Members Seminar

Act globally, compute locally: group actions, fixed points and localization

Tara Holm
Cornell University; von Neumann Fellow, School of Mathematics
October 20, 2014
Localization is a topological technique that allows us to make global equivariant computations in terms of local data at the fixed points. For example, we may compute a global integral by summing integrals at each of the fixed points. Or, if we know that the global integral is zero, we conclude that the sum of the local integrals is zero. This often turns topological questions into combinatorial ones and vice versa. I will give an overview of how this technique arises in symplectic geometry.

Hodge theory, coniveau and algebraic cycles

Claire Voisin
Centre national de la recherche scientifique; Distinguished Visiting Professor, School of Mathematics
October 6, 2014
My talk will be a broad introduction to what is the (mostly conjectural) higher dimensional generalization of Abel's theorem on divisors on Riemann surfaces, namely, the relationship between the structure of the group of algebraic cycles on a complex projective variety and the complexity of its so-called Hodge structures.

A Riemann-Roch theorem in Bott-Chern cohomology

Jean-Michel Bismut
Université Paris-Sud
April 21, 2014
If \(M\) is a complex manifold, the Bott-Chern cohomology \(H_{\mathrm{BC}}^{(\cdot,\cdot)}\left(M,\mathbf{C}\right)\) of \(M\) is a refinement of de Rham cohomology, that takes into account the \((p,q)\) grading of smooth differential forms. By results of Bott and Chern, vector bundles have characteristic classes in Bott-Chern cohomology, which will be denoted with the subscript \(\mathrm{BC}\). Let \(p:M\to S\) be a proper holomorphic submersion of complex manifolds. Let \(F\) be a holomorphic vector bundle on \(M\) and let \(Rp_{*}F\) be its direct image.

Toroidal Soap Bubbles: Constant Mean Curvature Tori in \(S^3\) and \(R^3\)

Emma Carberry
University of Sydney
April 14, 2014
Constant mean curvature (CMC) tori in \(S^3\), \(R^3\) or \(H^3\) are in bijective correspondence with spectral curve data, consisting of a hyperelliptic curve, a line bundle on this curve and some additional data, which in particular determines the relevant space form. This point of view is particularly relevant for considering moduli-space questions, such as the prevalence of tori amongst CMC planes. I will address these periodicity questions for the spherical and Euclidean cases, using Whitham deformations, which I will explain.

Gambling, Computational Information, and Encryption Security

Bruce Kapron
University of Victoria; Member, School of Mathematics
March 24, 2014
We revisit the question, originally posed by Yao (1982), of whether encryption security may be characterized using computational information. Yao provided an affirmative answer, using a compression-based notion of computational information to give a characterization equivalent to the standard computational notion of semantic security. We give two other equivalent characterizations.

Topologies of nodal sets of random band limited functions

Peter Sarnak
Institute for Advanced Study; Faculty, School of Mathematics
March 3, 2014
We discuss various Gaussian ensembles for real homogeneous polynomials in several variables and the question of the distribution of the topologies of the connected components of the zero sets of a typical such random real hypersurface. For the "real -Fubini -Study ensemble" and at the other end the "monochromatic wave ensemble ", one can show that these have universal laws. Some qualitative features of these laws are also established. Joint work with I. Wigman.

Zeros of polynomials via matrix theory and continued fractions

Olga Holtz
University of California, Berkeley; Member, School of Mathematics
February 24, 2014
After a brief review of various classical connections between problems of polynomial zero localization, continued fractions, and matrix theory, I will show a few ways to generalize these classical techniques to get new results about some interesting polynomials and entire functions.

Rigidity and Flexibility of Schubert classes

Colleen Robles
Texas A & M University; Member, School of Mathematics
January 27, 2014
Consider a rational homogeneous variety \(X\). (For example, take \(X\) to be the Grassmannian \(\mathrm{Gr}(k,n)\) of \(k\)-planes in complex \(n\)-space.) The Schubert classes of \(X\) form a free additive basis of the integral homology of \(X\). Given a Schubert class \([S]\), represented by a Schubert variety \(S\) in \(X\), Borel and Haefliger asked: aside from the Schubert variety, does \([S]\) admit any other algebraic representatives? I will discuss this, and related questions, in the case that \(X\) is Hermitian symmetric.

Eigenvalues and eigenvectors of spiked covariance matrices

Antti Knowles
December 9, 2013
I describe recent results on spiked covariance matrices, which model multivariate data containing nontrivial correlations. In principal components analysis, one extracts the leading contribution to the covariance by analysing the top eigenvalues and associated eigenvectors of the covariance matrix. I give non-asymptotic, sharp, high-probability estimates relating the principal components of the true covariance matrix to those of the sample covariance matrix.