Seminars and Colloquia by Series

Friday, October 21, 2011 - 15:05 , Location: Skiles 005 , Josephine Yu , School of Math, Ga Tech , Organizer: Prasad Tetali
The secondary polytope of a point configuration A is a polytope whose faces are in bijection with regular subdivions of A, e.g. the secondary polytope of the vertices of polygon is an associahedron. The resultant of a tuple of point configurations A_1, A_2, ..., A_k in Z^n is the set of coefficients for which the polynomials with supports A_1, A_2, ..., A_k have a common root with no zero coordinates over complex numbers, e.g. when each A_1 is a standard simplex and k = n+1, the resultant is defined by a determinant. The Newton polytope of a polynomial is the convex hull of the exponents, e.g. the Newton polytope of the determinant is the perfect matching polytope. In this talk, I will explain the close connection between secondary polytopes and Newton polytopes of resultants, using tropical geometry, based on joint work with Anders Jensen.
Friday, October 14, 2011 - 15:05 , Location: Skiles 005 , Thomas Valla , Charles University, Prague , Organizer: Prasad Tetali
Ramsey theory studies the internal homogenity of mathematical structures (i.e. graphs, number sets), parts of which (subgraphs, number subsets) are arbitrarily coloured. Often, the sufficient object size implies the existence of a monochromatic sub-object. Combinatorial games are 2-player games of skill with perfect information. The theory of combinatorial games studies mostly the questions of existence of winning or drawing strategies. Let us consider an object that is studied by a particular Ramsey-type theorem. Assume two players alternately colour parts of this object by two colours and their goal is to create certain monochromatic sub-object. Then this is a combinatorial game. We focus on the minimum object size such that the appropriate Ramsey-type theorem holds, called "Ramsey number", and on the minimum object size such that the first player has a winning strategy in the corresponding combinatorial game, called "game number". In this talk, we investigate the "restricted Ramsey-type theorems". This means, we show the existence of first player's winning strategies, and we show that game numbers are surprisingly small, compared to Ramsey numbers. (This is joint work with Jarek Nesetril.)
Friday, September 30, 2011 - 15:00 , Location: Skiles 005 , Alex Rice , University of Georgia , Organizer: Ernie Croot
 How large can a subset of the first N natural numbers be before it is guaranteed to contain two distinct elements which differ by a perfect square? What if I replaced "perfect square" with the image of a more general polynomial, or perhaps "one less than a prime number"? We will discuss results of this flavor, including recent improvements and generalizations.
Friday, September 9, 2011 - 15:05 , Location: Skiles 005 , Laszlo Vegh , School of Computer Science, Georgia Tech , , Organizer: Prasad Tetali
The generalized flow model is a classical and widely applicable extension of network flows, where on every arc, the flow leaving the arc is a linear function of the flow entering the arc. In the talk, I will investigate a nonlinear extension of this model, with the flow leaving an arc being a concave function of the entering flow. I exhibit the first combinatorial polynomial time algorithm for solving corresponding optimization problems. This model turns out to be a common framework for solving several market equilibrium problems, such as linear Fisher markets, and immediately enables to extend them to more general settings. I will also give a survey on generalized flow algorithms and previous nonlinear flow models.
Friday, May 6, 2011 - 15:05 , Location: Skiles 006 , Amanda Pascoe Streib , Georgia Tech , Organizer: Prasad Tetali
Colloids are mixtures of molecules  well-studied in material science that are not well-understood mathematically.  Physicists model colloids as a system of two types of tiles (type A and type B) embedded on a region of the plane, where no two tiles can overlap.  It is conjectured that at high density, the type A tiles tend to separate out and form large "clusters".   To verify this conjecture, we need methods for counting these configurations directly or efficient algorithms for sampling.  Local sampling algorithms are known to be inefficient. However, we provide the first rigorous analysis of a global "DK Algorithm" introduced by Dress and Krauth.  We also examine the clustering effect directly via a combinatorial argument. We prove for a certain class of colloid models that at high density the configurations are likely to exhibit clustering, whereas at low density the tiles are all well-distributed. Joint work with Sarah Miracle and Dana Randall.
Friday, April 29, 2011 - 15:05 , Location: Skiles 006 , Will Perkins , Courant Institute, NYU , , Organizer: Prasad Tetali
The Bohman-Frieze process is a simple modification of the Erdős-Rényi random  graph that adds dependence between the edges biased in favor of joining  isolated vertices. We present new results on the phase transition of the  Bohman-Frieze process and show that qualitatively it belongs to the same  class as the Erdős-Rényi process. The results include the size and structure  of small components in the barely sub- and supercritical time periods. We  will also mention a class of random graph processes that seems to exhibit  markedly different critical behavior.
Friday, April 22, 2011 - 15:05 , Location: Skiles 006 , Elena Grigorescu , College of Computing, Georgia Tech , Organizer: Xingxing Yu
In the Property Testing model an algorithm is required to distinguish between the case that an object has a property or is far from having the property. Recently, there has been a lot of interest in understanding which properties of Boolean functions admit testers making only a constant number of queries, and a common theme investigated in this context is linear invariance. A series of gradual results has led to a conjectured characterization of all testable linear invariant properties. Some of these results consider properties where the query upper bounds are towers of exponentials of large height dependent on the distance parameter. A natural question suggested by these bounds is whether there are non-trivial families with testers making only a polynomial number of queries in the distance parameter.In this talk I will focus on a particular linear-invariant property where this is indeed the case: odd-cycle freeness.Informally, a Boolean function fon n variables is odd-cycle free if there is no x_1, x_2, .., x_2k+1 satisfying f(x_i)=1 and sum_i x_i = 0.This property is the Boolean function analogue of bipartiteness in the dense graph model. I will discuss two testing algorithms for this property: the first relies on graph eigenvalues considerations and the second on Fourier analytic techniques. I will also mention several related open problems. Based on joint work with Arnab Bhattacharyya, Prasad Raghavendra, Asaf Shapira
Friday, April 15, 2011 - 15:00 , Location: Skiles 006 , Nathan Reading , North Carolina State University , Organizer: Christine Heitsch
A rectangulation is a tiling of a rectangle by rectangles.  The rectangulation is called generic if no four of its rectangles share a corner.  We will consider the problem of counting generic rectangulations (with n rectangles) up to combinatorial equivalence. This talk will present and explain an initial step in the enumeration: the fact that generic rectangulations are in bijection with permutations that avoid a certain set of patterns.  I'll give background information on rectangulations and pattern avoidance. Then I'll make the connection between generic rectangulations and pattern avoiding permutations, which draws on earlier work with Shirley Law on "diagonal" rectangulations. I'll also comment on two theories that led to this result and its proof: the lattice theory of the weak order on permutations and the theory of combinatorial Hopf algebras.
Friday, April 1, 2011 - 15:05 , Location: Skiles 006 , Sangjune Lee , Emory University , Organizer: Xingxing Yu
A set~$A$ of integers is a \textit{Sidon set} if all thesums~$a_1+a_2$, with~$a_1\leq a_2$ and~$a_1$,~$a_2\in A$, aredistinct.  In the 1940s, Chowla, Erd\H{o}s and Tur\'an determinedasymptotically the maximum possible size of a Sidon set contained in$[n]=\{0,1,\dots,n-1\}$.  We study Sidon sets contained in sparserandom sets of integers, replacing the `dense environment'~$[n]$ by asparse, random subset~$R$ of~$[n]$.Let~$R=[n]_m$ be a uniformly chosen, random $m$-element subsetof~$[n]$.  Let~$F([n]_m)=\max\{|S|\colon S\subset[n]_m\hbox{  Sidon}\}$.  An abridged version of our results states as follows.Fix a constant~$0\leq a\leq1$ and suppose~$m=m(n)=(1+o(1))n^a$.  Thenthere is a constant $b=b(a)$ for which~$F([n]_m)=n^{b+o(1)}$ almostsurely.  The function~$b=b(a)$ is a continuous, piecewise linearfunction of~$a$, not differentiable at two points:~$a=1/3$and~$a=2/3$; between those two points, the function~$b=b(a)$ isconstant.
Friday, March 4, 2011 - 15:05 , Location: Skiles 006 , Martin Loebl , Charles University, Prague, Czech Republic , Organizer: Robin Thomas
The Ising problem on finite graphs is usually treated by a reduction to the dimer problem. Is this a wise thing to do? I will show two (if time allows) recent results indicating that the Ising problem allows better mathematical analysis than the dimer problem. Joint partly with Gregor Masbaum and partly with Petr Somberg.