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Events for April – August 2013

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Apr. 09, 2013 2pm (MATH 350)	Lie Theory Mark Reeder (Boston College) X The character ring R(G) of a complex Lie group G has many interesting bases, each of whose elements is parametrized by dominant weights. The Weyl characters and the Macdonald-Hall polynomials are two examples of such bases. Each basis is given by a certain function F(t,d), defined on T x D, where T is a maximal torus of G and D is the lattice-cone of dominant weights of T. Normally one fixes a weight d in D and studies the analytic function t --> F(t,d). For example if F is the Weyl character then F(t,d) is the trace of t on the representation of G with highest weight d. It is also interesting to fix t and study the asymptotic behavior of the function d --> F(t,d). Such functions arise as matrix coefficients of representations of the p-adic group G* dual to G. Now the functions blow up, according to the singularity of t, and the blowing up is measured, at least in the two examples above, by the homology classes of certain subvarieties of the flag variety G/B of G. I will show this explicitly in the model of the homology of G/B given by harmonic polynomials on the Lie algebra of T, and will discuss implications for the representation theory of G*. Germs of matrix coefficients and their geometric interpretation
Apr. 17, 2013 4:15pm (MATH 3…	Grad Student Seminar Bryce Chriestenson X For a fixed group G I will define the group (co)homology of G with coefficients in an arbitrary G-module. After this I will calculate a few examples, and compute the low-dimensional (co)homology groups of G. These low-dimensional groups are related to: Group extensions of G, derivations of G-modules, the abelianization of G, universal central extensions of G, crossed G-modules, Schur multipliers, and more. I will say as much about these topics as I can fit into one hour. An introduction to group (co)homology
Apr. 22, 2013 4pm (DUAN G1B20)	DeLong Professor Yuval Peres X We compare several growth models on the two dimensional lattice. In some models, like internal DLA and rotor-router aggregation, the scaling limits are universal; in particular, starting from a point source yields a disk. In the abelian sandpile, particles are added at the origin and whenever a site has four particles or more, the top four particles topple, with one going to each neighbor. Despite similarities to other models, for the sandpile, the intriguing pattern that arises is not circular and depends on the particular lattice. A scaling limit exists for the sandpile, as was recently shown by Pegden and Smart, but it is not universal and is still mysterious. This research has been greatly influenced by pictures of the relevant sets, which I will show in the talk. They suggest a connection to conformal mapping which has not been established yet. (Talk based on joint works with Lionel Levine.) Following Monday's lecture, there will be a reception in honor of Professor Peres at the Koenig Alumni Center, 1202 University Avenue (the SE corner of Broadway and University). Laplacian Growth and the Mystery of the Abelian Sandpile: a Visual Tour
Apr. 23, 2013 4pm (DUAN G1B20)	DeLong Professor Yuval Peres X 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 (the area of K is of order 1/log(n)). Passing to the scaling limit yields a simple construction of a random Kakeya set with zero area from two Brownian motions. (Joint work with Y. Babichenko, R. Peretz, P. Sousi and P. Winkler.) Hunter, Cauchy Rabbit, and Optimal Kakeya Sets
Apr. 24, 2013 4:15pm (MATH 3…	Grad Student Seminar Amy Feaver X In 1801, Gauss proposed three conjectures about the class numbers of quadratic fields. Two of these conjectures have been proven. However, the conjecture that there are infinitely many real quadratic fields of class number 1, still remains open today. In this talk, we will explore some progress that has been made toward the understanding of this conjecture, including a series of more recent conjectures, brought about by the Cohen-Lenstra heuristics. Classy!
Apr. 30, 2013 2pm (MATH 350)	Lie Theory Martha Yip (U. Penn) X The set of n by n upper-triangular nilpotent matrices with entries in a finite field has Jordan canonical forms indexed by the partitions of n. We will look at a connection between these matrices and non-attacking rook placements, which leads to a combinatorial formula for the number of matrices of fixed Jordan type as a weighted sum over rook placements. q-Rook placements and upper-triangular nilpotent matrices Sponsored by the Meyer Fund
May. 01, 2013 4:15pm (MATH 3…	Grad Student Seminar Matt Moore X In group theory, the commutator \([x,y]\) induces a binary operation on the lattice of normal subgroups, which is used to define the important concepts of abelian group, solvable group, nilpotent group, etc. Together with the lattice operations, the commutator thus carries much of the information of the structure of a group. In this talk we will define a commutator for congruence lattices of general algebras and study some basic properties of the operation. A general commutator

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