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36 DM10 Abstracts 36 DM10 Abstracts IP0 IP3 D´enes K¨onig Prize Lecture: Ramsey Numbers of Extremal Problems for Convex Lattice Polytopes Graphs and Hypergraphs In this survey I will present several extremal problems, and Determining or estimating Ramsey numbers is one of the some solutions, concerning convex lattice polytopes. A typ- central problems in combinatorics. In this talk we discuss ical example is to determine the minimal volume that a recent progress on some longstanding conjectures in this convex lattice polytope can have if it has exactly n ver- area which have played an important role in the develop- tices. Other examples are the minimal surface area, or the ment of Ramsey theory. minimal lattice width in the same class of polytopes. These problems are related to a question of V I Arnold from 1980 Jacob Fox asking for the number of (equivalence classes of) lattice Princeton University polytopes of volume V in d-dimensional space, where two [email protected] convex lattice polytopes are equivalent if one can be carried to the other by a lattice preserving affine transformation. IP1 Imre B´ar´any Binary Matroid Minors R´enyi Institute University College London The graph minors project is a sequence of extraordinary [email protected] theorems proved by Neil Robertson and Paul Seymour in the 1980s. The banner theorem of that project is that every minor-closed family of graphs is characterized by a finite IP4 set of excluded minors. The project also has a number Optimizing in a Strategic World: A Survey of Re- of remarkable algorithmic consequences, for example, the cent Research in Algorithmic Game Theory membership testing problem can be efficiently solved for any minor-closed class of graphs. Much of the work in the The goal of discrete optimization is to design systems with graph minors project goes into proving a technical theorem optimal or near-optimal performance. In the age of the that constructively characterizes all minor-closed classes Internet, however, we must take into account the fact that of graphs. The graph minors project can be interpreted many of the users of our systems are driven by an eco- in the context of matroid theory by considering the class nomic goal, and interact with varying degrees of collab- of graphic matroids. Over the past decade, in joint work oration and competition. Moreover, the strategic nature with Bert Gerards and Geoff Whittle, we have extended of interactions in online dynamic marketplaces means that much of the graph minors project from the class of graphic the roll-out of a new algorithm designed with the expecta- matroids to the class of binary matroids. This talk will tion of improved performance can end up degrading perfor- be introductory, no prior knowledge of matroid theory is mance due to unanticipated responses by strategic users. assumed. We give a broad overview of the results and focus The field of algorithmic game theory addresses this issue, on potential applications to graph theory, coding theory, as well as a wide variety of other problems at the intersec- and quantum computing. tion of game theory, economics and computer science. In this talk, we survey recent research and open problems in Jim Geelen this field. University of Waterloo [email protected] Anna R. Karlin University of Washington [email protected] IP2 The Combinatorics of Discrete Random Matrices IP5 In random matrix theory, both continuous random matrix The Method of Multiplicities ensembles (e.g. the gaussian unitary ensemble, GUE) and discrete random matrix ensembles (e.g. the Bernoulli en- In 2008, Zeev Dvir achieved a breakthrough in combi- semble of random sign matrices, or the adjacency matrices natorial geometry by giving a stunningly simple, and n of random graphs) are of interest. However, the discrete sharp, bound on the size of ”Kakeya Sets” in Fq , the n- case contains additional difficulties that are not present in dimensional vector space over the finite field on q elements. the continuous case. For instance, it is obvious that contin- (A Kakeya set in any vector space is a set that contains a uous random square matrices are almost surely invertible, line in every direction.) Dvir proved this bound by setting but this is not true in the discrete case. Nevertheless, in re- up an n-variate low-degree non-zero polynomial that van- cent years several tools of an additive combinatorics nature ishedoneverypointintheset,andthenusedthealgebraic have been developed to close the gap between our under- nature of a Kakeya set to argue that this polynomial was standing of discrete random matrices and continuous ran- zero too often if the set was too small. In addition to re- dom matrices, and in particular inverse Littlewood-Offord solving a long-studied problem in combinatorial geometry, theory, which roughly speaking asserts that discrete ran- this method also led to new constructions of “randomness dom walks behave much like their continuous counterparts, extractors’. In this talk I will describe algebraic methods except in highly arithmetically structured cases, such as to improve the analysis of Dvir, by using polynomials that when the step sizes of the random walk all lie in an arith- vanish with “high multiplicity’ on every point on the given metic progression. We survey these developments, and set. This method, based on prior work with Guruswami their applications, in this talk. (1998), ends up yielding extremely tight (to within a fac- tor of 2) bounds on the size of Kakeya sets; and, in com- Terence Tao bination with a host of other techniques, state-of-the-art Department of Mathematics, UCLA ”extractors” (algorithms that purify randomness). In this [email protected] talk I will describe the (simple) idea behind the method of multiplicities and some of the applications. Based on joint DM10 Abstracts 37 works with Shubhangi Saraf (Analysis & PDE, 2009); and symmetric group Sn whose dimension is En. with Zeev Dvir, Swastik Kopparty, and Shubhangi Saraf (FOCS 2009). Richard Stanley Massachusetts Institute of Technology Madhu Sudan [email protected] Microsoft Research [email protected] CP1 A Solution to Alspach’s Problem for Complete IP6 Graphs of Large Odd Order Configurations in Large t-connected Graphs In 1981 Alspach posed the problem of proving that a I will discuss a technique that allows us to establish the complete graph of odd order n can be decomposed into existence of certain configurations in large t-connected edge-disjoint cycles of specified lengths m1,m2,...,mt graphs, even though such configurations need not exist in whenever the obvious necessary conditions that 3 ≤ small graphs. Two configurations of interest are complete ≤ ··· n m1,m2,...,mt n and m1 + m2 + + mt = 2 are minors and disjoint paths connecting prescribed pairs of satisfied. In this talk I will give a brief outline of a solution vertices. We prove that for every integer t there exists an to Alspach’s problem for sufficiently large odd values of n. integer N such that every t-connected graph on at least N vertices with no minor isomorphic to the complete graph on t vertices has a set of at most t-5 vertices whose dele- Daniel Horsley tion makes the graph planar. This is best possible, ex- Memorial University of Newfoundland ceptforthevalueofN.WealsoprovethatthekDisjoint [email protected] Paths Problem is feasible in every sufficiently big (2k+3)- connected graph. This is joint work with Sergey Norin. Darryn Bryant University of Queensland [email protected] Robin Thomas Georgia Tech [email protected] CP1 When Every k-Cycle Has at Least f(k) Chords IP7 Chordal graphs can be characterized by every k-cycle hav- Hypergraphs with Low Dimension − ≥ k−3 ing at least k 3 chords. Similarly, requiring 2 2 chords characterizes the house-hole-domino-free graphs, Any hypergraph can be viewed as an object with geomet- and requiring ≥ 2k − 7 chords characterizes graphs whose ric properties, by considering a geometric realisation of its blocks are trivially perfect. Moreover, these three functions associated abstract simplicial complex. It is well-known f(k) are optimum for their graph classes—there are always that any k-uniform hypergraph has such a realisation in graphs in each class that have k-cycles with exactly f(k) (2k − 1)-dimensional real space. We focus in particular on − chords. The functions 3 k 3 and 3k − 11 characterize k-uniform hypergraphs that have a geometric realisation in 3 similar graph classes without being optimum. k-dimensional space (so when k = 2 this is the class of pla- nar graphs). We consider some properties of planar graphs Terry McKee that naturally extend to this class of “low-dimensional’ hy- Wright State University pergraphs. Department of Mathematics & Statistics [email protected] Penny Haxell University of Waterloo [email protected] CP1 4-Cycle Systems of Kn with An Almost 2-Regular IP8 Leave A Survey of Alternating Permutations An almost 2-regular leave of Kn is a subgraph of Kn in which each vertex except one has degree two; the excep- A permutation a a ···a of 1, 2,...,n is alternating if 1 2 n tional vertex has arbitrary degree. In this talk, we will a >a <a >a <a > ···.IfE is the number of 1 2 3 4 5 n provide the necessary and sufficient conditions for the ex- alternating permutations of 1, 2,...,n,then istence of 4-cycle systems of Kn with an almost 2-regular leave. The solution to this problem has applications in xn E =secx +tanx. neighbor designs used by serology researchers. n n! ≥ n 0 Nidhi Sehgal,C.A.Rodger Auburn University We will discuss several aspects of the theory of alternating [email protected], [email protected] permutations.
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