Series: School of Mathematics Colloquium
Traditional Erdos Magic (a.k.a. The Probabilistic Method) proves the existence of an object with certain properties by showing that a random (appropriately defined) object will have those properties with positive probability. Modern Erdos Magic analyzes a random process, a random (CS take note!) algorithm. These, when successful, can find a "needle in an exponential haystack" in polynomial time. We'll look at two particular examples, both involving a family of n-element sets under suitable side conditions. The Lovasz Local Lemma finds a coloring with no set monochromatic. A result of this speaker finds a coloring with low discrepency. In both cases the original proofs were not implementable but Modern Erdos Magic finds the colorings in polynomial times. The methods are varied. Basic probability and combinatorics. Brownian Motion. Semigroups. Martingales. Recursions ... and Tetris!
Series: Research Horizons Seminar
The talk will include a crash course on infinite dimensional topology, with applications to various topological properties of the space of congruence classes of convex bodies in the Euclidean space.
Series: Analysis Seminar
The bispectral problem concerns the construction and the classification of operators possessing a symmetry between the space and spectral variables. Different versions of this problem can be solved using techniques from integrable systems, algebraic geometry, representation theory, classical orthogonal polynomials, etc. I will review the problem and some of these connections and then discuss new results related to the generic quantum superintegrable system on the sphere.
Monday, October 30, 2017 - 17:15 , Location: Skiles 005 , Spencer Bloch , University of Chicago , Organizer: Joseph Rabinoff
Golyshev and Zagier found an interesting new source of periods associated to (eventually inhomogeneous) solutions generated by the Frobenius method for Picard Fuchs equations in the neighborhood of singular points with maximum unipotent monodromy. I will explain how this works, and how one can associate "motivic Gamma functions" and generalized Beilinson style variations of mixed Hodge structure to these solutions. This is joint work with M. Vlasenko.
Monday, October 30, 2017 - 16:05 , Location: Skiles 005 , Bjorn Poonen , Massachusetts Institute of Technology , Organizer: Joseph Rabinoff
The function field case of the strong uniform boundedness conjecturefor torsion points on elliptic curves reduces to showing thatclassical modular curves have gonality tending to infinity.We prove an analogue for periodic points of polynomials under iterationby studying the geometry of analogous curves called dynatomic curves.This is joint work with John R. Doyle.
Series: Geometry Topology Seminar
Heegaard Floer theory provides a powerful suite of tools for studying 3-manifolds and their subspaces. In 2006, Ozsvath, Szabo and Thurston defined an invariant of transverse knots which takes values in a combinatorial version of this theory for knots in the 3—sphere. In this talk, we discuss a refinement of their combinatorial invariant via branched covers and discuss some of its properties. This is joint work with Mike Wong.
Series: Combinatorics Seminar
Cutoff is a remarkable property of many Markov chains in which they rapidly transition from an unmixed to a mixed distribution. Most random walks on the symmetric group, also known as card shuffles, are believed to mix with cutoff, but we are far from being able to proof this. We will survey existing cutoff results and techniques for random walks on the symmetric group, and present three recent results: cutoff for a biased transposition walk, cutoff for the random-to-random card shuffle (answering a 2001 conjecture of Diaconis), and pre-cutoff for the involution walk, generated by permutations with a binomially distributed number of two-cycles. The results use either probabilistic techniques such as strong stationary times or diagonalization through algebraic combinatorics and representation theory of the symmetric group. Includes joint work with Nayantara Bhatnagar, Evita Nestoridi, and Igor Pak.
Series: Math Physics Seminar
During the last few years there has been a systematic pursuit for sharp estimates of the energy components of atomic systems in terms of their single particle density. The common feature of these estimates is that they include corrections that depend on the gradient of the density. In this talk I will review these results. The most recent result is the sharp estimate of P.T. Nam on the kinetic energy. Towards the end of my talk I will present some recent results concerning geometric estimates for generalized Poincaré inequalities obtained in collaboration with C. Vallejos and H. Van Den Bosch. These geometric estimates are a useful tool to estimate the numerical value of the constant of Nam's gradient correction term.
Friday, October 27, 2017 - 15:00 , Location: Skiles 154 , Hassan Attarchi , Georgia Tech , Organizer:
This presentation is about the results of a paper by Y. Sinai in 1970. Here, I will talk about dynamical systems which resulting from the motion of a material point in domains with strictly convex boundary, that is, such that the operator of the second quadratic form is negative-definite at each point of the boundary, where the boundary is taken to be equipped with the field of inward normals. It was proved that such systems are ergodic and are K-systems. The basic method of investigation is the construction of transversal foliations for such systems and the study of their properties.
Friday, October 27, 2017 - 13:00 , Location: Skiles 006 , John Etnyre , Georgia Tech , Organizer: John Etnyre
Notice the seminar is back to 1.5 hours this week.
In this series of talks I will introduce branched coverings of manifolds and sketch proofs of most the known results in low dimensions (such as every 3 manifold is a 3-fold branched cover over a knot in the 3-sphere and the existence of universal knots). This week we should be able to finish our discussion of branched covers of surfaces and transition to 3-manifolds.