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.
(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.