UIUC Dept. of Mathematics Seminar Calendar

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Thursday, March 17, 2011

Group Theory Seminar
1:00 pm in Altgeld Hall 347, Thursday, March 17, 2011

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Submitted by kapovich.

Ilya Kapovich (UIUC Math)
Spectrally rigid subsets of free groups
Abstract: Marked length spectrum rigidity is an important phenomenon in the study of negatively curved and non-positively curved manifolds and in other related contexts. In particular, the Marked Length Spectrum Rigidity Conjecture states that for a closed negatively curved manifold the marked length spectrum (though of as a function on the fundamental group, assigning to each closed curve the length of the shortest element in its free homotopy class) uniqely determines the isometry type of the negatively curved Riemannian metric that gave rise to it. The Culler-Vogtmann Outer space consists of (equivariant F_N-isometry types of) real trees equipped with free discrete minimal isometric action of F_N. Each such tree defines a translation length function (or marked length spectrum) on F_N, and the tree can be uniquely recovered from its translation length function. We say that a subset S of $F_N$ is "spectrally rigid" in F_N, if whenever two trees from the Outer space have length functions that agree on $S$, then the two trees are equal in the Outer space. In this talk we discuss known results about examples opf spectrally rigid subsets of free groups, including those coming from random walks and from automorphic orbits.

Analysis Seminar
2:00 pm in 243 Altgeld Hall, Thursday, March 17, 2011

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Submitted by aimo.

Katrin Faessler (University of Bern)
Extremal quasiconformal mappings on the Heisenberg group
Abstract: A rich theory of quasiconformal mappings on the Heisenberg group has been developed by Pansu, Koranyi and Reimann, motivated by the use of such maps in the proof of Mostow's rigidity theorem. Similarly as in the Euclidean case, one can ask for "extremal" quasiconformal mappings, that is, maps which minimize a certain distortion functional within a given class of quasiconformal mappings. We use the modulus of curve families and a special set of coordinates in order to identify Heisenberg counterparts of classical extremal mappings in the complex plane. We will discuss both minimizers for the maximal distortion and for a weighted mean distortion functional. This is joint work with Zoltan Balogh and Ioannis Platis.

Commutative Algebra Seminar
3:00 pm in 243 Altgeld Hall, Thursday, March 17, 2011

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Submitted by asecele2.

Hai Long Dao (University of Kansas)
Solving equations in the semiring of vector bundles
Abstract: Let X be a connected Noetherian scheme. The set of isomorphism classes of vector bundles on X carries a semiring structure via direct sum and tensor product. In this talk we discuss two problems on the arithmetic of this semiring: canceling from product and existence of non-trivial finite bundles (a finite bundle, in the sense of Nori, is one satisfying f(V) = g(V) for two non-equal polynomials with non-negative integer coefficients). We will focus on two closely related cases: projective spaces and punctured spectrum of regular local rings (even in mixed characteristics). Some extensions and applications, for example on torsion-freeness of Picard groups of hypersurfaces, will be discussed if time permits.