Colloquiums and Public Lectures

2019

Date Information
​13 November 2019 ​​​​​Title: Fractional Cahn-Hilliard Equation(s): Analysis, Properties and Approximation

Professor Mark Ainsworth
Time: 3.30pm –4.30pm​

Abstract​
The classical Cahn-Hilliard equation [1] is a nonlinear, fourth order in space, parabolic partial differential equation which is often used as a diffuse interface model for the phase separation of a binary alloy. Despite the widespread adoption of the model, there are good reasons for preferring models in which fractional spatial derivatives appear [2,3]. We consider two such Fractional Cahn-Hilliard equations (FCHE). The first [4] corresponds to considering a gradient flow of the free energy functional in a negative order Sobolev space H^−α, α∈[0,1] where the choice α=1 corresponds to the classical Cahn-Hilliard equation whilst the choice α=0 recovers the Allen-Cahn equation. It is shown that the equation preserves mass for all positive values of fractional order and that it indeed reduces the free energy. The well-posedness of the problem is established in the sense that the H1-norm of the solution remains uniformly bounded. We then turn to the delicate question of the L^∞-boundedness of the solution and establish an L^∞ bound for the FCHE in the case where the non-linearity is a quartic polynomial. As a consequence of the estimates, we are able to show that the Fourier-Galerkin method delivers a spectral rate of convergence for the FCHE in the case of a semi-discrete approximation scheme. Finally, we present results obtained using computational simulation of the FCHE for a variety of choices of fractional order α. We then consider an alternative FCHE [3,5] in which the free energy functional involves a fractional order derivative (joint work with Zhiping Mao) .​

Host: Division of Mathematical Sciences, School of Physical and Mathematical Sciences​​
​29 August 20​19 ​​​​Title: ​​​Nonlinear Fokker-Planck equations and distribution dependent SDE​

Professor Michael Röckner
Time: 2.00 pm – 3.00 pm

Abstract​
It is a classical problem to present a solution of a PDE as the density of the time marginal distributions of a stochastic process. If the PDE is a linear Fokker-Planck equation, then by classical stochastic analysis this is known to be true under very general conditions. For nonlinear Fokker-Planck equations the situation is much more difficult and only known to be true under very restrictive assumptions on the regularity of the (nonlinear) dependence of the coefficients in the Fokker-Planck equations on the solutions. In this talk a new general concept is presented, how to find the desired stochastic process (similarly as in the linear case) through solving a corresponding stochastic differential equation (SDE), whose coefficients, however, depend on the marginal distributions of its solution (DDSDE). The point is that this new general concept does not require strong regularity assumptions on the coefficients (as e.g. fulfilled for McKean-Vlasov type equations) and thus does not rule out a lot of other nonlinear Forker-Planck equations of interest in Physics. As an example it will be shown that it can be applied to the case, where the nonlinear Fokker-Planck equation is a generalized porous media equation on d-dimensional Euclidean space (with d arbitrary), perturbed by a transport term. So its solution is the density of the time marginal distributions of a (tractable) stochastic process solving a corresponding DDSDE. Apart from its conceptual interest this result could lead to new numerical approximations of solutions to nonlinear Fokker-Planck equations through numerically solving the corresponding DDSDE. 

In the first part of the talk, we shall recall the general connection between stochastic differential equations and (both linear and nonlinear) Fokker-Planck equations.

Host: Division of Mathematical Sciences, School of Physical and Mathematical Sciences​
​15 August 2019 ​​Title: Statistical Challenges in Time Domain Astronomy

Professor Jogesh Babu
Time: 4.00pm to 5.00pm

Abstract
Objects in the sky exhibit a wide range of variability in brightness at different wavebands leading to time series data. Majority of data from gravitational wave detectors, pulsar timing array, exoplanet surveys, multi-messenger astronomy, and forthcoming data from Large Synoptic Survey Telescope, is in the form of irregularly spaced time series. Scientific interpretation of astronomical time series is complicated by the instrumental effects, variety of variable astronomical phenomena, the non-Gaussianity of the uninteresting noise, and the often irregular cadence of observations.  Brief review of statistical issues in time-domain astronomy will be presented.​

Host:
​Associate Professor Pan Guangming​
Division of Mathematical Sciences, School of Physical and Mathematical Sciences​
​06 August 2019 ​Title: Fractal: A high-performance, scalable, proof-of-stake Blockchain

Professor Jonathan Katz
Time: 3.30pm to 4.30pm

Abstract
I will present Fractal, a high-performance, scalable, provably secure blockchain. The Fractal system incorporates two components. The first is a novel proof-of-stake protocol that is provably secure and addresses notorious challenges such as nothing-at-stake and grinding attacks. The second is a new method for achieving scalability by decoupling data distribution and data ordering to improve network throughput. Via rigorous analysis and real-world benchmarking, we show that Fractal achieves excellent block-propagation time as well as near-optimal throughput (up to 80% of the network-physical-limit). Fractal's cross-layer design can scale to 10,000+ nodes across the globe and, under modest network assumptions, sustain a throughput of 3,000+ tps.

Host: Division of Mathematical Sciences, School of Physical and Mathematical Sciences​
16 January 2019 ​Title: Characterization of Intersecting Families of Maximum Size in PSL(2,q)

Professor Qing Xiang
Time: 1.30pm to 2.30pm

Abstract

Host: Professor Bernhard Schmidt
Division of Mathematical Sciences, School of Physical and Mathematical Sciences
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