Past Colloquia

12 May 2017
16:00
Abstract

Hinke Osinga, University of Auckland
joint work with: Bernd Krauskopf and Stefanie Hittmeyer (University of Auckland)

Dynamical systems of Lorenz type are similar to the famous Lorenz system of just three ordinary differential equations in a well-defined geometric sense. The behaviour of the Lorenz system is organised by a chaotic attractor, known as the butterfly attractor. Under certain conditions, the dynamics is such that a dimension reduction can be applied, which relates the behaviour to that of a one-dimensional non-invertible map. A lot of research has focussed on understanding the dynamics of this one-dimensional map. The study of what this means for the full three-dimensional system has only recently become possible through the use of advanced numerical methods based on the continuation of two-point boundary value problems. Did you know that the chaotic dynamics is organised by a space-filling pancake? We show how similar techniques can help to understand the dynamics of higher-dimensional Lorenz-type systems. Using a similar dimension-reduction technique, a two-dimensional non-invertible map describes the behaviour of five or more ordinary differential equations. Here, a new type of chaotic dynamics is possible, called wild chaos. 


 

 

28 April 2017
16:00
Catharina Stroppel
Abstract

Permutations of finitely many elements are often drawn as permutation diagrams. We take this point of view as a motivation to construct and describe more complicated algebras arising for instance from differential operators, from operators acting on (co)homologies, from invariant theory, or from Hecke algebras. The surprising fact is that these diagrams are elementary and simple to describe, but at the same time describe relations between cobordisms as well as categories of represenetations of p-adic groups. The goal of the talk is to give some glimpses of these phenomena and indicate which role categorification plays here.
 

3 March 2017
16:00
Ana Caraiani
Abstract

The law of quadratic reciprocity and the celebrated connection between modular forms and elliptic curves over Q are both examples of reciprocity laws. Constructing new reciprocity laws is one of the goals of the Langlands program, which is meant to connect number theory with harmonic analysis and representation theory.

In this talk, I will survey some recent progress in establishing new reciprocity laws, relying on the Galois representations attached to torsion classes which occur in the cohomology of arithmetic hyperbolic 3-manifolds. I will outline joint work in progress on better understanding these Galois representations, proving modularity lifting theorems in new settings, and applying this to elliptic curves over imaginary quadratic fields.

10 February 2017
16:00
Abstract

Self-organization is observed in systems driven by the “social engagement” of agents with their local neighbors. Prototypical models are found in opinion dynamics, flocking, self-organization of biological organisms, and rendezvous in mobile networks.

We discuss the emergent behavior of such systems. Two natural questions arise in this context. The underlying issue of the first question is how different rules of engagement influence the formation of clusters, and in particular, the emergence of 'consensus'. Different paradigms of emergence yield different patterns, depending on the propagation of connectivity of the underlying graphs of communication.  The second question involves different descriptions of self-organized dynamics when the number of agents tends to infinity. It lends itself to “social hydrodynamics”, driven by the corresponding tendency to move towards the local means. 

We discuss the global regularity of social hydrodynamics for sub-critical initial configurations.

27 January 2017
16:00
Paul Klemperer
Abstract

Mathematical methods are increasingly being used to design auctions. Paul Klemperer will talk about some of his own experience which includes designing the U.K.'s mobile phone licence auction that raised £22.5 billion, and a new auction that helped the Bank of England in the financial crisis. (The then-Governor, Mervyn King, described it as "a marvellous application of theoretical economics to a practical problem of vital importance".) He will also discuss further development of the latter auction using convex and "tropical" geometric methods.

2 December 2016
16:00
Steve Simon
Abstract

In two dimensional topological phases of matter, processes depend on gross topology rather than detailed geometry. Thinking in 2+1 dimensions, the space-time histories of particles can be interpreted as knots or links, and the amplitude for certain processes becomes a topological invariant of that link. While sounding rather exotic, we believe that such phases of matter not only exist, but have actually been observed (or could be soon observed) in experiments. These phases of matter could provide a uniquely practical route to building a quantum computer. Experimental systems of relevance include Fractional Quantum Hall Effects, Exotic superconductors such as Strontium Ruthenate, Superfluid Helium, Semiconductor-Superconductor-Spin-Orbit systems including Quantum Wires. The physics of these systems, and how they might be used for quantum computation will be discussed.

20 June 2016
16:00
Jacob Lurie (Hardy Lecture Tour)
Abstract

Let X be a complex algebraic variety containing a point x. One of the central ideas of deformation theory is that the local structure of X near the point x can be encoded by a differential graded Lie algebra. In this talk, Jacob Lurie will explain this idea and discuss some generalizations to more exotic contexts.

17 June 2016
16:00
David Vogan
Abstract

One of the big ideas in linear algebra is {\em eigenvalues}. Most matrices become in some basis {\em diagonal} matrices; so a lot of information about the matrix (which is specified by $n^2$ matrix entries) is encoded by by just $n$ eigenvalues. The fact that lots of different matrices can have the same eigenvalues reflects the fact that matrix multiplication is not commutative.

I'll look at how to make these vague statements (``lots of different matrices...") more precise; how to extend them from matrices to abstract symmetry groups; and how to relate abstract symmetry groups to matrices.

3 June 2016
16:00
Bernd Sturmfels
Abstract

Eigenvectors of square matrices are central to linear algebra. Eigenvectors of tensors are a natural generalization. The spectral theory of tensors was pioneered by Lim and Qi around 2005. It has numerous applications, and ties in closely with optimization and dynamical systems.  We present an introduction that emphasizes algebraic and geometric aspects

12 February 2016
16:00
Isabelle Gallagher
Abstract

The question of deriving Fluid Mechanics equations from deterministic
systems of interacting particles obeying Newton's laws, in the limit
when the number of particles goes to infinity, is a longstanding open
problem suggested by Hilbert in his 6th problem. In this talk we shall
present a few attempts in this program, by explaining how to derive some
linear models such as the Heat, acoustic and Stokes-Fourier equations.
This corresponds to joint works with Thierry Bodineau and Laure Saint
Raymond.

Pages