Seminars and Colloquia by Series

The Symmetric Rendezvous Problem: Codes and Lower Bounds

Series
Combinatorics Seminar
Time
Friday, September 18, 2015 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Tom Hayes – The University of New Mexico –
In the Rendezvous problem on the complete graph, two parties are trying to meet at some vertex at the same time, despite starting out with independent random labelings of the vertices. It is well known that the optimal strategy is for one player to wait at any vertex, while the other visits all n vertices in consecutive steps, which guarantees a rendezvous within n steps and takes (n + 1)/2 steps on average. This strategy is very far from being symmetric, however. E. Anderson and R. Weber presented a symmetric algorithm that achieves an expected meeting time of 0.829n, which has been conjectured to be optimal in the symmetric setting. We change perspective slightly: instead of trying to minimize the expected meeting time, we try to maximize the probability of successfully meeting within a specified number of timesteps. In this setting, for all time horizons that are linear in n, the Anderson-Weber strategy has a constant probability of failure. Surprisingly, we show that this is not optimal: there exists a different symmetric strategy that almost surely guarantees meeting within 4n timesteps. This bound is tight, in that the factor 4 cannot be replaced by any smaller constant. Our strategy depends on the construction of a new kind of combinatorial object that we dub”rendezvous code.”On the positive side, for T < n, we show that the probability of meeting within T steps is indeed (approximately) maximized by the Anderson-Weber strategy. Our results imply new lower bounds on the expected meeting time for any symmetric strategy, which establishes an asymptotic difference between the best symmetric and asymmetric strategies. Finally, we examine the symmetric rendezvous problem on other vertex-transitive graphs.

How unstable is our solar system?

Series
School of Mathematics Colloquium
Time
Thursday, September 17, 2015 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Dr. Jinxin Xue – University of Chicago –
Though the modern analytic celestial mechanics has been existing for more than 300 years since Newton, there are still many basic questions unanswered, for instance, there is still no rigorous mathematical proof explaining why our solar system has been stable for such a long time (five billion years) hence no guarantee that it would remain stable for the next five billion years. Instead, it is known that there are various instability behaviors in the Newtonian N-body problem. In this talk, we mention three types instability behaviors in Newtonian N-body problem. The first type we will talk about is simply chaotic motions, which include for instance the oscillatory motions, in which case, one body travels back and forth between neighborhoods of zero and infinity. The second type is “organized” chaotic motions, also known as Arnold diffusion or weak turbulence. Finally, we will talk about our work on the existence of the most wild unstable behavior, non collision singularities, also called finite time blow up solution. The talk is mostly expository. Zero background on celestial mechanism or dynamical systems is needed to follow the lecture.

Tiling with Arbitrary Tiles

Series
Combinatorics Seminar
Time
Wednesday, September 16, 2015 - 16:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Imre Leader – University of Cambridge –
Let $T$ be a finite subset of ${\Bbb Z}^n$. It may or may not tile ${\Bbb Z}^n$, in the sense of ${\Bbb Z}^n$ having a partition into copies of $T$. But is there a dimension $d$ such that $T$ does tile ${\Bbb Z}^d$ ? Our talk will focus on this question.

From classical mechanics to symplectic (and contact) geometry

Series
Research Horizons Seminar
Time
Wednesday, September 16, 2015 - 12:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Prof. John B. Etnyre – School of Mathematics, Georgia Institute of Technology –

Please Note: Food and Drinks will be provided after the seminar.

In this seminar, Prof. John Etnyre will begin this talk by discussing a classical question concerning periodic motions of particles in classical physics. In trying to better understand this question we will develop the notion of a symplectic structure. This is a fundamental geometric object that provides the "right way" to think about classical mechanics, and many many other things too. We will then indicate how modern ideas can be used to give, at least partial, answers to our initial naive questions about periodic motions.

Construction of quasi-periodic solutions of State-dependent delay differential equations by the parameterization method II: Details.

Series
Dynamical Systems Working Seminar
Time
Tuesday, September 15, 2015 - 17:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Xiaolong He – Georgia Tech (Math)/Hunan University
We investigate the existence of quasi-periodic solutions for state-dependent delay differential equationsusing the parameterization method, which is different from the usual way-working on the solution manifold. Under the assumption of finite-time differentiability of functions and exponential dichotomy, the existence and smoothness of quasi-periodic solutions are investigated by using contraction arguments We also develop a KAM theory to seek analytic quasi-periodic solutions. In contrast with the finite differentonable theory, this requires adjusting parameters. We prove that the set of parameters which guarantee the existence of analytic quasi-periodic solutions is of positive measure. All of these results are given in an a-posterior form. Namely, given a approximate solution satisfying some non-degeneracy conditions, there is a true solution nearby.

Deformation theory and cup products

Series
Algebra Seminar
Time
Monday, September 14, 2015 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Carl Wang Erickson – Brandeis University
We will introduce, through examples, the philosophy of Delignethat "in characteristic zero, a deformation problem is controlled by adifferential graded (or "dg-") Lie algebra." Focusing on the deformationtheory of representations of a group, we will give an extension of thisphilosophy to positive characteristic. This will be justified by thepresence of a dg-algebra controlling the deformations, and the fact thatthe cohomology of the dg-algebra has an A-infinity algebra structureexplicitly presenting the deformation problem. This structure can bethought of as "higher cup products" on group cohomology, extending theusual cup product and often computable as Massey products. We will writedown concrete, representation-theoretic questions that are answered bythese higher cup products. To conclude, we will show that the cup productstructure on Galois cohomology, which is the subject of e.g. the motivicBloch-Kato conjecture and its proofs, is enriched by these higher cupproducts, and that this enrichment reflects properties of the Galois group.Familiarity with dg-algebras and infinity-algebras will not be presumed.

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