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Series: Combinatorics Seminar

We will present the solution to a statistical mechanics model on random lattices. More precisely, we consider the Potts model on the set of planar triangulations (embedded planar graph such that every face has degree 3). The partition function of this model is the generating function of vertex-colored triangulations counted according to the number of monochromatic edges and dichromatic edges. We characterize this partition function by a simple system of differential equations. Some special cases, such as properly 4-colored triangulations, lead to particularly simple equations waiting for a more direct combinatorial explanation. This is joint work with Mireille Bousquet-Melou.

Series: Combinatorics Seminar

In this talk I will present the notion of a \delta-packing for set systems of bounded primal shatter dimension (closely related to the notion of finite VC-dimension). The structure of \delta-packing, which has been studied by Dudley in 1978 and Haussler in 1995, emerges from empirical processes and is fundamental in theoretical computer science and in computational geometry in particular. Moreover, it has applications in geometric discrepancy, range searching, and epsilon-approximations, to name a few. I will discuss a variant of \delta-packings where all the sets have small cardinality, we call these structures "shallow packings", and then present an upper bound on their size under additional natural assumptions on the set system, which correspond to several geometric settings, among which is the case of points and halfspaces in d-dimensions.

Series: Combinatorics Seminar

Bio: Georgios Piliouras is a postdoc at Caltech, Center for Mathematics and

Computation. He received his PhD in Computer Science from Cornell

University and has been a Georgia Tech postdoc at the EE department.

In a recent series of papers a strong connection has been established
between standard models of sexual evolution in mathematical biology and
Multiplicative Weights Updates Algorithm, a ubiquitous model of online
learning and optimization. These papers show that mathematical models of
biological evolution are tantamount to applying discrete replicator
dynamics, a close variant of MWUA on coordination games. We show that in
the case of coordination games, under minimal genericity assumptions,
discrete replicator dynamics converge to pure Nash equilibria for all but a
zero measure of initial conditions. This result holds despite the fact that
mixed Nash equilibria can be exponentially (or even uncountably) many,
completely dominating in number the set of pure Nash equilibria. Thus, in
haploid organisms the long term preservation of genetic diversity needs to
be safeguarded by other evolutionary mechanisms, such as mutation and
speciation.
This is joint work with Ruta Mehta and Ioannis Panageas.

Series: Combinatorics Seminar

Weighted Bipartite Edge Coloring problem is a generalization of two classical optimization problems: Bipartite Edge Coloring and Bin Packing. Given an edge-weighted bipartite multi-graph G, the goal is to color all edges with minimum colors such that the sum of the edges incident to any vertex of any color is at most one. Chung and Ross conjectured that given an instance of the weighted bipartite edge coloring problem, there is a proper weighted coloring using at most 2n-1 colors where n denotes the maximum over all the vertices of the number of unit-sized bins needed to pack the weights of edges incident at the vertex. In this talk I will present an algorithm that gives a proper weighted coloring using $20n/9$ colors and improved results for some special cases. I will also present an alternative proof of Konig's edge coloring theorem using skew-supermodular functions. The talk will have all three components of ACO: Approximation Algorithms, Graph Theory and Supermodular Optimization.

Series: Combinatorics Seminar

The famous Doignon-Bell-Scarf theorem is a Helly-type result about the existence of integer solutions on systems linear inequalities. The purpose
of this paper is to present the following ``weighted'' generalization: Given an integer k, we prove that there exists a constant c(k,n),
depending only on the dimension n and k, such that if a polyhedron {x : Ax <= b} contains exactly k integer solutions, then there exists a subset
of the rows of cardinality no more than c(k,n), defining a polyhedron that contains exactly the same k integer solutions. We work on both
upper and lower bounds for this constant.
This is joint work with Quentin Louveaux, Iskander Aliev and Robert Bassett.

Series: Combinatorics Seminar

Let G be an abelian group. A subset A of G is a Sidon set if A has the property that no sum of two elements of A is equal to another sum of two elements of A. These sets have a rich history in combinatorial number theory and frequently appear in the problem papers of Erdos. In this talk we will discuss some results in which Sidon sets were used to solve problems in extremal graph theory. This is joint work with Mike Tait and Jacques Verstraete.

Series: Combinatorics Seminar

(This seminar has been rescheduled for April 17 (Thursday) 12-1pm. Generalized triangle T_r is an r-graph with edges {1,2,…,r}, {1,2,…,r-1, r+1} and {r,r+1, r+2, …,2r-2}. The family \Sigma_r consists of all r-graphs with three edges D_1, D_2, D_3 such that |D_1\cap D_2|=r-1 and D_1\triangle D_2\subset D_3. In 1989 it was conjectured by Frankl and Furedi that ex(n,T_r) = ex(n,\Sigma_r) for large enough n, where ex(n,F) is the Tur\'{a}n function. The conjecture was proven to be true for r=3, 4 by Frankl, Furedi and Pikhurko respectively. We settle the conjecture for r=5,6 and show that extremal graphs are blow-ups of the unique (11, 5, 4) and (12, 6, 5) Steiner systems. The proof is based on a technique for deriving exact results for the Tur\'{a}n function from “local stability" results, which has other applications. This is joint work with Sergey Norin.

Series: Combinatorics Seminar

The celebrated Green-Tao theorem states that there are arbitrarily long arithmetic progressions in the primes. In this talk, I will explain the ideas of the proof and discuss our recent simplifications. One of the main ingredients in the proof is a relative Szemerédi theorem, which says that every relatively dense subset of a pseudorandom set of integers contains long arithmetic progressions. Our main advance is both a simplification and a strengthening of the relative Szemerédi theorem, showing that a much weaker pseudorandomness condition suffices. I will explain the transference principle strategy used in the proof. Also see our recent exposition of the Green-Tao theorem: http://arxiv.org/abs/1403.2957 Based on joint work with David Conlon and Jacob Fox.

Series: Combinatorics Seminar

We will explain the concept of aproximate well-supported Nash equilibrium
and show that one must consider equilibria with large supports to achieve
good approximation ratio. Our arguments use tools from probabilistic,
extremal and additive combinatorics.
Joint work with Y. Anbalagan, R. Savani and A. Vetta.

Series: Combinatorics Seminar

We study an old problem of Linial and Wilf to find the graphs with n vertices and m edges which maximize the number of proper q-colorings of their vertices. In a breakthrough paper, Loh, Pikhurko and Sudakov asymptotically reduced the problem to an optimization problem. We prove the following structural result which tells us how the optimal solutionlooks like: for any instance, each solution of the optimization problem corresponds to either a complete multipartite graph or a graph obtained from a complete multipartite graph by removing certain edges. We then apply this result on optimal graphs to general instances, including a conjecture of Lazebnik from 1989 which asserts that for any q>=s>= 2, the Turan graph T_s(n) has the maximum number of q-colorings among all graphs with the same number of vertices and edges. We disprove this conjecture by providing infinity many counterexamples in the interval s+7 <= q <= O(s^{3/2}). On the positive side, we show that when q= \Omega(s^2) the Turan graph indeed achieves the maximum number of q-colorings. Joint work with Humberto Naves.