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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

Series: School of Mathematics Colloquium

Motivated by a problem in the synthesis
of nanowires, a sequential space filling design, called Sequential
Minimum Energy Design (SMED), is proposed for exploring and searching
for the optimal conditions in complex black-box functions. The
SMED is a novel approach to generate designs that are model
independent, can quickly carve out regions with no observable
nanostructure morphology, allow for the exploration of complex
response surfaces, and can be used for sequential experimentation. It
can be viewed as a sequential design procedure for stochastic
functions and a global optimization procedure for
deterministic functions. The basic idea has been developed into an
implementable algorithm, and guidelines for choosing the parameters
of SMED have been proposed. Convergence of the algorithm has been
established under certain regularity conditions. Performance of the
algorithm has been studied using experimental data on nanowire
synthesis as well as standard test functions.(Joint
work with V. R. Joseph, Georgia Tech and T. Dasgupta, Harvard U.)

Series: School of Mathematics Colloquium

In this talk, I will introduce a notion of geometric complexity to study topological rigidity of manifolds. This is joint work with Erik Guentner and Romain Tessera. I will try to make this talk accessible to graduate students and non experts.

Series: School of Mathematics Colloquium

Consider any dynamical system with the phase space (set of all states)
M. One gets an open dynamical system if M has a subset H (hole) such
that any orbit escapes ("disappears") after hitting H. The question in
the title naturally appears in dealing with some experiments in
physics, in some problems in bioinformatics, in coding theory, etc.
However this question was essentially ignored in the dynamical systems
theory. It occurred that it has a simple and counter intuitive answer.
It also brings about a new characterization of periodic orbits in
chaotic dynamical systems.
Besides, a duality with Dynamical Networks allows to introduce
dynamical characterization of the nodes (or edges) of Networks, which
complements such static characterizations as centrality, betweenness,
etc. Surprisingly this approach allows to obtain new results about such
classical objects as Markov chains and introduce a hierarchy in the set
of their states in regard of their ability to absorb or transmit an
"information".
Most of the results come from a finding that one can make finite
(rather than traditional large) time predictions on behavior of
dynamical systems even if they do not contain any small parameter.
It looks plausible that a variety of problems in dynamical systems, probability,
coding, imaging ... could be attacked now.
No preliminary knowledge is required. The talk will be accessible to students.

Series: School of Mathematics Colloquium

The talk will introduce basic mathematical concepts of General Relativity and review the progress, main challenges and open problems, viewed through the prism of the evolution problem. I will illustrate interaction of Geometry and PDE methods in the context of General Relativity on examples ranging from incompleteness theorems and formation of trapped surfaces to geometric properties of black holes and their stability.

Series: School of Mathematics Colloquium

Information encoded in a bar code can be read using a laser scanner or a camera-based scanner. For one-dimensional bar codes, which are in most prevalent use, the information that needs to be extracted are the widths of the black and white bars. The collection of black and white bars may be viewed as a binary one-dimensional image. The signal measured at the scanner amounts to the convolution of the binary image with a smoothing kernel. The challenge is that the smoothing kernel, in addition to the binary image, is also unknown. This presentation will review the technology behind bar code scanning and present several approaches to the decoding problem.