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Events for December 2012 – April 2013

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Dec. 11, 2012 2pm (MATH 350)	Lie Theory Vivek Dhand (MSU) X Young's partition lattice L(m,n) consists of unordered partitions having m parts where each part is at most n, ordered by inclusion of the corresponding Young diagrams. Many years ago, Richard Stanley conjectured that this poset has a decomposition into saturated rank-symmetric chains. Despite many efforts, this conjecture has only been proved when min(m,n) is less than or equal to 4. In this talk, we decompose L(m,n) into level sets for certain tropical polynomials derived from the secant varieties of the rational normal curve in projective space, and we show that resulting subposets have an elementary raising and lowering algorithm. As a corollary, we obtain a symmetric chain decomposition for the subposet of L(m,n) consisting of "sufficiently generic" partitions. Tropical Decomposition of Young's Partition Lattice
Jan. 30, 2013 4:15pm (MATH 3…	Grad Student Seminar Nathan Wakefield X Heights are an important component of the number theorist's toolbox. This presentation will describe the basic elements of heights, and how they play a role in many major areas of number theory; ranging from elliptic curves to dynamical systems. The canonical height is a particularly important tool in arithmetic dynamics. In this talk, we will discuss recent developments in the area of canonical heights, and explain an extension to generalized iteration. We will conclude with an open conjecture and discuss how heights may contribute to its solution. Canonical Heights in Generalized Iterations
Feb. 05, 2013 2pm (MATH 350)	Lie Theory Steve Doty (Loyola) X Generalized Schur algebras were introduced in the late 1980s by S. Donkin. They are one way of extending the classical theory from Type A root systems to arbitrary type, and of course q-analogues exist. In 2002 Giaquinto and I found two presentations of the classical Schur algebras and their q-analogues; one of them generalized to arbitrary type. One of the most important facts about generalized q-Schur algebras is that they are quasihereditary (in the sense of Cline-Parshall-Scott). I proved this in 2003 as an application of known facts about the Lusztig-Kashiwara canonical-crystal bases for quantized enveloping algebras of semisimple Lie algebras. Recently, Giaquinto and I found an elementary proof that does not depend on the deep existence results of Lusztig and Kashiwara. Instead, the proof starts from the defining presentation and uses only known results on Lie algebras and elementary calculations. Generalized (q)-Schur algebras and their quasihereditary structure I
Feb. 12, 2013 2pm (MATH 350)	Lie Theory Steve Doty (Loyola) X Generalized Schur algebras were introduced in the late 1980s by S. Donkin. They are one way of extending the classical theory from Type A root systems to arbitrary type, and of course q-analogues exist. In 2002 Giaquinto and I found two presentations of the classical Schur algebras and their q-analogues; one of them generalized to arbitrary type. One of the most important facts about generalized q-Schur algebras is that they are quasihereditary (in the sense of Cline-Parshall-Scott). I proved this in 2003 as an application of known facts about the Lusztig-Kashiwara canonical-crystal bases for quantized enveloping algebras of semisimple Lie algebras. Recently, Giaquinto and I found an elementary proof that does not depend on the deep existence results of Lusztig and Kashiwara. Instead, the proof starts from the defining presentation and uses only known results on Lie algebras and elementary calculations. Generalized (q)-Schur algebras and their quasihereditary structure II
Feb. 13, 2013 4:15pm (MATH 3…	Grad Student Seminar Amy Feaver X The Mahler measure of a complex-valued polynomial P(z) is the geometric mean of the modulus of the polynomial evaluated over the unit circle. At first glance, this quantity appears to be a harmless integral, but it turns out that it invades several areas of number theory. In this talk, we will discuss the relationship between Mahler measure, the golden ratio and algebraic integers. We will also extend the definition of Mahler measure to multivariable functions, and will conclude with evaluating the Mahler measure of a polynomial in three variables, finding that it can be expressed in terms of the Riemann zeta function evaluated at 3. Mahler measure
Feb. 26, 2013 2pm (MATH 350)	Lie Theory Shawn Burkett (CU) X Motivated by the problem of bounding the order of a group with a large character degree, Martin Isaacs predicted that the largest degree among irreducible complex representations of a finite nonabelian simple group could be bounded in terms of the smaller degrees. Using the Jordan decomposition of characters, we answer this question in the affirmative for a few infinite families of finite simple classical groups. We also discuss a recent theorem of Michael Larson, Gunter Malle and Pham Tiep regarding the largest irreducible representations of simple groups, as well as Isaacs's application of this theorem to the problem of bounding the orders of certain groups. Bounding the Largest Irreducible Character Degree of a Finite Simple Group
Feb. 27, 2013 4:15pm (MATH 3…	Grad Student Seminar Bryce Chriestenson X This talk will present certain homological invariants attached to a stratified space \(X\), called the Whitney-deRham cohomology. This cohomology is defined as the cohomology of a chain complex, \(\Omega^{ * }_{\mathrm{ W } } \left( X \right)\), associated to \(X\) in a somewhat ad-hoc way. The main result is to show that though the definition of \(\Omega^{ * }_{\mathrm{ W } } \left( X \right)\) depends on several choices, when certain conditions are imposed on \(X\), the Whitney-deRham cohomology only depends on the homotopy type of \(X\). This is achieved by showing that \(\Omega^{ * }_{\mathrm{ W } } \left( X \right)\) can be realized as a fine complex of sheaves which is a resolution of the locally constant sheaf on \(X\). An application of this work is in the area of homotopy theory. One can canonically define a commutative differential graded algebra(CDGA), \(A_{PL}\left(X\right)\), on \(X\) in such a way that any CDGA which is quasi-isomorphic to it determines the real homotopy type of \(X\). It will be shown that the complex \(\Omega^{ * }_{\mathrm{ W } } \left( X \right)\) is quasi-isomorphic to \(A_{PL}\left(X\right)\), and thus determines the real homotopy type of \(X\). The real homotopy type of singular spaces via the Whitney-deRham Complex
Mar. 19, 2013 2pm (MATH 350)	Lie Theory Chris Bowman (Paris 7) X We discuss the partition algebra as a setting for the study of row-removal phenomena for the symmetric group. We describe its representation theory in terms of a type A facet geometry. We then discuss applications to the Kronecker problem, in particular a uniform description of the Kronecker coefficients when one of the indexing partitions is a hook or a two-part partition. The representation theory of the partition algebra
Mar. 20, 2013 4:15pm (MATH 3…	Grad Student Seminar Matthew Grimes X Answering global questions with local data is a common theme in geometry, and sheaves are a powerful tool for organizing such problems. Cech cohomology aids in the study of sheaves on topological spaces, and helps us understand when local data is enough to answer a global question. We will discuss the basics of sheaves, and then go over the construction of Cech cohomology. We will examine basic theorems with an emphasis on motivation, examples, and accessibility. Applications to topology and geometry will also be covered, and so a cursory review of simplicial homology and complex analysis prior to the talk may be useful. Sheaves and Cech Cohomology
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

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