
Title: Frequencyseverity 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 frequencyseverity 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 longtail loss distributions and generalized additive models, as time permits.

Title: 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 krational 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.

Title: 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 riskfree 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 limitorder books for highfrequency 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.

Title: 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 socalled 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 soilscience factors in maize yield.

Title: Coalgebras and their Invariants
Research Team: Minhua Cheng, Noah Ankney
Professors: Teena Gerhardt and Gabe AngeliniKnoll
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.

Title: 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 lowdimensional
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 nonuniform noise are allowed for a nearlyexact recovery.

Title: 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 wellstudied 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.

Title: 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 AB. 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 finitedimensional spaces.

Title: 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.

Title: Landscape Theory for TightBinding 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 Nobelprize 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 infinitedimensional 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.

Title: 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 6080% and there are still a
lot of missing variants that need to be identified. The goal of this project is to implement various genomewide
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, highdimensional
variable selection) to identify SNP markers associated with AD.