Abstracts of 2012-2013 UW-PIMS Mathematics Colloquia

 May 17, 2013 at 2:30pm DEM 004 Yuval Peres Principal Researcher and Theory Group Manager at Microsoft Research Search games and Optimal Kakeya Sets A planar set that contains a unit segment in every direction is called a Kakeya set. These sets have been studied intensively in geometric measure theory and harmonic analysis since the work of Besicovich (1928); we find a new connection to game theory and probability. A hunter and a rabbit move on an n-vertex cycle without seeing each other until they meet. At each step, the hunter moves to a neighboring vertex or stays in place, while the rabbit is free to jump to any node. Thus they are engaged in a zero sum game, where the payoff is the capture time. We show that every rabbit strategy yields a Kakeya set; the optimal rabbit strategy is based on a discretized Cauchy random walk, and it yields a Kakeya set K consisting of 4n triangles, that has minimal area among such Kakeya sets. Passing to the scaling limit yields a simple construction of a random Kakeya set with zero area from two Brownian motions. (Talk based on joint work with Y. Babichenko, R. Peretz, P. Sousi and P. Winkler).
 May 10, 2013 at 2:30pm DEM 004 S.R.Srinivasa Varadhan New York University's Courant Institute Counting Graphs Let ${H_i}, i=1,2,...,k$ be a finite collection of finite graphs. Let $v_i$ be the number of vertices in $H_i$. We want to count the number of graphs G with N vertices in which $H_i$ appears roughly $c_i N_i^v$ times. The tools are a combination of large deviations in probability theory and Szemeredi's regularity theorem for dense graphs. March 20, 2013 at 2:30pm CMU 230 Cathy O'Neil mathbabe.org, Data Scientist at Johnson Research Labs How Math is Used Outside Academia In this talk we'll look into how modelers use and compromise mathematics in industry, specifically finance, internet advertising and social media, and education. The goal of the talk is to encourage mathematicians to be more engaged with the way math is used by and against the public. February 22, 2013 at 2:30pm DEN 216 Gerald Folland University of Washington From the Applicable to the Abstruse: An Example in Representation Theory The operations of time shift (ƒ(t) → ƒ(t+1)) and frequency shift (ƒ(t) → e2πiωtƒ(t)) are fundamental ingredients of applied Fourier analysis, and the group of operators on L2(ℝ) that they generate gives a unitary representation of the so-called discrete Heisenberg group. How does this representation de- compose into irreducible representations? The answer provides illustrations of (i) some useful tools of modern harmonic analysis, when ω is rational, and (ii) some pathological phenomena from the dark side of representation theory, when ω is irrational. We shall discuss these results after providing a bit of background on unitary representation theory. February 4, 2013 at 2:30pm THO 135 Mark Behrens Massachusetts Institute of Technology Exotic spheres and topological modular forms An exotic sphere is a smooth manifold which is homeomorphic, but not diffeomorphic, to a standard sphere. In which dimensions do there exist exotic spheres? I will discuss what we know about the answer to this question in terms of the classical work of Kervaire and Milnor, the recent solution of the Kervaire invariant problem by Hill, Hopkins, and Ravenel, and a current project with Hill, Hopkins and Mahowald, where we use topological modular forms to detect exotic spheres. February 1, 2013 at 2:30pm LOW 102 Thomas Rothvoß Massachusetts Institute of Technology Approximating Bin Packing within O(log OPT * log log OPT) bins For bin packing, the input consists of n items with sizes s_1,...,s_n in [0,1] which have to be assigned to a minimum number of bins of size 1. The seminal Karmarkar-Karp algorithm from 1982 produces a solution with at most OPT + O(log^2 OPT) bins in polynomial time, where OPT denotes the value of the optimum solution. I will describe the first improvement in 3 decades and show that one can find a solution of cost OPT + O(log OPT * log log OPT) in polynomial time. This is achieved by rounding a fractional solution to the Gilmore-Gomory LP relaxation using the Entropy Method from discrepancy theory. I will also survey other results of mine from discrete mathematics and combinatorial optimization concerning: an approach to the Hirsch conjecture on the diameter of polytopes; the Chvatal rank of polytopes; the extension complexity of 0/1 polytopes; and the approximability of the Steiner tree problem. January 29, 2013 at 2:30pm MEB 246 Dmitriy Drusvyatskiy Cornell University Slope and geometry in variational mathematics Various notions of the "slope" of a real-valued function pervade optimization, and variational mathematics more broadly. In the semi-algebraic setting - an appealing model for concrete variational problems - the slope is particularly well-behaved. This talk sketches a variety of surprising applications, illustrating the unifying power of slope. Highlights include error bounds for level sets, the existence and regularity of steepest descent curves in complete metric spaces (following Ambrosio et al.), transversality and the convergence of von Neumann's alternating projection algorithm, and the geometry underlying nonlinear programming active-set algorithms. This talk will be self-contained, requiring no familiarity with variational analysis, optimization theory, or semi-algebraic geometry. Joint work with A. Daniilidis (Barcelona), A.D. Ioffe (Technion), M. Larsson (Lausanne), A.S. Lewis (Cornell). January 28, 2013 at 2:30pm THO 135 Katya Krupchyk University of Helsinki Inverse boundary value problems for elliptic operators In an inverse boundary value problem one is interested in determining the internal properties of a medium by making measurements on the boundary of the medium. In mathematical terms, one wishes to recover the coefficients of a partial differential equation inside the medium from the knowledge of the Cauchy data of the solutions on the boundary. These problems have numerous applications, ranging from medical imaging to exploration geophysics. We shall discuss some recent progress in the analysis of inverse problems for elliptic equations, starting with the celebrated Calderon problem. The case of inverse problems with rough coefficients and with measurements performed only on a portion of the boundary will also be addressed. January 25, 2013 at 2:30pm LOW 102 Fredrik Viklund Columbia University On geometric properties of the Schramm-Loewner evolution The Schramm-Loewner evolution (SLE) is a family of random fractal curves that arise in a natural manner as scaling limits of interfaces in planar critical lattice models from statistical mechanics. The SLE curves exhibit many interesting geometric structures also present in the related models, now rigorously accessible through a combination of complex analytic and probabilistic techniques. I will give an overview of some known properties and open questions regarding the fine geometry of SLE curves. The talk is in part based on joint works with G. Lawler, with S. Rohde and C. Wong, and with T. Alberts and I. Binder. January 22, 2013 at 2:30pm LOW 101 Reinier Bröker Brown University Geometric applications of class field theory One of the major developments in number theory during the 20th century is the invention of class field theory. In this talk we will recall some highlights of this theory, and explain how to apply it to solve geometric problems. In particular, we will show how to efficiently compute isogenies between elliptic curves and how to construct elliptic curves of prescribed order. Many examples will be given. January 18, 2013 at 2:30pm DEN 216 Mathias Drton UW Statistics Department Applications of Algebraic Geometry in Statistics Statistical modeling amounts to specifying a set of candidates for what the probability distribution of an observed random quantity might be. Many models used in practice are of an algebraic nature in that they are defined in terms of a polynomial parametrization. The goal of this talk is to exemplify how techniques from computational algebraic geometry may be used to solve statistical problems that concern algebraic models. The focus will be on applications in hypothesis testing and parameter identification, for which we will survey some of the known results and open problems. November 16, 2012 at 2:30pm Sieg 225 Max Warshauer Texas State University Collaborations Between University Math Departments and Public Schools In this talk, I will describe challenges in working with public schools and our experiences with math camps, a curriculum project, and teacher training. This discussion will outline steps in setting up collaborations, how they can benefit all the parties, and problems that can occur. I will also describe lessons learned, and where we are now in some of our current projects. October 5, 2012 at 2:30pm Sieg 225 Matthew Kahle Ohio State University Topology of Random Flag Complexes Random flag complexes are a natural generalization of random graphs to higher dimensions, and since every simplicial complex is homeomorphic to a flag complex this puts a measure on a wide range of possible topologies.  In this talk, I will discuss the recent proof that according to the Erdős–Rényi measure, asymptotically almost all $d$-dimensional flag complexes only have nontrivial (rational) homology in middle degree $\lfloor d/2 \rfloor$. The highlighted technique is originally due to Garland -- what he called "$p$-adic curvature" in a somewhat different context. This method allows one to prove cohomology-vanishing theorems by showing that certain discrete Laplacians have sufficiently large spectral gap. This reduces certain questions in probabilistic topology to questions about random matrices. Some of this depends on new results for random matrices, in joint work with Chris Hoffman and Elliot Paquette. Proving central limit theorems for Betti numbers in the non-vanishing regime was done in joint work with Elizabeth Meckes. The talk will aim to be self contained--I will not assume any particular probability or topology prerequisites.

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