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Series: School of Mathematics Colloquium

Thresholds for increasing properties are a central concern
in probabilistic combinatorics and elsewhere.
(An increasing property, say F, is a superset-closed family
of subsets of some (here finite) set X;
the threshold question for such an F asks, roughly, about how many
random elements of X should one choose to make it likely that the
resulting set lies in F?
For example: about how many random edges from the complete graph K_n
are typically required to produce a Hamiltonian cycle?)
We'll discuss recent progress and lack thereof on a few threshold-type
questions, and try to say something about a
ludicrously general conjecture of G. Kalai and the speaker
to the effect that there is always
a pretty good naive explanation for a threshold being what it is.

Series: School of Mathematics Colloquium

I will review the well known method (pushed mainly by Karlin and McGregor) to study birth-and-death processes with the help of orthogonal polynomials. I will then look at several extensions of this idea, including ¨poker dice¨ (polynomials in several variables) and quantum walks (polynomials in the unit circle).

Series: School of Mathematics Colloquium

In many practical situations one has to make
decisions
sequentially based on data available at the time of the
decision and facing uncertainty of the future. This leads to
optimization problems which can be formulated in a framework of
multistage stochastic programming. In this talk we
consider risk neutral and risk averse approaches to multistage
stochastic programming. We discuss conceptual and computational
issues involved in formulation and solving such problems. As an
example we give numerical results based on the Stochastic Dual
Dynamic Programming method applied to planning of the Brazilian
interconnected power system.

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.