When deforming a material body (e.g. heating, bending, stretching or otherwise stressing), physics
(basically the principle of least action) informs us that the final deformation rearranges itself so as to
minimise some energy or action functional. Modelling is often about trying to find the correct functional from other physical first principles.
The theory of nonlinear elasticity has been developed over a century to try and say important and generic things about the structure and regularity of
minimisers or minimal energy configurations for various classes of functionals. Of course minimisers of functionals satisfy differential equations (Euler-Lagrange) e There have been significant advances in solving these differential equations over the
years, particularly when they are nice (the technical term is elliptic). But to model more interesting
phenomena, like critical phase and transitions, supersonically moving objects and so forth, the
equations develop singularities and nonlinear terms can’t be ignored. In dealing with these ugly
equations (the technical term is nonlinear degenerate elliptic) it’s sometimes easier to go back to
the functionals themselves.

In this talk I will discuss some recent work with others about a special interesting case modelling nonlinear phenomena in elastic media by
minimising a scale invariant measure of the anisotropic properties of the material in the simplest 2D
case (with 3D applications). Surprisingly this is connected with a conjecture from J.C.C. Nitsche in
1962 (solved this year) concerning harmonic mappings and minimal surfaces. There is a wonderful
dichotomy in the solutions to these equations as one passes through a critical phase when one can
identify conformal invariants of the material (= geometric quantities derived from infinitesimal
information). This dichotomy shows, for instance, that materials can only be stretched so far before
breaking or tearing. There appear to be other applications in modelling cellular structures, foam
physics and tissues as well.

Contact person:
Vaughan Jones

Let K be a closed subgroup of a Lie group G. The contraction of G to K
is a
Lie group, usually more elementary in structure than G itself, that
approximates G to first order near K. The terminology is due to the
mathematical physicists, who examined the group of Galilean
transformations
as a contraction of the group of Lorentz transformations. My focus will
be
on a related but different class of examples, the prototype of which is
the
group of isometric motions of Euclidean space, viewed as a contraction
of
the group of isometric motions of hyperbolic space. It is natural to
expect some sort of limiting relation between representations of the
contraction and representations of G. But in the 1970s George Mackey
carried out a few calculations pointing to an interesting rigidity
phenomenon: as the contraction group is deformed back to G, the
representation theory remains in some sense unchanged. In particular
the
irreducible representations of the contraction group parametrize the
irreducible representations of G. I shall formulate a reasonably
precise
conjecture that was inspired by subsequent developments in C*-algebra
theory
and noncommutative geometry, and describe the evidence in support of it,
which is by now substantial. However a conceptual explanation for
Mackey’s
rigidity phenomenon remains elusive.

Contact person:
Dietmar Bisch

This expository lecture will lead to a new fixed point
theorem and discuss the relation with derivations. As an application,
we present the optimal answer to the “derivation problem” for group
algebras which originated in the 1960s.

Contact person:
Denis Osin

This was an open problem for more than 20 years in the communities of Numerical
Analysis and Scientific Computing. Only the most important Monge-
Amore equation of order 2 in R^2 had been discussed without proofs, e.g. by
Dean/Glowinski. My paper in SINUM 2008 proved for the first time for the
general case of fully nonlinear elliptic differential equations (and systems of order
2m, m \geq 1 on C^2m) domains in R^2, the necessary stability and convergence
results. This includes convergence for the numerical method for solving these
equations and quadrature approximations. In the mean time several papers
have appeared for the most important spcial case, the Monge Ampere equation.
Brenner, S.C. and her group Gudi, T. and Neilan, M. and Sung, L. Y.studied
C^0 penalty methods and proved convergence for Monge- Amore equations.
A simplified approach for second order equations on convex polygonal domains
in Rn is presented in this lecture. It allows indicating the essential ideas
for the general case. We suggest a nonstandard nonconforming C^1 finite element
method. The classical theory of discretization methods is applied to the
differential operator. The consistency error vanishes, but the stability has to be
proved in an unusual way. This is the hard core of the paper. Essential tools
are linearization, a compactness argument, the interplay between the weak and
strong form of the linearized operator and a new regularity result for solutions
of finite element equations. An essential basis for our proofs are Davydov’s C^1
finite elements on polygonal. The method applies to non divergent quasilinear
elliptic problems as well. Algorithms are formulated to calculate the nonlinear
system and to solve it by a combination of continuation and discrete Newton
methods. The latter converges locally quadratically, essentially independently
of the actual grid size by the mesh independence principle. EXplicit numerical
results are due to Davydov/Saeed.

Contact person:
Larry Schumaker

Contact person:
Vaughan Jones

In this talk, I will analyze this problem from a Ramsey-theoretic
perspective. In particular, the problem is related to generalizations of
Ellis’s Lemma and Hindman’s Theorem to the setting of nonassociative
binary systems. The amenability of $F$ is itself equivalent to the
existence of certain finite Ramsey numbers. I will also discuss the
growth rate of the F\olner function for $F$ (if it exists).

Contact person:
Ralph N. McKenzie

After giving an overview of finite frame theory, I will consider the question of modifying a general frame to generate a tight frame by rescaling its frame vectors. A frame that positively answers this question will be called scalable. I will give various characterizations of the set of scalable frames, and present some topological descriptions of this set. (This talk is based on joint work with G. Kutyniok, F. Philipp and E. Tuley).

Contact person:
Akram Aldroubi

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Contact person:
Edward B. Saff

Fourier series provide a way of writing almost any signal as a
superposition of pure tones, or musical notes. Unfortunately, this
representation is not local, and it does not reflect the way that music
is actually generated by instruments playing individual notes at different
times. We will discuss time-frequency representations, which are a type
of local Fourier representation of signals. While such representations
have limitations when it comes to music, they are powerful mathematical
tools that appear widely throughout mathematics (e.g., partial differential
equations and pseudodifferential operators), physics (e.g., quantum
mechanics), and engineering (e.g., time-varying filtering). We ask one
very basic question: are the notes in this representation linearly
independent? This seemingly trivial question leads to surprising
mathematical difficulties. This talk is intended to be introductory
and accessible to beginning graduate students.

Contact person:
Alex Powell

Contact person:
Akram Aldroubi

Contact person:
Vaughan Jones

Contact person:
Gieri Simonett

Contact person:
Ioana Suvaina

Contact person:
Denis Osin

Contact person:
Vaughan Jones

Contact person:
Doug Hardin