Tue, 14 Oct 2014
15:45
L4

Exotic spheres and the topology of the symplectomorphism group

Georgios Rizell
(Cambridge)
Abstract

Using the fact that certain exotic spheres do not admit Lagrangian embeddings into $T^*{\mathcal S}^{n+1}$, as proven by Abouzaid and Ekholm-Smith, we produce non-trivial homotopy classes of the group of compactly supported symplectomorphisms of $T^*{\mathcal S}^n$. In particular, we show that the Hamiltonian isotopy class of the symplectic Dehn twist depends on the parametrisation used in the construction.  Related results are also obtained for $T^*({\mathcal S}^n \times {\mathcal S}^1)$.

Joint work with Jonny Evans.

 

Tue, 25 Nov 2014
15:45
L4

Complex Geometry and the Hele-Shaw flow

Julius Ross
(Cambridge)
Abstract

The goal of this talk is to discuss a link between the Homogeneous Monge Ampere Equation in complex geometry, and a certain flow in the plane motivated by some fluid mechanics.   After discussing and motivating the Dirichlet problem for this equation I will focus to what is probably the first non-trivial case that one can consider, and prove that it is possible to understand regularity of the solution in terms of what is known as the Hele-Shaw flow in the plane. As such we get, essentially explicit, examples of boundary data for which there is no regular solution, contrary to previous expectation.  All of this is joint work with David Witt Nystrom.

Tue, 28 Oct 2014

15:45 - 16:45
L4

Infinitely many monotone Lagrangian Tori in CP^2

Renato Vianna
(Cambridge)
Abstract
In previous work, we constructed an exotic monotone Lagrangian torus in $\mathbb{CP}^2$ (not Hamiltonian isotopic to the known Clifford and Chekanov tori) using techniques motivated by mirror symmetry. We named it $T(1,4,25)$ because, when following a degeneration of $\mathbb{CP}^2$ to the weighted projective space $\mathbb{CP}(1,4,25)$, it degenerates to the central fibre of the moment map for the standard torus action on $\mathbb{CP}(1,4,25)$. Related to each degeneration from $\mathbb{CP}^2$ to $\mathbb{CP}(a^2,b^2,c^2)$, for $(a,b,c)$ a Markov triple -- $a^2 + b^2 + c^2 = 3abc$ -- there is a monotone Lagrangian torus, which we call $T(a^2,b^2,c^2)$.  We employ techniques from symplectic field theory to prove that no two of them are Hamiltonian isotopic to each other.
Thu, 01 May 2014
14:00
L5

Adjoint sensitivity analysis in Thermoacoustics

Dr Matthew Juniper
(Cambridge)
Abstract

Thermoacoustic oscillations occur in combustion chambers when heat release oscillations lock into pressure oscillations. They were first observed in lamps in the 18th century, in rockets in the 1930s, and are now one of the most serious problems facing gas turbine manufacturers.

This theoretical and numerical study concerns an infinite-rate chemistry diffusion flame in a tube, which is a simple model for a flame in a combustion chamber. The problem is linearized around the non-oscillating state in order to derive the direct and adjoint equations governing the evolution of infinitesimal oscillations.

The direct equations are used to predict the frequency, growth rate, and mode shape of the most unstable thermoacoustic oscillations. The adjoint equations are then used to calculate how the frequency and growth rate change in response to (i) changes to the base state such as the flame shape or the composition of the fuel (ii) generic passive feedback mechanisms that could be added to the device. This information can be used to stabilize the system, which is verified by subsequent experiments.

This analysis reveals that, as expected from a simple model, the phase delay between velocity and heat-release fluctuations is the key parameter in determining the sensitivities. It also reveals that this thermo-acoustic system is exceedingly sensitive to changes in the base state. This analysis can be extended to more accurate models and is a promising new tool for the analysis and control of thermo-acoustic oscillations.

Thu, 06 Feb 2014

16:00 - 17:30
L2

Tractable interest rate and volatility models

Mike Tehranchi
(Cambridge)
Abstract

There are many financial models used in practice (CIR/Heston, Vasicek,

Stein-Stein, quadratic normal) whose popularity is due, in part, to their

analytically tractable asset pricing. In this talk we will show that it is

possible to generalise these models in various ways while maintaining

tractability. Conversely, we will also characterise the family of models

which admit this type of tractability, in the spirit of the classification

of polynomial term structure models.

Tue, 26 Nov 2013

15:45 - 16:45
L4

Contact property of symplectic magnetic flows on the two-sphere.

Gabriele Benedetti
(Cambridge)
Abstract

In this talk we aim to study periodic orbits on the energy levels of a symplectic magnetic flow on the two-sphere using methods from contact geometry. In particular we show that, if the energy is low enough, we either have two or infinitely many closed orbits. The second alternative holds if there exists a prime contractible periodic orbit. Finally we present some generalisations and work in progress for closed orientable surfaces of higher genus.

Subscribe to Cambridge