# Research Projects, Spring Semester 2019

## Frequency-severity Insurance Ratemaking Using Modern Techniques

**Research Team:**Zihao Gu, Charlie Crampton, Ahmad Zaini

**Professor:**Gee Y. Lee

**Subject Areas:**Actuarial Science

**Description:**Accurate insurance rates are required for the solvency of an insurance provider, as well as the affordability of insurance coverage for the policyholder. In this project, the student will obtain handson ratemaking experience under the supervision of an actuarial science faculty member. We will analyze a problem, and learn how to build and utilize models to provide a solution to an actuarial problem. We will develop models for a hypothetical ratemaking problem, and utilize historic frequencies and severities to build and test our models. Data from the Medical Expenditure Panel Survey (MEPS) will be utilized to implement the frequency-severity approach to ratemaking. Throughout the course of the project, we will attempt to improve existing models, and in particular we will explore the possibility of using penalized likelihoods in combination with long-tail loss distributions and generalized additive models, as time permits.

## Brauer Groups of Elliptic Curves

**Research Team:**Gengzhuo Liu, Jon Miles

**Professor:**Rajesh Kulkarni

**Subject Areas:**Algebra, specifically Galois theory

**Description:**Given an elliptic curve over a number field k and a Brauer class on it, there is an associated group homomorphism from the group of k-rational points E(k) to the Brauer group Br(k) of k. Our goal is to understand the underlying objects and then compute many examples to understand the size of the image of this homomorphism.

## Financial Mathematics and Actuarial Science

**Research Team:**Zeyuan Li, Aaron Bawol, Zoe Zhang

**Professor:**Frederi Viens

**Subject Areas:**Applied Probability

**Description:**The question of how to allocate funds for risky and risk-free investments is notoriously plagued by the inability to estimate the rates of returns of risky stocks. Beyond the industrial question of "discovering alpha", the student will have the option to investigate mathematical methods based on the concept of robust optimization, where an investor takes into account aversion to financial and insurance risks as well as aversion to modeling ambiguity. The latter ambiguity is the inability to be sure that a model is better than all other models. It is strongly related, in principle, with challenges in statistical estimation. This project will use tools from stochastic control, and may involve working with financial data, either historically for backtesting, or live data for testing algorithms in real time. Part of this project may delve into the structure of limit-order books for high-frequency financial algorithms on US stock markets. Part of it may investigate statistical arbitrage opportunities, as those currently found in Chinese and Hong Kong stock markets.

## Bayes in Environmental, Agricultural, and Earth Sciences

**Research Team:**Meiqi Liu, Yifei Li, Anna Weixel, Xinyao Yu

**Professor:**Frederi Viens

**Subject Areas:**Computational Bayesian Statistics

**Description:**The use of Bayesian statistics is becoming more widespread because of the possibility of implementing complex Bayesian posterior calculations and their associated samples, thanks to modern computational platforms. Prof Viens and his team are involved in various applied projects, some of which have substantial associated theoretical questions, all of which involve a need for implementing approximate Bayesian computation. The exchange student will learn about classical linear Bayesian hierarchical modeling, about its numerical implementation using the so-called Gibbs sampler, and will have opportunities to engage in one or more of the following applied statistics topics:

(a) understanding the factors which drive the hydrology of the Lake Chad Basin in the Eastern Sahel: a study towards optimizing ecosystem services and preserving the environment in one of the world's least developed regions.

(b) developing a model for paleoclimatology in the late Holocene, which includes accounting for the role of the oceans: a study towards accurate assessment of uncertainty for climate projection over the next two centuries.

(c) agricultural productivity and applied economics: estimating structural equations for hierarchical models at the farmer to region level in least developed countries in Africa and Asia, including estimating environmental, social, and soil-science factors in maize yield.

## Coalgebras and their Invariants

**Research Team:**Minhua Cheng, Noah Ankney

**Professors:**Teena Gerhardt and Gabe Angelini-Knoll

**Subject Areas:**Algebra and Topology

**Description:**Coalgebras are algebraic objects equipped with an operation, called a comultiplication, which arise naturally in topology. In this project, students will study coalgebras and their properties. Students will learn background in homological algebra and algebraic topology, and use tools from these areas to compute an invariant of coalgebras called coHochschild homology. Understanding this invariant is an essential step towards computing topological coHochschild homology, an exciting new object of study in algebraic topology.

## Improving Image Quality via Compressed Sensing

**Research Team:**Shuai Yuan, Jonathan Fleck, Changxiong Liu, Hongbo Lu

**Professor:**Rongrong Wang

**Subject Areas:**Applied Mathematics

**Description:**Robust PCA is a powerful method that can accurately separate data from noise when the noise obeys heavy tail distributions. However, this method only works for data points sampled from a low-dimensional linear subspace, which greatly limits its application. To make the method widely applicable, it is necessary to consider dataset sampled from a general manifold with nonlinear structures. In this project, we explore what happens when the manifold is smooth and Robust PCA is applied to a collection of local patches of the manifold simultaneously. We are particularly interested in establishing theoretical conditions under which non-uniform noise are allowed for a nearly-exact recovery.

## Constructing Large Ideal Class Groups

**Research Team:**Shengkuan Yan, Luke Wiljanen

**Professor:**Aaron Levin

**Subject Areas:**Number Theory, Algebraic Geometry

**Description:**The ideal class group of a number field is a fundamental and well-studied object in number theory which gives a measure of the extent to which unique factorization in a ring of integers fails. Recently, a geometric approach to constructing number fields with a large ideal class group has been developed. This approach relies on finding curves with certain properties, and the project will explore theoretical aspects of this problem, as well as the possibility of constructing algorithms to search for suitable curves.

## Functions of Perturbed Matrices

**Research Team:**Dingjia Mao,Yuan Luo

**Professor:**Vladimir Peller

**Subject Areas:**Analysis, Linear Algebra

**Description:**The project will deal with comparing functions f(A) and f(B) for square matrices A and B. In particular, the problem is to estimate the norm of f(A)-f(B) in terms of the norm of A-B. Such estimates depend on properties of the functions f. Such problems are problems in perturbation theory in the case when we deal with linear operators on finite-dimensional spaces.

## Machine Learning and Applications

**Research Team:**Che Yang, Neel Modi, Billy Pan

**Professor:**Guowei Wei

**Subject Areas:**Computational Mathematics

**Description:**We are interested in designing advance machine learning and deep learning architectures for realistic applications to finance, insurance, actuarial science and other industries. Our goal is to carry out mathematical analysis and improvement of existing ensemble methods (i.e., random forest, gradient boosted decision trees, extra trees, etc.), multitask learning and deep neural networks (convolutional neural network, recurrent neural network, Boltzmann machine, etc). The student will work with my PhD students or postdocs to learn related machine learning theory and algorithm, and applications.

## Landscape Theory for Tight-Binding Hamiltonians

**Research Team:**Xingyan Liu, John Buhl, Isaac Cinzori, Isabella Ginnett, Mark Landry, Yikang Li

**Professors:**Ilya Kachkovskiy and Shiwen Zhang

**Subject Areas:**Analysis, Spectral Theory, Mathematical Physics

**Description:**Anderson localization is one of the central phenomena studied in modern mathematical physics, especially in dimensions 2 and 3, starting from Nobel-prize winning discovery by P. W. Anderson. Recently, a new approach for the 1D discrete model was proposed by Lyra, Mayboroda, and Filoche, which shows interesting relations with the Dirichlet problem on the lattice and also allows to significantly reduce complexity of some numerics related to the problem. The main goal of the project is to extend this this approach to higher (and most interesting physically) dimensions. The project will involve advanced reading, possible new results in finite and infinite-dimensional spectral theory, understanding physics behind some problems in linear algebra, and novel numerical experiments. Original research results are expected as an outcome for successful students.

## Statistical Identification of Genetic Variants Associated with Alzheimer Disease

**Research Team:**Yang Liu, Deontae Hardnett, Gaeun Lee, Glenna Wang

**Professor:**Yuehua Cui

**Subject Areas:**Biostatistics

**Description:**Alzheimer disease (AD) is the most common causes of neurodegenerative disorder in the elderly individuals. Currently reported genetic variants in gene APP, PSEN1, PSEN2 and APOE4 only contribute less than 30% of the genetic variation of AD. However, the total predicted genetic variation is about 60-80% and there are still a lot of missing variants that need to be identified. The goal of this project is to implement various genome-wide association study strategies to identify genetic variants (e.g., single nucleotide polymorphisms (SNPs)) which are associated with brain volumes in six brain regions, with data obtained from the Alzheimer Disease Neuroimaging Initiative (ADNI) project. There are over 800K SNP markers. Students will learn some basic concepts in statistical genetics and use various analytical strategies (e.g., linear regression, multiple testing adjustment, high-dimensional variable selection) to identify SNP markers associated with AD.