April update: We are delighted to confirm that DOmath 2020 is going ahead. The program will be held virtually in accordance with current university policy. The dates and projects for DOmath remain unchanged.
In 2020, DOmath will run from May 18 to July 10. We are pleased to announce the following five projects for DOmath 2020.
- Neural network dimension reduction of data with topological constraint, led by Professor Xiuyuan Cheng
In many applications, observed high dimensional data may lie on or near to low dimensional manifolds that are embedded in the high dimensional space. Dimension reduction aims at revealing such hidden geometrical structures without using data labels, and the learned data representation can be used for downstream data analysis tasks such as clustering. Due to the process of data production, many data in practice lie on manifolds which are not isomorphic to any simply connected bounded domain in the Euclidean space. Such topological constraints may pose an issue for some popular neural network dimension reduction methods.
The project will investigate this problem in more detail, starting from numerical experiments with existing neural network dimension reduction algorithms. We will then study the possible remedy by imposing the proper topological constraint, if known a priori, in the neural network model. Time permitting, we will also study the automatic discovery of such hidden structure by spectral methods based on affinity graphs built from data samples, and the combination of the latter with the neural network method. Such an approach will then inherit the theoretical guarantee of the learned low-dimensional representation provided by the spectral method. The project involves computation on simulated and real-world datasets as well as mathematical analysis.
Potentially helpful background includes: linear algebra and multivariate calculus; basic knowledge of manifold and topology in Euclidean space; working knowledge of machine learning, particularly unsupervised learning/dimension-reduction methods in data analysis, and optimization; coding experience of neural networks. (We welcome applications from students with some subset of this background.)
- Statistics on class groups and representation theory, led by Professors Samit Dasgupta and Jiuya Wang
The class group of a number field is one of the most important invariants in algebraic number theory. In this project we will explore the distribution of the sizes of class groups in extensions with fixed Galois group. The project will include concepts from the representation theory of finite groups. Computer explorations will be useful, and some experience in coding may be helpful.
- Using probability to understand cancer, led by Professor Rick Durrett
Cancer is the end result of processes that take many years. It is not possible to observe its development or how it spreads to other organs, a process called metastasis, so mathematical models are needed to make inferences about the underlying mechanisms. The models that we investigate will be motivated by recent publications such as the “Big Bang Model” and the paradoxical finding that cancer mutations show a signature of neutral evolution.
Math 230 (or 340) is a necessary (but not sufficient) prerequisite for this project. Students should be prepared to do some “on the job training” on Markov chains and the basics of population genetics. Simulation will be a tool, so some previous programming experience will be useful. However, we are also happy to have students that want to study the models analytically.
- PDE modeling of collective motion, led by Professors Alexander Kiselev and Siming He
We will develop models based on differential equations to describe biological processes such as diffusion, aggregation, and collective motion. The goal will be to study formation of patterns and behavioral features essential for survival. We are looking for students ideally with background in ODE and PDE. The project will consist of both analytic and computer simulation parts, so some exposure to numerical methods and software such as Matlab is useful.
- Materials science problems: numerical methods and techniques, led by Professor Saulo Orizaga
Mathematical models associated with phase separation process are of high importance to the area of materials science and engineering. Phase separation of materials is a phenomenon that can be observed at the liquid or solid state of a mixture (during solidification process). One of the simplest situations for phase separation of two materials is in the mixing of water and oil. For this case, water and oil will separate and segregate into small regions of either water or oil alone and this process continues until full separation occurs. There are more complicated processes in which materials do not fully separate and as a result different patterns, configurations and structures are possible due to micro-segregation.
The models describing phase separation are called phase field models. These are often given in the form of partial differential equations (PDEs). We will study the basic properties of well-known models (Cahn-Hilliard and possible practical extensions) and their applications. Since phase field models are found to have numerous applications across different fields, it is important to develop accurate and efficient numerical methods to solve such problems.
In this project we will develop accurate numerical methods with energy decreasing property to solve problems commonly arising in phase separation processes. Our main goal will be to develop several time-stepping techniques (coupled with an energy splitting technique) and test the performance in terms of accuracy and computation time of such numerical methods.
We are looking for students interested in learning more about applied and computational mathematics. Useful background may include some experience or interest in ODEs and PDEs, and basic programming experience in Matlab may be helpful.
The DOmath 2019 web page has information about completed projects from our 2019 program, including videos featuring our student researchers.