.

A monodromy invariant in the space of knots

Series
Geometry Topology Seminar
Time
Monday, September 13, 2010 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 114
Speaker
Jason McGibbon – University of Massachusetts
Knot contact homology (KCH) is a combinatorially defined topological invariant of smooth knots introduced by Ng. Work of Ekholm, Etnyre, Ng and Sullivan shows that KCH is the contact homology of the unit conormal lift of the knot. In this talk we describe a monodromy result for knot contact homology,namely that associated to a path of knots there is a connecting homomorphism which is invariant under homotopy. The proof of this result suggests a conjectural interpretation for KCH via open strings, which we will describe.

On randomizing two derandomized greedy algorithms

Series
Combinatorics Seminar
Time
Friday, September 10, 2010 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Kevin Costello – SoM, Georgia Tech
Many of the simplest and easiest implemented approximation algorithms can be thought of as derandomizations of the naive random algorithm.  Here we consider the question of whether performing the algorithm on a random reordering of the variables provides an improvement in the worst case expected performance. (1) For Johnson's algorithm for Maximum Satisfiability, we show this is indeed the case: While in the worst case Johnson's algorithm only provides a 2/3 approximation, the additional randomization step guarantees a 2/3+c approximation for some positive c. (2) For the greedy algorithm for MAX-CUT, we show to the contrary that the randomized version does NOT provide a 1/2+c approximation for any c on general graphs. This is in contrast to a result of Mathieu and Schudy showing it provides a 1-epsilon approximation on dense graphs. Joint with Asaf Shapira and Prasad Tetali.

Non-loose torus knots.

Series
Geometry Topology Working Seminar
Time
Thursday, September 9, 2010 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Amey Kaloti – Georgia Tech. –

Please Note: This talk is part of the oral exam for the speaker. Please note the special time, place. Also the talk itself will be 45 min long.

Non-loose knots is a special class of knots studied in contact geometry. Last couple of years have shown some applications of these kinds of knots. Even though defined for a long time, not much is known about their classification except for the case of unknot. In this talk we will summarize what is known and tell about the recent work in which we are trying to give classification in the case of trefoil.

The Point Mass Problem on the Real Line

Series
Analysis Seminar
Time
Wednesday, September 8, 2010 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Manwah Wong – Georgia Tech
In this talk, I will talk about recent developments on the point mass problem on the real line. Starting from the point mass formula for orthogonal polynomials on the real line, I will present new methods employed to compute the asymptotic formulae for the orthogonal polynomials and how these formulae can be applied to solve the point mass problem when the recurrence coefficients are asymptotically identical. The technical difficulties involved in the computation will also be discussed.

The size of crossings in antichains

Series
Research Horizons Seminar
Time
Wednesday, September 8, 2010 - 12:00 for 1 hour (actually 50 minutes)
Location
Skiles 171 (NOTICE THE CHANGE OF ROOM)
Speaker
Tom Trotter – School of Mathematics - Georgia Institute of Technology

Please Note: Hosted by: Yao Li and Ricardo Restrepo

Combinatorial mathematics exhibits a number of elegant, simply stated problems that turn out to be surprisingly challenging. In this talk, I report on a problem of this type on which I have been working with Noah Streib, Stephen Young and Ruidong Wang from Georgia Tech, as well as Piotr Micek, Bartek Walczak and Tomek Krawczyk, all computer scientists from Poland. Given positive integers $k$ and $w$, what is the largest integer $t = f(k,w)$ for which there exists a family $\mathcal{F}$ of $t$ vectors in $N^{w}$ so that: \begin{enumerate} \item Any two vectors in the family $\mathcal{F}$ are incomparable in the product ordering; and \item There do not exist two vectors $A$ and $B$ in the family for which there are distinct $i$ and $j$ so that $a_i\ge k +b_i$ and $b_j \ge k + a_j$. \end{enumerate} The Polish group posed the problem to us at the SIAM Discrete Mathematics held in Austin, Texas, this summer. They were able to establish the following bounds: \[ k^{w-1} \le t \le k^w \] We were able to show that the lower bound is essentially correct by showing that there is a constant $c_w$ so that $t \l c_w k^{w-1}$. But recent work suggests that the lower bound might actually be tight.

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