Seminars and Colloquia by Series

Residual Torsion-Free Nilpotence and Two-Bridge Knot Groups

Series
Geometry Topology Seminar
Time
Monday, December 2, 2019 - 14:00 for 1 hour (actually 50 minutes)
Location
Speaker
Jonathan JohnsonThe University of Texas at Austin

I will discuss how a graph theoretic construction used by Hirasawa and Murasugi can be used to show that the commutator subgroup of the knot group of a two-bridge knot is a union of an ascending chain of parafree groups. Using a theorem of Baumslag, this implies that the commutator subgroup of a two-bridge knot group is residually torsion-free nilpotent which has applications to the anti-symmetry of ribbon concordance and the bi-orderability of two-bridge knots. In 1973, E. J. Mayland gave a conference talk in which he announced this result. Notes on this talk can be found online. However, this result has never been published, and there is evidence, in later papers, that a proper proof might have eluded Mayland.

The Underlying Contact and Symplectic Topology of Anosov Flows in Dimension 3

Series
Geometry Topology Student Seminar
Time
Wednesday, November 27, 2019 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Surena HozooriGeorgia Institute of Technology

Anosov flows provide beautiful examples of interactions between dynamics, geometry and analysis. In dimension 3 in particular, they are known to have a subtle relation to topology as well. Motivated by a result of Mitsumatsu from 1995, I will discuss their relation to contact and symplectic structures and argue why contact topological methods are natural tools to study the related global phenomena.

Thinking Outside the Circle

Series
Undergraduate Seminar
Time
Monday, November 25, 2019 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 171
Speaker
Dr. Evans HarrellGeorgia Tech

Did you know that a wheel or a ball bearing does not need to be round? Convex regions that can roll smoothly come in many remarkable shapes and have practical applications in engineering and science. Moreover, the mathematics used to describe them, known as convex geometry, is a subject that beautifully ties together analysis and geometry. I'll bring some of these objects along and tell the class how to describe them effectively and recount their interesting history.

Classifying incompressible surfaces in hyperbolic 4-punctured sphere mapping tori

Series
Geometry Topology Seminar
Time
Monday, November 25, 2019 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Sunny Yang XiaoBrown University

One often gains insight into the topology of a manifold by studying its sub-manifolds. Some of the most interesting sub-manifolds of a 3-manifold are the "incompressible surfaces", which, intuitively, are the properly embedded surfaces that can not be further simplified while remaining non-trivial. In this talk, I will present some results on classifying orientable incompressible surfaces in a hyperbolic mapping torus whose fibers are 4-punctured spheres. I will explain how such a surface gives rise to a path satisfying certain combinatorial properties in the arc complex of the 4-punctured sphere, and how we can reconstruct such surfaces from these paths. This extends and generalizes results of Floyd, Hatcher, and Thurston.

Asymptotic Homotopical Complexity of a Sequence of 2D Dispersing Billiards

Series
CDSNS Colloquium
Time
Monday, November 25, 2019 - 11:15 for 1 hour (actually 50 minutes)
Location
Skyles 005
Speaker
Nandor SimanyiUniversity of Alabama at Birgminham

We are studying the asymptotic homotopical complexity of a sequence of billiard flows on the 2D unit torus T^2 with n
circular obstacles. We get asymptotic lower and upper bounds for the radial sizes of the homotopical rotation sets and,
accordingly, asymptotic lower and upper bounds for the sequence of topological entropies. The obtained bounds are rather
close to each other, so this way we are pretty well capturing the asymptotic homotopical complexity of such systems.

Note that the sequence of topological entropies grows at the order of log(n), whereas, in sharp contrast, the order of magnitude of the sequence of metric entropies is only log(n)/n.


Also, we prove the convexity of the admissible rotation set AR, and the fact that the rotation vectors obtained from
periodic admissible trajectories form a dense subset in AR.

 

Towards the sunflower conjecture

Series
ACO Colloquium
Time
Monday, November 25, 2019 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Shachar LovettUniversity of California, San Diego, CA

A sunflower with $r$ petals is a collection of $r$ sets so that the intersection of each pair is equal to the intersection of all. Erdos and Rado in 1960 proved the sunflower lemma: for any fixed $r$, any family of sets of size $w$, with at least about $w^w$ sets, must contain a sunflower. The famous sunflower conjecture is that the bound on the number of sets can be improved to $c^w$ for some constant $c$. Despite much research, the best bounds until recently were all of the order of $w^{cw}$ for some constant c. In this work, we improve the bounds to about $(\log w)^{w}$.

There are two main ideas that underlie our result. The first is a structure vs pseudo-randomness paradigm, a commonly used paradigm in combinatorics. This allows us to either exploit structure in the given family of sets, or otherwise to assume that it is pseudo-random in a certain way. The second is a duality between families of sets and DNFs (Disjunctive Normal Forms). DNFs are widely studied in theoretical computer science. One of the central results about them is the switching lemma, which shows that DNFs simplify under random restriction. We show that when restricted to pseudo-random DNFs, much milder random restrictions are sufficient to simplify their structure.

Joint work with Ryan Alweiss, Kewen Wu and Jiapeng Zhang.

Fast convergence of fictitious play

Series
ACO Student Seminar
Time
Friday, November 22, 2019 - 13:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Kevin A. LaiCS, Georgia Tech

Fictitious play is one of the simplest and most natural dynamics for two-player zero-sum games. Originally proposed by Brown in 1949, the fictitious play dynamic has each player simultaneously best-respond to the distribution of historical plays of their opponent. In 1951, Robinson showed that fictitious play converges to the Nash Equilibrium, albeit at an exponentially-slow rate, and in 1959, Karlin conjectured that the true convergence rate of fictitious play after k iterations is O(k^{-1/2}), a rate which is achieved by similar algorithms and is consistent with empirical observations. Somewhat surprisingly, Daskalakis and Pan disproved a version of this conjecture in 2014, showing that an exponentially-slow rate can occur, although their result relied on adversarial tie-breaking. In this talk, we show that Karlin’s conjecture holds if ties are broken lexicographically and the game matrix is diagonal. We also show a matching lower bound under this tie-breaking assumption. This is joint work with Jake Abernethy and Andre Wibisono.

A solution to the Burr-Erdos problems on Ramsey completeness

Series
Joint School of Mathematics and ACO Colloquium
Time
Thursday, November 21, 2019 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Jacob FoxStanford University

A sequence A of positive integers is r-Ramsey complete if for every r-coloring of A, every sufficiently large integer can be written as a sum of the elements of a monochromatic subsequence. Burr and Erdos proposed several open problems in 1985 on how sparse can an r-Ramsey complete sequence be and which polynomial sequences are r-Ramsey complete. Erdos later offered cash prizes for two of these problems. We prove a result which solves the problems of Burr and Erdos on Ramsey complete sequences. The proof uses tools from probability, combinatorics, and number theory. 

Joint work with David Conlon.

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