## Seminars 2017

Title: | Assigning a Small Agreeable Set of Indivisible Items to Multiple Agents |

Speaker: | Warut Suksompong |

Date: | 13 December 2017 |

Time: | 3.00pm - 4.00pm |

Venue: | Colloquium Seminar Room 2 SPMS-MAS-05-35 |

Host: | Dr Bei Xiaohui |

Abstract: | We study the problem of assigning a small subset of indivisible items to a group of agents so that the subset is agreeable to all agents, meaning that all agents value the subset as least as much as its complement. For an arbitrary number of agents and items, we derive a tight worst-case bound on the number of items that may need to be included in such a subset. We then present polynomial-time algorithms that find an agreeable subset whose size matches the worst-case bound when there are two or three agents. Furthermore, we investigate the problem of efficiently computing an agreeable subset whose size approximates the size of the smallest agreeable subset for any given instance. We consider three well-known models for representing the preferences of the agents---ordinal preferences on single items, the value oracle model, and additive utilities---and establish tight bounds on the approximation ratio that can be obtained by algorithms running in polynomial time in each of these models. |

Title: | Interface problems and an adaptive time-step scheme |

Speaker: | Professor Shuwang Li |

Date: | 7 December 2017 |

Time: | 10.30am - 11.30am |

Venue: | MAS Executive Classroom 2, MAS-03-07 |

Host: | Assistant Professor Kelin Xia |

Abstract: | Many physical and biological problems involve interfaces separating different domains. To efficiently compute the dynamics of the interface, we develop an adaptive time-step scheme. The idea is to map the original time and space onto a new time and space such that the interface can evolve at an arbitrary speed in the new rescaled frame. In particular, for the expanding or shrinking interface problem, we choose (1) the space scaling function so that the expanding or shrinking interface is always mapped back to its initial size, i.e. the interface does not expand or shrink in the rescaled frame; (2) the time scaling function to speed up or slow down the motion of the interface, especially at later times when the interface expands slowly or shrinks extremely fast. We will show some examples. |

Title: | Direct and Inverse Algebraic Approximation and Application to the p and h-p Version of Finite Element Method |

Speaker: | Professor Benqi Guo |

Date: | 25 September 2017 |

Time: | 10.30 am – 11.30 am |

Venue: | MAS Executive Classroom 2, MAS-03-07 |

Host: | Associate Professor Wang Li-Lian |

Abstract: | We present an important progress in past two decades on direct and inverse algebraic approximation theory in the framework of the Jacobi-weighted spaces. The direct and inverse algebraic approximation are investigated in various functional spaces, e.g., the Sobolev spaces. Comparing algebraic approximation with trigonometric approximation, we conclude that Sobolev spaces are appropriate tools to establish the trigonometric approximation theory, but not for the algebraic approximation. In late 1990s we introduced the Jacobi-weighted Sobolev and Besov spaces and have investigated algebraic approximation in framework of these weighted spaces, which leads to the optimal direct approximation errors as well as the sharpest inverse results for algebraic approximation. The match-up between the direct and inverse results of algebraic approximation was achieved for the first time. Applying the new algebraic approximation theory in framework of the Jacobi-weighted spaces, we have been solving some important issues in approximation theory of the p and h-p version of the finite element method in two and three dimensions. |

Title: | Matrix spaces as a linear algebraic analogue of graphs |

Speaker: | Dr Youming Qiao |

Date: | 20 September 2017 |

Time: | 3.30 pm – 4.30 pm |

Venue: | MAS Executive Classroom 2, MAS-03-07 |

Host: | Dr Bei Xiaohui |

Abstract: | In the last few years, two research projects of mine (joint with collaborators) suggest that it is fruitful to view linear spaces of matrices as a linear algebraic analogue of graphs. In this talk I will first formally propose such an analogue, and then review how this helps in some decades-old problems from theoretical computer science and mathematics. We will see how this serves as a platform for interactions among algorithm design, combinatorics, and algebraic geometry. Finally I will mention some future directions. Based on joint works with Gabor Ivanyos, K. V. Subrahmanyam, and Yinan Li. |

Title: | An Exact Spectrum Formula for the Maximum Size of Finite Length Block Codes |

Speaker: | Dr Vincent Tan |

Date: | 14 August 2017 |

Time: | 3.00pm - 4.00pm |

Venue: | MAS Executive Classroom 2, MAS-03-07 |

Host: | Dr Kiah Han Mao |

Abstract: | An exact information spectrum- type formula for the maximum size of finite length block codes subject to a minimum pairwise distance constraint is presented. This formula can be applied to codes for a broad class of distance measures. As revealed by the formula, the largest code size is fully characterized by the information spectrum of the distance between two independent and identically distributed (i.i.d.) random codewords drawn from an optimal distribution. A new family of lower bounds to the maximal code size is thus established, and the well -known Gilbert -Varshamov (GV) lower bound is a special case of this family. By duality, an explicit expression for the largest minimum distance of finite length block codes o f a fixed code size is also obtained. Under an arbitrary uniformly bounded symmetric distance measure, the asymptotic largest code rate (in the block length n) attainable for a sequence of (n,M,nd) - codes is given exactly by the maximum large deviation rate function of the normalized distance between two i.i.d. random codewords. The exact information spectrum- type formula also yields bounds on the second -order terms in the asymptotic expansion of the optimum finite length rate for block codes with a fixed no rmalized minimum distance. This is joint work with Ling- Hua Chang (National Chiao Tung University, Taiwan), Carol Wang (NUS), Po -Ning Chen (NCTU, Taiwan), and Yunghsiang S. Han (Dongguan University of Technology, China). It can be found on arXiv:1706.04709. |

Title: | Lightweight Symmetric - Key Cryptography |

Speaker: | Dr Guo Jian |

Date: | 12 July 2017 |

Time: | 10.00am - 11.00am |

Venue: | MAS Executive Classroom 2, MAS-03-07 |

Host: | Division of Mathematical Sciences, School of Physical and Mathematical Sciences |

Abstract: | Recent years have witnessed massive and wide deployment of IoT devices, ranging from smart cards to implanted medical devices. It is estimated that 50 billion IoT devices will be connected by year 2020. The diverse feature of IoT devices results in many special requirements to cryptographic mechanisms over traditional ones, such as low hardware area when implemented on small devices or low energy consumption when running on devices powered by limited battery. We show, by examples of concrete designs, how effective cryptographic mechanisms are still possible under these constraints without affecting the security strengths. It is also interesting to note that a single algorithm could be implemented in several ways to fit very different IoT usecase scenarios while keeping the functionality and security strength unaffected. |

Title: | Viruses and Geometry: New Insights into Virus structure, Assembly and Evolution |

Speaker: | Professor Reidun Twarock |

Date: | 14 June 2017 |

Time: | 11.00am - 12.00pm |

Venue: | MAS Executive Classroom 2, MAS-03-07 |

Host: | Assistant Professor Kelin Xia |

Abstract: | Viruses are remarkable examples of order at the nanoscale. The capsids of many viruses, enclosing and protecting their genomes, are organised in lattice like arrangements with overall icosahedral symmetry. Mathematical techniques from group, graph and tiling theory can therefore be used to characterise their architectures. In this talk, I will introduce our mathematical approach to the modelling of viral capsids, and demonstrate its applications in vaccine design. I will then present our Hamiltonian path approach to the modelling of genome packing in RNA viruses that underpins the discovery of an RNA-encoded assembly instruction manual in a wide range of viruses, including Picornavirusus, Hepatitis C and Hepatitis B virus. Finally, I will introduce our models of virus assembly and demonstrate how they can be used to develop implicit fitness functions that shed new light on viral evolution and anti-viral drug therapy. |

Title: | Identities of A. I. Popov |

Speaker: | Professor Bruce C. Berndt |

Date: | 2 June 2017 |

Time: | 10.30am - 11.30am |

Venue: | MAS Executive Classroom 2, MAS-03-07 |

Host: | Associate Professor Chan Song Heng |

Abstract: | A. I. Popov was a Russian mathematician who wrote 13 papers in the 1930s. His mathematical papers have been forgotten, but his work on toponymics (the study of the origins of place names in countries) provided him with fame that has lasted through the present. Popov's mathematical contributions centered on formulas involving rk(n), the number of representations of the positive integer n as the sum of k squares. Special functions, in particular Bessel functions, appear in many of his formulas. Several of Popov's theorems are discussed in this lecture. In particular, in 1934 Popov stated, but did not prove, a beautiful series transformation involving rk(n) and certain Bessel functions. We provide a proof of this identity for the first time as well as for another identity, which can be regarded as both an analogue of Popov's identity and an identity involving r2(n) from Ramanujan's lost notebook. |

Title: | Geometric Understanding and analysis of unstructured data |

Speaker: | Professor Zhao Hongkai |

Date: | 1 June 2017 |

Time: | 4.50pm - 5.50pm |

Venue: | MAS Executive Classroom 2, MAS-03-07 |

Host: | Associate Professor Chan Song Heng |

Abstract: | One of the simplest and most natural ways of representing geometry and information in three and higher dimensions is using point clouds, such as scanned 3D points for shape modeling and feature vectors viewed as points embedded in high dimensions for general data analysis. Geometric understanding and analysis of point cloud data poses many challenges since they are unstructured, for which a global mesh or parametrization is difficult if not impossible to obtain in practice. Moreover, the embedding is highly non-unique due to rigid and non-rigid transformations. In this talk, I will present some of our recent work on geometric understanding and analysis of point cloud data. I will first discuss a multi-scale method for non-rigid point cloud registration based on the Laplace-Beltrami eigenmap and optimal transport. The registration is defined in distribution sense which provides both generality and flexibility. If time permits I will also discuss solving geometric partial differential equations directly on point clouds and show how it can be used to 'connect the dots' to extract intrinsic geometric information for the underlying manifold. |

Title: | Open dynamical systems: -expansion map on an interval with a hole |

Speaker: | Assistant Professor Nikita Agarwal |

Date: | 2 May 2017 |

Time: | 3.00pm - 4.00pm |

Venue: | MAS Executive Classroom 2, MAS-03-07 |

Host: | Division of Mathematical Sciences, School of Physical and Mathematical Sciences |

Abstract: | The study of open dynamical systems, first proposed by Pianigiani and Yorke in 1979 in the context of billiards, has attracted attention on account of both its dynamical interest and applications. We discuss some recent results on expanding interval maps with a hole. |

Title: | Torus knots and quantum modular forms |

Speaker: | Dr Jeremy Lovejoy |

Date: | 28 April 2017 |

Time: | 10.30am - 11.30am |

Venue: | SPMS LT 4 (SPMS-03-09) |

Host: | Associate Professor Chan Song Heng |

Abstract: | In the first part of this talk I will explain how the theory of Bailey pairs leads to an explicit formula for the cyclotomic coefficients of the colored Jones polynomial of the torus knots (2,2t+1). In the second part, I will describe how this gives a kind of "analytic continuation" of a family of quantum modular forms (the generalized Kontsevich-Zagier series). If time permits, I will also discuss how formulas for the colored Jones polynomial lead to qhypergeometric and Hecke-type formulas for certain families of unified WRT invariants. |

Title: | Basic Properties of the Blockchain |

Speaker: | Dr Juan Garay |

Date: | 27 April 2017 |

Time: | 11.00am - 12.00pm |

Venue: | MAS Executive Classroom 2, MAS-03-07 |

Host: | Division of Mathematical Sciences, School of Physical and Mathematical Sciences |

Abstract: | As the first decentralized cryptocurrency, Bitcoin has ignited much excitement, not only for its novel realization of a central bank-free financial instrument, but also as an alternative approach to classical distributed computing problems, such as reaching agreement distributedly in the presence of misbehaving parties, as well as to numerous other applications―contracts, reputation systems, name services, etc. The soundness and security of these applications, however, hinges on the thorough understanding of the fundamental properties of its underlying blockchain data structure, which parties (“miners”) maintain and try to extend by generating “proofs of work” (POW, aka “cryptographic puzzle”). In this talk we formulate such fundamental properties of the blockchaincommon prefix, chain quality, chain growthand show how applications such as consensus and a robust public transaction ledger can be built ``on top'' of them, assuming the adversary’s hashing power (our analysis holds against arbitrary attacks) is strictly less than ½ and high network synchrony: The above properties hold assuming that all parties―honest and adversarial―”wake up” and start computing at the same time, or, alternatively, that they compute on a common random string (the “genesis” block) only made available at the exact time when the protocol execution is to begin. In this talk we also consider the question of whether such a trusted setup/behavioral assumption is necessary, answering it in the negative by presenting a Bitcoin-like blockchain protocol that is provably secure without trusted setup, and, further, overcomes such lack in a scalable way―i.e., with running time independent of the number of parties [4]. A direct consequence of our construction above is that consensus can be solved directly by a blockchain protocol without trusted setup assuming an honest majority (in terms of computational power). |

Title: | Gradient Boosting for Partially Linear Additive Models in Survival Analysis |

Speaker: | Dr Tang Xingyu |

Date: | 21 April 2017 |

Time: | 9.30am - 10.30am |

Venue: | MAS Executive Classroom 2, MAS-03-07 |

Host: | Division of Mathematical Sciences, School of Physical and Mathematical Sciences |

Abstract: | The partially linear additive model is a special form of additive models, which combines the strengths of linear and nonlinear models by allowing linear and nonlinear predictors to coexist. One of the most interesting questions associated with the partially linear additive model is to identify nonlinear, linear, and non-informative covariates with no such pre-specification given, and to simultaneously recover underlying component functions which indicate how each covariate affects the response. Survival analysis is a popular topic in statistics, and the Cox's model is one of the most commonly used models in survival analysis. Here the partially linear additive model is adapted to survival analysis, and gradient boosting approaches are applied to optimize the Cox's log partial likelihood, with simple linear regressions and univariate penalized splines are together used as base learners. Twin boosting is incorporated as well to achieve better variable selection accuracy. Simulation studies as well as real data applications illustrate the strength of our proposed algorithms. |

Title: | Genome-wide association studies: Applications and insights gained in Ophthalmology |

Speaker: | Dr Zhao Wanting |

Date: | 30 March 2017 |

Time: | 9.30am - 10.30am |

Venue: | MAS Executive Classroom 2, MAS-03-07 |

Host: | Division of Mathematical Sciences, School of Physical and Mathematical Sciences |

Abstract: | Genome-wide association studies (GWAS) use high-throughput genotyping technologies to genotype and impute millions of single-nucleotide polymorphisms (SNPs) and relate them to the development of clinical and quantitative traits. Their use has been highly successful in the field of ophthalmology, and since the advent of GWAS in 2005, many genes not previously suspected of having a role in disease have been identified and the findings replicated. In this seminar, Dr. Zhao will report findings from the large multi-ethnic meta-analysis of genomewide association studies of complex ocular diseases by the International Cataract Genetics Consortium. They identified several new loci associated with these eye diseases and replicated the association through large-scale multi-ethnic meta-analysis of GWAS. These findings provide additional candidate genes for follow-up work and may lead to uncovering of previously unknown mechanisms in ocular diseases formation. |

Title: | Clinical Drug Development with Statistics: A Personal Perspective |

Speaker: | Dr Yeo Kwee Poo |

Date: | 21 March 2017 |

Time: | 9.30am - 10.30am |

Venue: | MAS Executive Classroom 2, MAS-03-07 |

Host: | Division of Mathematical Sciences, School of Physical and Mathematical Sciences |

Abstract: | In this talk, I want to give the audience an idea of what the work of a statistician in a pharmaceutical company might look like using my own experiences. I will give an overview of the drug development; an examples of applied statistics during early phase drug development; and discuss the desirable skill sets that help statisticians in the industry to succeed. I will conclude with some general comments that might be helpful for young statisticians considering to work in the pharmaceutical industry. |

Title: | Parameter estimation and multilevel clustering with mixture and hierarchical models |

Speaker: | Mr Nhat Ho |

Date: | 28 February 2017 |

Time: | 10.30am - 11.30am |

Venue: | MAS Executive Classroom 2, MAS-03-07 |

Host: | Division of Mathematical Sciences, School of Physical and Mathematical Sciences |

Abstract: | This talk addresses statistical inference with mixture and hierarchical models: efficiency of parameter estimation in finite mixtures, and scalable clustering of multilevel structured data. It is well-known that due to weak identifiability and singularity structures of latent variable models' parameter space, the convergence behaviors of parameter estimation procedures for mixture models remain poorly understood. In the first part of the talk, we describe a general framework for characterizing impacts of weak identifiability and singularity structures on the convergence behaviors of the maximum likelihood estimator in finite mixture models. This allows us to resolve several open questions regarding popular models such as Gaussian and Gamma mixtures, as well as to explicate the behaviors of complex models such as mixtures of skew normal distributions. In the second part of the talk, we address a clustering problem with multilevel structured data, with the goal of simultaneously clustering a collection of data groups and partitioning the data in each group. By exploiting optimal transport distance as a natural metric for distributions and a collection of distributions, we propose an optimization formulation that allows to discover the multilevel clustering structures in grouped data in an efficient way. We illustrate the performance of our clustering method in a number of application domains, including computer vision . |

Title: | Net-Trim: A Layer-wise Convex Pruning of Deep Neural Networks |

Speaker: | Dr Alireza Aghasi |

Date: | 14 February 2017 |

Time: | 10.30am - 11.30am |

Venue: | MAS Executive Classroom 2, MAS-03-07 |

Host: | Division of Mathematical Sciences, School of Physical and Mathematical Sciences |

Abstract: | Model reduction is a highly desirable process for deep neural networks. While large networks are theoretically capable of learning arbitrarily complex models, overfitting and model redundancy negatively affects the prediction accuracy and model variance. Net-Trim is a layer-wise convex framework to prune (sparsify) deep neural networks. The method is applicable to neural networks operating with the rectified linear unit (ReLU) as the nonlinear activation. The basic idea is to retrain the network layer by layer keeping the layer inputs and outputs close to the originally trained model, while seeking a sparse transform matrix. We present both the parallel and cascade versions of the algorithm. While the former enjoys computational distributability, the latter is capable of achieving simpler models. In both cases, we mathematically show a consistency between the retrained model and the initial trained network. We also derive the general sufficient conditions for the recovery of a sparse transform matrix. In the case of standard Gaussian training samples of dimension N being fed to a layer, and s being the maximum number of nonzero terms across all columns of the transform matrix, we show that O(s.logN) samples are enough to accurately learn the layer model. |

Title: | Ramanujan congruences for bipartitions |

Speaker: | Associate Professor Bernard Lishuang Lin |

Date: | 13 February 2017 |

Time: | 10.30am - 11.30am |

Venue: | MAS Executive Classroom 1, MAS-03-06 |

Host: | Associate Professor Chan Song Heng |

Abstract: | We present some work on bipartitions. In the first part, we recall some results for the partition function p(n). In the second part, we focus on the partition function Bl(n), which counts the number of l-regular bipartitions of n. Several infinite families of congruences for l = 4, 7. 13 are established. In the third part, we investigate the arithmetic properties of A3(n), the number of bipartitions of n with 3-cores. |

Title: | Growth of homology torsion in finite covering |

Speaker: | Professor Thang Le |

Date: | 23 January 2017 |

Time: | 1.30pm - 2.30pm |

Venue: | MAS Executive Classroom 2, MAS-03-07 |

Host: | Associate Professor Andrew James Kricker |

Abstract: | We discuss the growth of homology torsion of finite coverings. For 3- manifolds we show that the growth rate of the homology torsion is bounded from above by the hyperbolic volume (or Gromov norm). This upper bound is conjectured to be exact. |

Title: | Recent Advances in the Theory of Hilbert C*-Modules over Reduced Twisted Crossed-Product C*-Algebras |

Speaker: | Leonard Huang |

Date: | 5 January 2017 |

Time: | 2.30pm - 3.30pm |

Venue: | MAS Executive Classroom 1, MAS-03-06 |

Host: | Associate Professor Wu Guohua |

Abstract: | In this talk, I will discuss the main result of my PhD dissertation, which is a classification ' up to isomorphism ' of Hilbert C*-modules over reduced twisted crossed-product C*-algebras. This classification result generalizes an earlier one established by Ralf Meyer (Mathematisches Institut, Georg-August-Universit't G'ttingen) for Hilbert C*-modules over reduced crossed-product C*-algebras. I will then explain how a certain powerful Fourier- analytic technique used to construct Hilbert C*-modules over non-commutative tori - introduced by Marc Rieffel (University of California, Berkeley) and later refined by Franz Luef (Norwegian University of Science and Technology) can be viewed as a natural example of my result. |