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Series: School of Mathematics Colloquium

An intesting class of bounded operators or algebras of bounded operators
on Hilbert spaces, particularly on Hilbert spaces of holomorphic
functions, have a natural interpretation in terms of concepts from
complex geometry. In particular, there is an intrinsic hermitian
holomorphic vector bundle and many questions can be answered in terms of
the Chern connection and the associated curvature.
In this talk we describe this setup and some of the results obtained
in recent years using this approach. The emphasis will be on concrete
examples, particularly in the case of Hilbert spaces of holomorphic
functions such as the Hardy and Bergman spaces on the unit sphere in
C^n.

Series: School of Mathematics Colloquium

Pardon the inconvenience. We plan to reschedule later...

Series: School of Mathematics Colloquium

There will be a tea 30 minutes before the colloquium.

Tom Church, Jordan Ellenberg and I recently discovered that the i-th Betti number of the space of configurations of n points on any manifold is given by a polynomial in n. Similarly for the moduli space of n-pointed genus g curves. Similarly for the dimensions of various spaces of homogeneous polynomials arising in algebraic combinatorics. Why? What do these disparate examples have in common? The goal of this talk will be to answer this question by explaining a simple underlying structure shared by these (and many other) examples in algebra and topology.

Series: School of Mathematics Colloquium

Hosts are Ernie Croot and Dan Margalit.

We survey some new and classic recreations in the fields of mathematics,
magic and mystery in the style of Martin Gardner, Prince of Recreational
Mathematics, whose publishing career recently ended after an astonishing
80 years. From card tricks and counter-intuitive probability results to
new optical illusions, there will be plenty of reasons to celebrate the
ingenuity of the human mind.

Series: School of Mathematics Colloquium

This is a joint ARC-SoM colloquium, and is in conjunction with the ARC Theory Day on November 11, 2011

Man has grappled with the meaning and utility of randomness for centuries. Research in the Theory of Computation in the last thirty years has enriched this study considerably. I'll describe two main aspects of this research on randomness, demonstrating respectively its power and weakness for making algorithms faster. I will address the role of randomness in other computational settings, such as space bounded computation and probabilistic and zero-knowledge proofs.

Series: School of Mathematics Colloquium

One of the basic problems of Harmonic analysis is to determine ifa given collection of functions is complete in a given Hilbert space. Aclassical theorem by Beurling and Malliavin solved such a problem in thecase when the space is $L^2$ on an interval and the collection consists ofcomplex exponentials. Two closely related problems, the so-called Gap andType Problems, studied by Beurling, Krein, Kolmogorov, Levinson, Wiener andmany others, remained open until recently.In my talk I will present solutions to the Gap and Type problems anddiscuss their connectionswith adjacent fields.

Series: School of Mathematics Colloquium

Optimization problems involving sparse vectors or low-rank matrices are of great importance in applied mathematics and engineering. They provide a rich and fruitful interaction between algebraic-geometric concepts and convex optimization, with strong synergies with popular techniques like L1 and nuclear norm minimization. In this lecture we will provide a gentle introduction to this exciting research area, highlighting key algebraic-geometric ideas as well as a survey of recent developments, including extensions to very general families of parsimonious models such as sums of a few permutations matrices, low-rank tensors, orthogonal matrices, and atomic measures, as well as the corresponding structure-inducing norms.Based on joint work with Venkat Chandrasekaran, Maryam Fazel, Ben Recht, Sujay Sanghavi, and Alan Willsky.

Series: School of Mathematics Colloquium

In many integer factoring algorithms, one produces a
sequence of integers (created in a pseudo-random way), and wishes
to rapidly determine a subsequence whose product is a square
(which we call a `square product'). In his lecture at the 1994
International Congress of Mathematicians, Pomerance observed that the
following problem encapsulates all of the key issues: Select integers
a1, a2, ..., at random from the interval [1,x], until some (non-empty)
subsequence has product equal to a square. Find good esimates for
the expected stopping time of this process. A good solution to this
problem should help one to determine the optimal choice of parameters
for one's factoring algorithm, and therefore this is a central question.
In this talk I will discuss the history of this problem, and its somewhat
recent solution due to myself, Andrew Granville, Robin Pemantle, and
Prasad Tetali.

Series: School of Mathematics Colloquium

Problem: describe the asymptotic behavior of the coefficients a_{ij} of the Taylor series for 1/Q(x,y) where Q is a polynomial. This problem is the simplest of a number of such problems arising in analytic combinatorics whose answer was not until recently known. In joint work with J. van der Hoeven and T. DeVries, we give a solution that is completely effective and requires only assumptions that are met in the generic case. Symbolic algebraic computation and homotopy continuation tools are required for implementation.

Series: School of Mathematics Colloquium

This will be a survey talk on some aspects of the geometry and topology of moduli spaces of representations of surface groups into Lie groups. I will discuss recent generalizations of the techniques of Atiyah and Bott on equivariant Morse theory. These extend results on stable bundles to Higgs bundles and associated moduli spaces, which correspond to representation varieties into noncompact Lie groups