Forthcoming Seminars

Please note that the list below only shows forthcoming events, which may not include regular events that have not yet been entered for the forthcoming term. Please see the past events page for a list of all seminar series that the department has on offer.

Past events in this series
12 February 2018
14:15
JURGEN ANGST
Abstract

The study of the Geometry of random nodal domains has attracted a lot of attention in the recent past, in particular due to their connection with famous conjectures such as Yau's conjecture on the nodal volume of eigenfunctions of the Laplacian on compact manifolds, and Berry's conjecture on the relation between the geometry of the nodal sets associated to these eigenfunctions and the geometry of the nodal sets associated to toric random waves.

At first, the randomness involved in the definition of random nodal domains is often chosen of Gaussian nature. This allows in particular the use of explicit techniques, such as Kac--Rice formula, to derive the asymptotics of many observables of interest (nodal volume, number of connected components, Leray's measure etc.). In this talk, we will raise the question of the universality of these asymptotics, which consists in deciding if the asymptotic properties of random nodal domains do or do not depend on the particular nature of the randomness involved. Among other results, we will establish the local and global universality of the asymptotic volume associated to the set of real zeros of random trigonometric polynomials with high degree.

 

  • Stochastic Analysis Seminar
13 February 2018
12:00
to
13:15
Tim Palmer
Abstract

Hardy's axiomatic approach to quantum theory revealed that just one axiom
distinguishes quantum theory from classical probability theory: there should
be continuous reversible transformations between any pair of pure states. It
is the single word `continuous' that gives rise to quantum theory. This
raises the question: Does there exist a finite theory of quantum physics
(FTQP) which can replicate the tested predictions of quantum theory to
experimental accuracy? Here we show that an FTQP based on complex Hilbert
vectors with rational squared amplitudes and rational phase angles is
possible providing the metric of state space is based on p-adic rather than
Euclidean distance. A key number-theoretic result that accounts for the
Uncertainty Principle in this FTQP is the general incommensurateness between
rational $\phi$ and rational $\cos \phi$. As such, what is often referred to
as quantum `weirdness' is simply a manifestation of such number-theoretic
incommensurateness. By contrast, we mostly perceive the world as classical
because such incommensurateness plays no role in day-to-day physics, and
hence we can treat $\phi$ (and hence $\cos \phi$) as if it were a continuum
variable. As such, in this FTQP there are two incommensurate Schr\"{o}dinger
equations based on the rational differential calculus: one for rational
$\phi$ and one for rational $\cos \phi$. Each of these individually has a

simple probabilistic interpretation - it is their merger into one equation
on the complex continuum that has led to such problems over the years. Based
on this splitting of the Schr\"{o}dinger equation, the measurement problem
is trivially solved in terms of a nonlinear clustering of states on $I_U$.
Overall these results suggest we should consider the universe as a causal
deterministic system evolving on a finite fractal-like invariant set $I_U$
in state space, and that the laws of physics in space-time derive from the
geometry of $I_U$. It is claimed that such a  deterministic causal FTQP will
be much easier to synthesise with general relativity theory than is quantum
theory.

  • Quantum Field Theory Seminar
13 February 2018
14:00
Man-Chung Yue
Abstract

In this talk, we revisit the cubic regularization (CR) method for solving smooth non-convex optimization problems and study its local convergence behaviour. In their seminal paper, Nesterov and Polyak showed that the sequence of iterates of the CR method converges quadratically a local minimum under a non-degeneracy assumption, which implies that the local minimum is isolated. However, many optimization problems from applications such as phase retrieval and low-rank matrix recovery have non-isolated local minima. In the absence of the non-degeneracy assumption, the result was downgraded to the superlinear convergence of function values. In particular, they showed that the sequence of function values enjoys a superlinear convergence of order 4/3 (resp. 3/2) if the function is gradient dominated (resp. star-convex and globally non-degenerate). To remedy the situation, we propose a unified local error bound (EB) condition and show that the sequence of iterates of the CR method converges quadratically a local minimum under the EB condition. Furthermore, we prove that the EB condition holds if the function is gradient dominated or if it is star-convex and globally non-degenerate, thus improving the results of Nesterov and Polyak in three aspects: weaker assumption, faster rate and iterate instead of function value convergence. Finally, we apply our results to two concrete non-convex optimization problems that arise from phase retrieval and low-rank matrix recovery. For both problems, we prove that with overwhelming probability, the local EB condition is satisfied and the CR method converges quadratically to a global optimizer. We also present some numerical results on these two problems.

  • Numerical Analysis Group Internal Seminar
13 February 2018
14:30
Abstract

The object of this talk is a class of generalised Newtonian fluids with implicit constitutive law.
Both in the steady and the unsteady case, existence of weak solutions was proven by Bul\'\i{}\v{c}ek et al. (2009, 2012) and the main challenge is the small growth exponent $q$ and the implicit law.
I will discuss the application of a splitting and regularising strategy to show convergence of FEM approximations to weak solutions of the flow. 
In the steady case this allows to cover the full range of growth exponents and thus generalises existing work of Diening et al. (2013). If time permits, I will also address the unsteady case.
This is joint work with Endre Suli.

  • Numerical Analysis Group Internal Seminar

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