Georgia Institute of Technology College of Sciences

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.

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.

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.

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.

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.

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!

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!

Let G be a graph containing 5 different vertices a0, a1, a2, b1 and b2. We say that (G, a0, a1, a2, b1, b2) is feasible if G contains disjoint connected subgraphs G1, G2, such that {a0, a1, a2}⊆V(G1) and {b1, b2}⊆V(G2). In this talk, we will introduce ideal frames, slim connectors and fat connectors. We will first deal with the ideal frames without fat connectors, by studying 3-edge and 5-edge configurations. Joint work with Changong Li, Robin Thomas, and Xingxing Yu.

The Sherrington-Kirkpatirck (SK) model is a mean-field spin glass introduced by theoretical physicists in order to explain the strange behavior of certain alloys, such as CuMn. Despite of its seemingly simple formulation, it was conjectured to possess a number of profound properties. This talk will be focused on the energy landscapes of the SK model and the mixed p-spin model with both Ising and spherical configuration spaces. We will present Parisi formule for their maximal energies followed by descriptions of the energy landscapes near the maximum energy. Based on joint works with A. Auffinger, M. Handschy, G. Lerman, and A. Sen.

We continue our discussion of free resolutions and Stanley-Reisner ideals. We introduce Hochster's formula and state results on the behavior of Betti tables under clique-sums.

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 sstart discussing branched covers of 3-manifolds.

The search for the asymptotics of the Ramsey function R(3,k) has a long and fascinating history. It begins in the hill country surrounding Budapest and winding over the decades through Europe, America, Korea and Rio de Janiero. We explore it through a CS lens, giving algorithms that provide the various upper and lower bounds. The arguments are various more or less sophisticated uses of Erdos Magic and, indeed, many of the most important advances in the Probabilistic Method have come from these investigations.

This presentation is about the results of a paper by L. Bunimovich in 1974. One considers dynamical systems generated by billiards which are perturbations of dispersing billiards. It was shown that such dynamical systems are systems of A. N. Kolmogorov (K-systems), if the perturbation satisfies certain conditions which have an intuitive geometric interpretation.

All of us have seen talks where the speaker uses slides. Some are great, and some are awful. Come and learn how to make great slide decks and how to avoid making awful ones. We will share a number of pieces of software that are easy to use and that can help you to improve your slide decks. We will also discuss best practices and dissect several short slide decks together. Next week there will be a follow-up, hands-on workshop on using the software Inkscape to create mathematical figures for talks, posters, and papers.