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Series: School of Mathematics Colloquium

The purpose of this talk is to describe a variational approach to the problemof A.D. Aleksandrov concerning existence and uniqueness of a closed convexhypersurface in Euclidean space $R^{n+1}, ~n \geq 2$ with prescribed integral Gauss curvature. It is shown that this problem in variational formulation is closely connected with theproblem of optimal transport on $S^n$ with a geometrically motivated cost function.

Series: School of Mathematics Colloquium

There are presently different approaches to definealgebraic geometry over the mysterious "field with one element".I will focus on two versions, one by Soule' and one by Borger,that appear to have a direct connection to NoncommutativeGeometry via the quantum statistical mechanics of Q-latticesand the theory of endomotives. I will also relate to endomotivesand Noncommutative Geometry the analytic geometry over F1,as defined by Manin in terms of the Habiro ring.

Series: School of Mathematics Colloquium

The lecture will outline how the method of characteristics can
be used in the context of solutions to hyperbolic conservation laws that
are merely continuous functions. The Hunter-Saxton equation will be used
as a vehicle for explaining the approach.

Series: School of Mathematics Colloquium

Light refreshments will be available in Room 236 at 10:30 am.

A single round soap bubble provides the least-area way to enclose a given volume. How does the solution change if space is given some density like r^2 or e^{-r^2} that weights both area and volume? There has been much recent progress by undergraduates. Such densities appear prominently in Perelman's paper proving the Poincare Conjecture. No prerequisites, undergraduates welcome.

Series: School of Mathematics Colloquium

Much research in modern, quantitative seismology is motivated -- on
the one hand -- by the need to understand subsurface structures and
processes on a wide range of length scales, and -- on the other hand
-- by the availability of ever growing volumes of high fidelity
digital data from modern seismograph networks or multicomponent
acquisition systems developed for hydro-carbon exploration, and access
to increasingly powerful computational facilities. We discuss
(elastic-wave) inverse scattering of reflection seismic data,
wave-equation tomography, and their interconnection using techniques
from microlocal analysis and applied harmonic analysis. We introduce a
multi-scale approach and present a framework of partial reconstruction
in connection with limited boundary acquisition geometry. The formation of caustics
leads to one of the complications which will be discussed. We illustrate various
aspects of this research program with examples from global seismology and mineral
physics coupled to thermo-chemical convection.

Series: School of Mathematics Colloquium

Let k be a p-adic field and K/k function field in one variable
over k. We discuss Hasse principle for existence of rational points
on homogeneous spaces under connected linear algebraic groups.
We illustrate how a positive answer to Hasse principle leads for instance to the result:
every quadratic form in nine variables over K has a nontrivial zero.

Series: School of Mathematics Colloquium

Refreshments at 4PM in Skiles 236

The Pentagram map is a projectively natural iteration on
plane polygons. Computer experiments show that the Pentagram map has
quasi-periodic behavior. I shall explain that the Pentagram map is a
completely integrable system whose continuous limit is the Boussinesq
equation, a well known integrable system of soliton type. As a
by-product, I shall demonstrate new configuration theorems of
classical projective geometry.

Series: School of Mathematics Colloquium

Understanding the folding of RNA sequences into three-dimensional structures is one of the fundamental challenges in molecular biology. In this talk, we focus on understanding how an RNA viral genome can fold into the dodecahedral cage known from experimental data. Using strings and trees as a combinatorial model of RNA folding, we give mathematical results which yield insight into RNA structure formation and suggest new directions in viral capsid assembly. We also illustrate how the interaction between discrete mathematics and molecular biology motivates new combinatorial theorems as well as advancing biomedical applications.

Series: School of Mathematics Colloquium

Real life networks are usually large and have a very complicated
structure. It is tempting therefore to simplify or reduce the associated
graph of interactions in a network while maintaining its basic structure
as well
as some characteristic(s) of the original graph. A key question is which
characteristic(s) to conserve while reducing a graph. Studies of
dynamical networks reveal that an important characteristic of a
network's structure is a spectrum of its adjacency matrix.
In this talk we present an approach which allows for the reduction of
a general
weighted graph in such a way that the spectrum of the graph's (weighted)
adjacency matrix is maintained up to some finite set that is known in
advance. (Here, the possible weights belong to the set of complex
rational functions, i.e. to a very general class of weights).
A graph can be isospectrally reduced to a graph on any subset of its
nodes, which could be an important property for various applications. It
is also possible to introduce a new equivalence relation in the set of
all networks. Namely, two networks are spectrally equivalent if each of
them can be isospectrally reduced onto one and the same (smaller) graph.
This result should also be useful for analysis of real networks.
As the first application of the isospectral graph reduction we
considered a problem of estimation of spectra of matrices. It happens
that our procedure allows for improvements of the estimates obtained by
all three classical methods given by Gershgorin, Brauer and Brualdi.
(Joint work with B.Webb)
A talk will be readily accessible to undergraduates familiar with
matrices and complex functions.

Series: School of Mathematics Colloquium

After a brief account of some of
the history of this classical subject,
we indicate how such models are derived.
Rigorous theory justifying the models
will be discussed and the conversation
will then turn to some applications.
These will be drawn from questions
arising in geophysics and coastal
engineering, as time permits.