Events for the week of February 27 2012

Abstract. For a given prime number p, Hida, Coleman and Mazur constructed the p-adic families of elliptic modular forms having a fixed finite slope. In this talk, starting from such p-adic families, I shall present how to construct one-parameter p-adic families of Siegel modular forms, that is, automorphic forms on the symplectic group of higher genus given by means of a functorial lifting à la Duke-Imamoglu and Ikeda. Key ingredients in the proof are: (1) A certain p-stabilization process (a kind of p-adic normalization) for Siegel modular forms above and its effect on the Fourier expansions; (2) The vertical control theorem for p-adic Siegel modular forms which is shown by Hida and Pilloni. Moreover, if circumstances allow, I'd also like to mention about some applications of the above result, for instance, to the study of a generalized lifting of cuspidal automorphic representations of PGL(2) over a totally real number field.

Abstract. Let R be a ring with identity 1 but not necessarily associative. The level (resp. sublevel) of R is the least number n such that -1 (resp. 0) can be written (nontrivially) as a sum of n (resp. n+1) squares in R, provided such a sum of squares representation is possible at all. The Pythagoras number of R is the least number n such that any sum of squares in R can be written as a sum of n squares provided such an n exists. These invariants and the relations between them have been studied extensively. For example, a famous result by Pfister states that the level of a field, if finite, is always a 2-power, and that any such value can be realized as level of a suitable field. In the field case, level and sublevel coincide, and and if the level of a field is finite, say s, then the Pythagoras number is s or s+1 (both are possible for a given 2-power s). We give a survey of some old and new results and present various open problems concerning these three invariants in the case of certain algebras over fields and in the case of integral domains.

Abstract. Tug-of-War is a zero sum, two player game played by moving a token in a domain until it hits its boundary. At each step Player II pays to Player I a certain value determined by the current position of the token, and the order of moves is determined by fair coin tosses. In a work of Peres, Schramm, Sheffield and Wilson these games were used to obtain new results on the existence and uniqueness of solutions for certain Infinity Laplace equations with Dirichlet boundary conditions. In this talk we will study the limiting behavior of the game values for Tug-of-War of prescribed horizon, and use it to prove the existence results for the Infinity Laplace equation with vanishing Neumann boundary conditions. This is a joint work with Yuval Peres, Scott Sheffield and Stephanie Somersille.

Abstract. Let f be a newform of weight 2 associated with an elliptic curve E. We relate certain global points on E with the value at 2 of the Fourier coefficient (or its derivative) of the Lambda-adic Saito-Kurokawa lifting of the p-adic family passing through f. This is joint work with Matteo Longo (Padova).

Abstract. A common theme of enumerative combinatorics is formed by counting functions that are polynomials. For example, one proves in any introductory graph theory course that the number of proper k-colorings of a given graph G is a polynomial in k, the chromatic polynomial of G. Combinatorics is abundant with polynomials that count something when evaluated at positive integers, and many of these polynomials have a (completely different) interpretation when evaluated at negative integers: these instances go by the name of combinatorial reciprocity theorems. For example, when we evaluate the chromatic polynomial of G at -1, we obtain (up to a sign) the number of acyclic orientations of G, that is, those orientations of G that do not contain a coherently oriented cycle.

Reciprocity theorems appear all over combinatorics. This talk will attempt to show some of the charm (and usefulness!) these theorems exhibit. Our goal is to weave a unifying thread through various old and new combinatorial reciprocity theorems, by looking at them through the lens of geometry.

Abstract. We will give a self-contained proof of a local rigidity property of the usual action of SL_2(Z) on R^2, and use this to obtain new information concerning the Borel reducibility hierarchy just beyond the (measure) hyperfinite equivalence relations.

Abstract. In 1974 Robert T. Powers proved that the reduced $C^*$-algebra for the free group on two generators is simple and has a unique tracial state. We will explore the generalization of this result to "Powers groups" given in a 2005 paper by Pierre de la Harpe. Starting with the definition of weak containment in the context of unitary representations, we will prove the equivalence of being $C^*$-simple and having a simple reduced $C^*$-algebra. The relationship between these conditions and amenability will be established. Lastly, we will show that Powers groups are $C^*$-simple and have a unique tracial state.

Abstract. We consider eigenvalues of finite rank deformation to Wigner matrices that lie outside of the support of the semicircle. It was first shown by M.Capitaine, C. Donati-Martin, and D. Feral that the fluctuations of these eigenvalues are non-universal. We can generalize these results by relating the outliers to quadratic forms of the resolvent of random matrices.

Abstract. By the Glimm-Effros dichotomy, every non-smooth countable Borel equivalence relation contains a copy of the non-smooth hyperfinite Borel equivalence relation. We will discuss how local rigidity properties of the usual action of SL_2(Z) on R^2 can be used to rule out analogous results for the class of non-measure-hyperfinite countable Borel equivalence relations. I hope to make the talk accessible to a broad audience.

Abstract. This talk explores relations between colored Jones polynomials and the topology of incompressible spanning surfaces in knot and link complements. Under mild diagrammatic hypotheses, we prove that the growth of the degree of the colored Jones polynomials is a boundary slope of an essential surface in the knot complement. We also show that certain coefficients of the Jones and colored Jones polynomials measure how far this surface is from being a fiber in the knot complement. This is joint work with Effie Kalfagianni and Jessica Purcell.

Abstract. We will start by describing Khovanov's categorification of the Jones polynomial from a cube of resolutions of a link diagram. We will then introduce the notion of a framed flow category, as defined by Cohen, Jones and Segal. We will see how a cube of resolutions produces a framed flow category for the Khovanov chain complex, and how the framed flow category produces a space whose reduced cohomology is the Khovanov homology. We will show that the stable homotopy type of the space is a link invariant. Time permitting, we will show that the space is often non-trivial, i.e., not a wedge sum of Moore spaces. This work is joint with Robert Lipshitz.