Past

We shall discuss the problem of the 'trend to equilibrium' for a 

degenerate kinetic linear Fokker-Planck equation. The linear equation is 

assumed to be degenerate on a subregion of non-zero Lebesgue measure in the 

physical space (i.e., the equation is just a transport equation with a 

Hamiltonian structure in the subregion). We shall give necessary and 

sufficient geometric condition on the region of degeneracy which guarantees 

the exponential decay of the semigroup generated by the degenerate kinetic 

equation towards a global Maxwellian equilibrium in a weighted Hilbert 

space. The approach is strongly influenced by C. Villani's strategy of 

'Hypocoercivity' from Kinetic theory and the 'Bardos-Lebeau-Rauch' 

geometric condition from Control theory. This is a joint work with Frederic 

Herau and Clement Mouhot.

“Narrative and Proof”, is an interdisciplinary discussion where one of the UK's leading scientists, Marcus du Sautoy, will argue that mathematical proofs are not just number-based, but also rely on narrative. He will be joined by author Ben Okri, mathematician Roger Penrose, and literature expert Laura Marcus, to consider how narrative shapes the sciences as well as the arts.

The discussion will be chaired by Elleke Boehmer, Professor of World Literature in English, University of Oxford, and will be followed by audience questions and a drinks reception.

The event will take place from 5 to 6:30 pm on Tuesday 20 January 2015 at the Mathematical Institute, Radcliffe Observatory Quarter, Woodstock Road, Oxford. The event is free and open to all, but registration is recommended. 

Please click here to register.

This event is co-hosted by the Mathematical Institute and The Oxford Research Centre in the Humanities (TORCH), and celebrates the launch of TORCH’s Annual Headline Series 2014-15, Humanities and Science.
 

There is a well established body of research on quadratic optimization problems based on reformulations of the original problem as a conic program over the cone of completely positive matrices, or its conic dual, the cone of copositive matrices. As a result of this reformulation approach, novel solution schemes for quadratic polynomial optimization problems have been designed by drawing on conic programming tools, and the extensively studied cones of completely positive and of copositive matrices. In particular, this approach has been applied to address key combinatorial optimization problems. Along this line of research, we consider quadratically constrained quadratic programs and provide sufficient and necessary conditions for
this type of problems to be reformulated as a conic program over the cone of completely positive matrices. Thus, recent related results for quadratic problems can be further strengthened. Moreover, these results can be generalized to optimization problems involving higher order polynomias.

A key question to develop our understanding of turbulence in shear flows is: what is the smallest perturbation to the laminar flow that causes a transition to turbulence, and how does this change with the Reynolds number, R?  Finding this so-called ``minimal seed'' is as yet unachievable in direct numerical simulations of the Navier-Stokes equations. We search for the minimal seed in a low-dimensional model analogue to the full Navier-Stokes in plane sinusoidal flow, developed by Waleffe (1997). A previous such calculation found the minimal seed as the least distance (energy norm) from the origin (laminar flow) to the basin of attraction of another fixed point (turbulent attractor).  However, using a non-linear optimization technique, we found an internal boundary of the basin of attraction of the origin that separates flows which directly relaminarize from flows which undergo transient turbulence. It is this boundary which contains the minimal seed, and we find it to be smaller than the previously calculated minimal seed. We present results over a range of Reynolds numbers up to 2000 and find an R^{-1} scaling law fits reasonably well. We propose a new scaling law which asymptotes to R^{-1} for large R but, using some additional information, matches the minimal seed scaling better at low R.

Axions are ubiquitous in string theory compactifications. They are
pseudo goldstone bosons and can be extremely light, contributing to
the dark sector energy density in the present-day universe. The
mass defines a characteristic length scale. For 1e-33 eV<m< 1e-20
eV this length scale is cosmological and axions display novel
effects in observables. The magnitude of these effects is set by
the axion relic density. The axion relic density and initial
perturbations are established in the early universe before, during,
or after inflation (or indeed independently from it). Constraints
on these phenomena can probe physics at or beyond the GUT scale. I
will present multiple probes as constraints of axions: the Planck
temperature power spectrum, the WiggleZ galaxy redshift survey,
Hubble ultra deep field, the epoch of reionisation as measured by
cmb polarisation, cmb b-modes and primordial gravitational waves,
and the density profiles of dwarf spheroidal galaxies. Together

these probe the entire 13 orders of magnitude in axion mass where
axions are distinct from CDM in cosmology, and make non-trivial
statements about inflation and axions in the string landscape. The
observations hint that axions in the range 1e-22 eV<m<1e-20 eV may
play an interesting role in structure formation, and evidence for
this could be found in the future surveys AdvACT (2015), JWST, and
Euclid (>2020). If inflationary B-modes are observed, a wide range
of axion models including the anthropic window QCD axion are
excluded unless the theory of inflation is modified. I will also
comment briefly on direct detection of QCD axions.

In this talk I will describe Arthur's classification of automorphic representations of symplectic and orthogonal groups using automorphic representations of $\mathrm{GL}_N$.

It is well known that there are topological obstructions to a manifold $M$ admitting a Riemannian metric of everywhere positive scalar curvature (psc): if $M$ is Spin and admits a psc metric, the Lichnerowicz–Weitzenböck formula implies that the Dirac operator of $M$ is invertible, so the vanishing of the $\hat{A}$ genus is a necessary topological condition for such a manifold to admit a psc metric. If $M$ is simply-connected as well as Spin, then deep work of Gromov--Lawson, Schoen--Yau, and Stolz implies that the vanishing of (a small refinement of) the $\hat{A}$ genus is a sufficient condition for admitting a psc metric. For non-simply-connected manifolds, sufficient conditions for a manifold to admit a psc metric are not yet understood, and are a topic of much current research.

I will discuss a related but somewhat different problem: if $M$ does admit a psc metric, what is the topology of the space $\mathcal{R}^+(M)$ of all psc metrics on it? Recent work of V. Chernysh and M. Walsh shows that this problem is unchanged when modifying $M$ by certain surgeries, and I will explain how this can be used along with work of Galatius and myself to show that the algebraic topology of $\mathcal{R}^+(M)$ for $M$  of dimension at least 6 is "as complicated as can possibly be detected by index-theory". This is joint work with Boris Botvinnik and Johannes Ebert.

Motivated by stochastic models for order books in stock exchanges we consider stochastic partial differential equations with a free boundary condition. Such equations can be considered generalizations of the classic (deterministic) Stefan problem of heat condition in a two-phase medium. 

Extending results by Kim, Zheng & Sowers we allow for non-linear boundary interaction, general Robin-type boundary conditions and fairly general drift and diffusion coefficients. Existence of maximal local and global solutions is established by transforming the equation to a fixed-boundary problem and solving a stochastic evolution equation in suitable interpolation spaces. Based on joint work with Marvin Mueller.

'We study an optimal switching problem with a state constraint: the controller is only allowed to choose strategies that keep the controlled diffusion in a closed domain. We prove that the value function associated to the weak formulation of this problem is the limit of the value function associated to an unconstrained switching problem with penalized coefficients, as the penalization parameter goes to infinity. This convergence allows to set a dynamic programming principle for the constrained switching problem. We then prove that the value function is a constrained viscosity solution to a system of variational inequalities (SVI for short). We finally prove that the value function is the maximal solution to this SVI. All our results are obtained without any regularity assumption on the constraint domain.’

The 2015 Oxford Brain Mechanics Workshop 19 and 20 January, 2015 in St Hugh’s College, Oxford

Everybody is welcome to attend but (free) registration is required.

The event will include speakers from both CMU and Oxford working on Brain Mechanics and Trauma, as well as some chosen international members from the IBMTL* (www.brainmech.ox.ac.uk).

As well as focusing on various aspects of brain mechanics research, the 2015 Oxford Brain Mechanics Workshop will include the UK launch of the Carnegie Mellon University (CMU) – University of Oxford ‘Brain Alliance’. We are delighted that Dr Subra Suresh, President of CMU will launch the workshop, introduced by Oxford Vice-Chancellor Prof. Andrew Hamilton.

The aim of the workshop is to foster new collaborative partnerships and facilitate the dissemination of ideas from researchers in different fields related to the study of brain mechanics, including pathology, injury and healing. The IBMTL is delighted to be a global partner in CMU’s ‘BrainHub’ initiative and further extend the truly interdisciplinary, collaborative network of IBMTL and its associated researchers in Medical Sciences, Neuroscience, Biology, Engineering, Physics and Mathematics.

• Speakers:
• Professor Andrew Hamilton, University of Oxford, UK
• Dr Subra Suresh, Carnegie Mellon University, USA
• Mr Nick de Pennington, University of Oxford, UK
• Professor Michel Destrade, National University of Ireland, Galway
• Dr Kristian Franze, University of Cambridge, UK
• Professor Alain Goriely, University of Oxford, UK
• Professor Gerhard Holzapfel, Graz University of Technology, Austria
• Professor Jimmy Hsia, University of Illinois
• Mr Jayaratnam Jayamohan, University of Oxford, UK
• Professor Antoine Jerusalem, University of Oxford, UK
• Professor Ellen Kuhl, Stanford University, USA
• Professor Philip R LeDuc, Carnegie Mellon University, USA
• Professor Riyi Shi, Purdue University, USA

The workshop is generously supported by the Oxford Centre for Collaborative Applied Mathematics (OCCAM), which is led by IBMTL Co-Director, Prof Alain Goriely.

Those mentioned in the title are integral functionals of the Calculus of Variations

characterized by the fact of having an integrand switching between two different

kinds of degeneracies, dictated by a modulating coefficient. They have introduced

by Zhikov in the context of Homogenization and to give new examples of the related

Lavrentiev phenomenon. In this talk I will present some recent results aimed at

drawing a complete regularity theory for minima.