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Series: School of Mathematics Colloquium

Series: School of Mathematics Colloquium

There is an extensive applied mathematics literature developed for problems in the biological and physical sciences. Our understanding of social science problems from a mathematical standpoint is less developed, but also presents some very interesting problems. This lecture uses crime as a case study for using applied mathematical techniques in a social science application and covers a variety of mathematical methods that are applicable to such problems. We will review recent work on agent based models, methods in linear and nonlinear partial differential equations, variational methods for inverse problems and statistical point process models. From an application standpoint we will look at problems in residential burglaries and gang crimes. Examples will consider both "bottom up" and "top down" approaches to understanding the mathematics of crime, and how the two approaches could converge to a unifying theory.

Series: School of Mathematics Colloquium

We show that the LIA approximation of the incompressible Euler equation describes the skew-mean-curvature flow on vortex membranes in any dimension. This generalizes the classical binormal, or vortex filament, equation in 3D. We present a Hamiltonian framework for higher-dimensional vortex filaments and vortex sheets as singular 2-forms with support of codimensions 2 and 1, respectively. This framework, in particular, allows one to define the symplectic structures on the spaces of vortex sheets.

Series: School of Mathematics Colloquium

Building on work of Jordan from 1870, in 1979 Harris
showed that a geometric monodromy group associated to
a problem in enumerative geometry is equal to the Galois
group of an associated field extension. Vakil gave a
geometric-combinatorial criterion that implies a Galois
group contains the alternating group. With Brooks and
Martin del Campo, we used Vakil's criterion to show that
all Schubert problems involving lines have at least
alternating Galois group.
My talk will describe this background and sketch a
current project to systematically determine Galois groups
of all Schubert problems of moderate size on all small
classical flag manifolds, investigating at least several
million problems. This will use supercomputers employing
several overlapping methods, including combinatorial
criteria, symbolic computation, and numerical homotopy
continuation, and require the development of new
algorithms and software.

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.