Seminars 2014

Title: On some invariants of hyperbolic 3-manifolds
Speaker: Professor Sergei Duzhin
Date:15, 18 December 2014
Time:2.00pm - 3.00pm
Venue:15 Dec - SPMS MAS Executive Classroom 2 (SPMS-MAS-03-07) 
18 Dec - SPMS MAS Executive Classroom 1 (SPMS-MAS-03-06)
 Abstract: I will give two review lectures about some important invariants of 3-dimensional hyperbolic manifolds.  In the first lecture, I will give a general introduction into the subject and then speak about the hyperbolic volume and the procedures to compute it.  In the second lecture, I will explain the definition of the number field (a finite extension of the rationals) associated with a hyperbolic manifold and give some examples of its computation.  I hope that the exposition will be accessible to people who had never heard of (the incredible beauty of) hyperbolic manifolds.


Title: Univalent Formalization of Mathematics with Proof Assistant Coq
Speaker: Professor Vladimir Voevodsky
Date:8, 15, 22 August 2014 
Time:10.30 am – 11.30am
Venue:SPMS MAS Executive Classroom 2 (SPMS-MAS-03-07) 
 Abstract: Professor Voevodsky will give 3 seminars on "Univalent Formalization of Mathematics with Proof Assistant Coq". The main reading of the topic is the paper at . It would also be useful to download and install on your computers Proof Assistant Coq from here: .


Title: Breaking the Diffraction Limit via Inverse Scattering
Speaker: Associate Professor Peijun Li
Date:9 May 2014
Time:3.30pm – 4.30pm
Venue:MAS Executive Classroom 1, MAS-03-06, School of Physical and Mathematical Sciences
 Abstract: Scattering problems are concerned with how an inhomogeneous medium scatters an incident field. The direct scattering problem is to determine the scattered field from the incident field; the inverse scattering problem is to determine the nature of the inhomogeneity from the measured scattered field. These problems have played a fundamental role in diverse scientific areas such as radar and sonar, geophysical exploration, medical imaging, near-field and nano- optics. According to the Rayleigh criterion, there is a resolution limit to the sharpness of details that can be observed by conventional far-field imaging, one half the wavelength, referred to as the diffraction limit. It presents challenging mathematical and computational questions to solve the underlying inverse scattering problems with increased resolution due to the nonlinearity, ill-posedness, and large scale computation.
In this talk, our recent progress on inverse surface scattering problems will be discussed. I will present new approachs to achieve subwavelength resolution for the inverse problems of the Helmholtz and Maxwell equations. Based on transformed field expansion, the methods convert the problems with complex scattering surfaces into successive sequences of two-point boundary value problems, where explicit reconstruction formulas are made possible. A spectral cut-off regularization is adopted to suppress the exponential growth of the noise in the evanescent wave components, which carry high spatial frequency of the surfaces and contribute to the super-resolution. The methods require only a single incident field and are realized by using the fast Fourier transform. The error estimates of the solutions for the model equations will be addressed. I will also highlight ongoing projects in rough surface imaging, random medium imaging, and near-field and nano- optics modeling.


Title: Fixed-point Algorithms for Emission Computed Tomography Reconstruction
Speaker: Professor Yuesheng Xu
Date:2 May 2014
Time:3.30pm – 4.30pm
Venue:MAS Executive Classroom 1, MAS-03-06, School of Physical and Mathematical Sciences
 Abstract: Emission computed tomography (ECT) is a noninvasive molecular imaging method that finds wide clinical applications. It provides estimates of the radiotracer distribution inside a patient’s body through tomographic reconstruction from the detected emission events. In this talk, we propose a fixed-point algorithm - preconditioned alternating projection algorithm (PAPA) for solving the maximum a posteriori (MAP) ECT reconstruction problem. Specifically, we formulate the reconstruction problem as a constrained convex optimization problem with the total variation (TV) regularization via the Bayes law. We then characterize the solution of the optimization problem and show that it satisfies a system of fixed-point equations defined in terms of two proximity operators of the convex functions that define the TV-norm and the constraint involved in the problem. This characterization naturally leads to an alternating projection algorithm (APA) for finding the solution. For efficient numerical computation, we introduce to the APA a preconditioning matrix (the EM-preconditioner) for the large-scale and dense system matrix. We prove theoretically convergence of the PAPA. In numerical experiments, performance of our algorithms, with an appropriately selected preconditioning matrix, is compared with performance of the conventional expectation-maximization (EM) algorithm with TV regularization (EM-TV) and that of the recently developed nested EM-TV algorithm for ECT reconstruction. Based on the numerical experiments performed in our work, we observe that the APA with the EM-preconditioner outperforms significantly the conventional EM-TV in all aspects including the convergence speed and the reconstruction quality. It also outperforms the nested EM-TV in the convergence speed while providing comparable reconstruction quality.


Title: The Dirichlet problem for the prescribed Ricci curvature equation
Speaker: Dr Artem Pulemotov
Date:2 May 2014
Time:2.00pm – 3.00pm
Venue:MAS Executive Classroom 1, MAS-03-06, School of Physical and Mathematical Sciences
 Abstract: We will discuss the following question: is it possible to recover the shape of a Riemannian manifold M from the Ricci curvature of M? To answer this question, one must analyze a second-order geometric PDE. In the first part of the talk, we will review the relevant background and the history of the subject. We will also state several classical theorems. After that, our focus will be on new results concerning the case where M has nonempty boundary ∂M and the shape of ∂M is prescribed.


Title: Spectral-Collocation Methods for Volterra Type Integral Equations
Speaker: Professor Yanping Chen
Date:15 April 2014
Time:3.00pm – 4.00pm
Venue:MAS Executive Classroom 2, MAS-03-07, School of Physical and Mathematical Sciences
 Abstract: In this work, spectral-collocation methods are developed for Volterra type integral equations of the second kind with a weakly singular kernel. We use some function transformations and variable transformations to change the equation into a new Volterra integral equation defined on the standard interval [−1,1], so that the solution of the new equation possesses better regularity and the orthogonal polynomial theory can be applied conveniently. In order to obtain high-order accuracy for the approximation, the integral term in the resulting equation is approximated by using Gauss quadrature rules. The convergence analysis of this novel method is based on the Lebesgue constants corresponding to the Lagrange interpolation polynomials, polynomial approximation theory for orthogonal polynomials and operator theory. The spectral rate of convergence for the proposed method is established in the L^{\infty}-norm and the weighted L^2-norm. Numerical results are presented to demonstrate the effectiveness of the proposed method. We extended our work to the Volterra integro-differential equations (VIDEs), including the Legendre spectral-collocation method for high order VIDEs and VIDEs with delay.


Title: A Nested Stochastic Simulation Algorithm for An Insulin-Signaling Network
Speaker: Dr Can Huang
Date:15 April 2014
Time:4.00pm – 5.00pm
Venue:MAS Executive Classroom 2, MAS-03-07, School of Physical and Mathematical Sciences
 Abstract: I will first present a new mathematical model for insulin signaling network, which can predict some important features of the network observed from in vitro experiments.
Secondly, I will add Poisson noise to the model, apply an efficient stochastic algorithm to simulate it, and prove the strong convergence rate of our algorithm.


Title: Specification and Testing of Multiplicative Time Varying GARCH Models with Applications
Speaker: Professor Timo Terasvirta 
Date:14 April 2014
Time:4.00pm – 5.00pm
Venue:MAS Executive Classroom 2, MAS-03-07, School of Physical and Mathematical Sciences 
 Abstract: This paper develops a specification technique for building multiplicative time-varying GARCH models of Amado and Teräsvirta (2008, 2013). In the model the variance is decomposed into an unconditional and a conditional component such that the unconditional deterministic component is allowed to evolve smoothly over time. The deterministic component is defined as a linear combination of logistic transition functions with time as the transition variable. The appropriate number of transition functions is determined by applying a sequence of specification tests. For that purpose, a coherent modelling strategy based on statistical inference is presented. It is heavily dependent on Lagrange multiplier type misspecification tests. The tests are easily implemented as they are entirely based on auxiliary regressions. Finite-sample properties of the strategy and tests are studied by simulations. The modelling strategy is illustrated with two examples, an application to daily exchange rate returns and another one to daily coffee futures returns.


Title: Computational Methods for Crystalline Defects: Construction, Analysis, and Benchmarking
Speaker: Dr Alexander Shapeev 
Date:7 April 2014
Time:10.30am – 11.30am 
Venue:MAS Executive Classroom 2, MAS-03-07, School of Physical and Mathematical Sciences
Defects, defined as irregularities in the periodic arrangement of atoms, determine many important properties of crystalline materials, such as plasticity or failure. Computing defects is often challenging, as the spatial and temporal scales accessible for direct molecular simulations are limited.

My talk will be devoted to efficient methods for computing crystalline defects. I will focus on atomistic-to-continuum (AtC) coupling, a popular approach utilizing atomistic resolution near the defect core while using the continuum model to resolve the elastic far-field. In my talk I will
    • give a brief introduction to crystalline defects and AtC coupling,
    • report one of the recent developments in construction of a consistent energy-based AtC coupling method, and
    • present a theory of how to optimize and compare the performance of existing methods.


Title: On CLT type results for spectral linear statistics of large random matrices 
Speaker: Professor Alexander Soshnikov 
Date:2 April 2014
Time:2.00pm – 3.00pm 
Venue:MAS Executive Classroom 1, MAS-03-06, School of Physical and Mathematical Sciences
 Abstract: I will talk about some old and new results about Gaussian fluctuation for linear statistics of eigenvalues of large random matrices. The talk will be based on my recent works with Lingyun Li, Sean O'Rourke, and Matthew Reed. If time permits, I will shortly discuss work in progress with Sean O'Rourke, David Renfrew, and Van Vu on spectral properties of products of elliptic random matrices.


Title: The KAM theorem
Speaker: Professor John H. Hubbard 
Date:1 April 2014 
Time:12.30pm – 1.30pm 
Venue:MAS Executive Classroom 2, MAS-03-07, School of Physical and Mathematical Sciences
 Abstract: If a mechanical system with n degrees of freedom (so the phase space has dimension 2n ), and if it admits n conservation 
laws, then by a theorem of Liouville, under fairly general conditions the motions are linear flow on an n-dimensional torus. 

Suppose that the system is perturbed so that the conservation laws are destroyed. Certainly one would expect generic 
motions to go wherever any remaining laws will allow. The KAM theorem (Kolmogorov-Arnold-Moser) asserts that this is not 
the case: for sufficiently small perturbations, the perturbed system admits invariant tori, in fact all those on which the linear 
motions is "sufficiently irrational irrational". Spelling out exactly what this means is already a challenge. 

In an introductory lecture I will give a precise statement, and illustrate what it means for some mechanical systems, and also 
for some numerical methods.


Title: Parodi Stability Relation in the Liquid Crystal Flow
Speaker: Dr Xiang Xu 
Date:9 March 2014
Time:2.00pm – 3.00pm  
Venue:MAS Executive Classroom 2, MAS‐03‐07, School of Physical and Mathematical Sciences 
The hydrodynamic theory of nematic liquid crystals was developed around 1960's. There are many physical parameters in this system, on which certain constraints are imposed. One of these constraints is a physical relation called Parodi's relation which remained poorly understood to the liquid crystal community.
In my talk I will discuss the role of Parodi's relation in the liquid crystal flow.


Speaker: Professor Michael Wolf
Date:7 February 2014
Time:2.30pm - 3.30pm
Venue:MAS Executive Classroom 2, MAS‐03‐07, School of Physical and Mathematical Sciences 
 Abstract: Covariance matrix estimation and principal component analysis (PCA) are two cornerstones of multivariate analysis. Classic textbook solutions perform poorly when the dimension of the data is of a magnitude similar to the sample size, or even larger. In such settings, there is a common remedy for both statistical problems: nonlinear shrinkage ofthe eigenvalues of the sample covariance matrix. The optimal nonlinear shrinkage formula depends on unknown population quantities and is thus not available. It is, however, possible to consistently estimate an oracle nonlinear shrinkage, which is motivated on asymptotic grounds. A key tool to this end is consistent estimation of the set of eigenvalues of the population covariance matrix (also known as the spectrum), an interesting and challenging problem in its own right. Extensive Monte Carlo simulations demonstrate that our methods have desirable finite-sample properties and outperform previous proposals.