Michigan Research Experience for Graduates 2022
Information About MREG 2023 can be found here.
Important Update: MREG 2022 will be entirely virtual.
Michigan Research Experience for Graduates (MREG) is a workshop designed to provide early graduate students in pure, applied, and interdisciplinary mathematics an opportunity to engage in research with their peers, and to network with early-career faculty and professionals working in these fields. At the same time, we aim to promote and help start conversations around various initiatives and issues centered on diversity, equity and inclusion (DEI).
The two-week program splits students into small groups who will work closely with a project mentor. Both weeks will be conducted virtually. During the first week, students will learn the background material as guided by their group's mentor. The second week will be focused on working on the project at hand. At the end of the workshop, groups will give presentations about their project and methodology. The projects will be in pure, applied, and interdisciplinary mathematics, and we have invited mentors working in commutative algebra, algebraic geometry, topology, dynamics, differential equations, probability and machine learning. We plan to have a graduate student or postdoc assistant in each group.
The 2022 program will take place online from June 20th to July 1st.
Click here to view the program poster.
Research Experience for Early Graduate Students
There will be working groups of 3-5 graduate students led by early career mathematicians with some groups also having UM faculty as mentors. The participants will be working on accessible research problem(s) and are expected to commit to the project for the full two week period. The exact workload and the meeting times will vary for each project and will be flexible based on the preferences of the group members and the group leader.
The list of available projects is given below.
List of Research Projects
- Jack Burkart - Generalizations of the Traveling Salesman Theorem
Home Institution: University of Wisconsin - Madison
Website: https://www.jackburkart.com/
Project Description: Click here.
Prerequisites: Basic measure theory/real analysis is essential and complex analysis provides additional context, but is more supplementary than necessary.
Final Report: Click here.
TA: April Nellis
Participants:
Younghyun Kim
Melina Galilea Jimenez Deniz
Danae Sarahi Galan Covarrubias
Zikang Jia
- Rankeya Datta - Frobenius Splitting of Valuation Rings
Home Institution: Michigan State University
Website: https://www.rankeyadatta.com/
Project Description: In the absence of geometric techniques stemming from Hironaka’s seminal theorem on resolution of singularities in characteristic 0, the Frobenius map of a commutative ring of prime characteristic p>0 is a powerful tool that encodes information about the singularities of the ring. One of the most well-studied aspects of the Frobenius map is the question of when it admits a section, commonly known as a Frobenius splitting. The broad aim of this project is to understand Frobenius splittings in the highly non-noetherian setting of valuation rings. Valuation rings have deep historical roots in the study of singularities and have become increasingly important in many contemporary theories like Berkovich spaces, perfectoid spaces and the theory of algebraic K-stability. Our specific goal will be to ascertain if the only Frobenius split valuation rings of finitely generated field extensions of transcendence degree 2 (i.e. function fields of algebraic surfaces) in prime characteristic are the ones that are natural generalizations of valuation rings associated to prime divisors on normal varieties.
Prerequisites: Graduate Sequence in Algebra. Some comfort with commutative algebra.
Final Report: Coming soon.
TA: Andrés Martinez Servellon
Participants:
Sandra Nair
Sandra Sandoval
Dawei Shen
Ying Wang
- Yusheng Luo - Complex Dynamics and Blaschke Products
Home Institution: SUNY Stony Brook
Website: https://sites.google.com/view/yushengmath
Project Description: Click here.
Prerequisites: We assume a first graduate course in complex analysis and topology. Some basic knowledge in hyperbolic geometry will be helpful for the project, but not necessary.
Final Report: Coming soon.
TA: Caroline Davis
Participants:
Yucong Lei
Sicheng Zhao
Jiayi Shen
Guoran Ye
Lingxiao Wang
Tasnim Iqbal
- Neel Patel - Parabolic PDEs
Home Institution: ICMAT
Website: https://sites.google.com/view/neelpatelmath/home
Project Description: We will study parabolic PDEs and consider the effect of nonlinear source terms on the growth of solutions to these PDE in various norms. Some interesting questions to investigate will be the types of nonlinearity that can be controlled and how sensitive certain normed spaces are for studying solutions. Given time, we will read through: https://arxiv.org/abs/2104.06921 connecting number theory/polynomial roots to analysis and PDE. Background material to be covered: Fourier transform; defining normed spaces (Sobolev spaces, Wiener algebras); various inequalities (interpolation, Holder's, Young's, Jensen, etc.), applications of parabolic equations (heat, porous media interfaces, polynomial root distribution).
Prerequisites: First year graduate analysis sequence
Final Report: Click here.
TA: Katja Vassilev
Participants:
Zhengjun (Jasper) Liang
Lizhe Wan
Ning Tang
Abhishek Adimurthi
Zachary Deiman
Santiago Cordero-Misteli
- Allechar Serrano López - Counting Elliptic Curves with an Isogeny of Degree Two
Home Institution: Harvard University
Website: http://www.math.utah.edu/~serrano/
Project Description: Click here.
Prerequisites: A course in abstract algebra
Final Report: Coming soon.
Participants:
Asma Alrashidi
William Li
Yuxin Xue
Evan Cole
Luke Seaton
- Rishi Sonthalia - Math for Machine Learning and Data Science
Home Institution: University of California Los Angeles
Website: https://sites.google.com/umich.edu/rsonthal/
Project Description: Click here.
Prerequisites: Familiarity with Python, linear algebra, probability and analysis.
Final Report: Coming soon.
TA: Yutong Wang
Participants:
Jiashu Han
Kashvi Srivastava
Chinmaya Kausik
Gabriel Raposo
Madelyn Esther Cruz
- Rachel Webb - Geometric Invariant Theory
Home Institution: University of California Berkeley
Website: https://sites.google.com/view/rachel-webb
Project Description: Click here. NOTE: Due to scheduling conflicts, this project will run during the weeks of June 13-17 and June 24-July 1.
Prerequisites: Basic knowledge of group actions and group representation theory, affine and projective varieties. Interest in learning about geometric invariant theory, representation theory of reductive groups, and some basic aspects of quotient stacks.
Report: Click here.
TA: Kimoi Jeptoo Kemboi
Participants:
Riku Kurama
Ruoxi Li
Marwa Mosallam
Henry Talbott
Vignesh Jagathese
Jiazhen Tan
- Becca Winarski - Braids and Polynomials
Home Institution: College of the Holy Cross
Website: https://sites.google.com/site/rebeccawinarski/
Project Description: Braids are fundamental objects in surface topology that have been study by Artin since the 1920’s. A braid is a continuous 1-1 map of the plane (or sphere) that preserves a finite set of points (that is, a homeomorphism). Polynomials are basic objects in many areas in math, and in this project well be interested in them from a complex dynamics perspective. We will study functions of the plane (actually, the complex numbers) that are d-to-1 (here d is some positive integer) except at a finite set of points; such maps are called branched covers (of the plane). Thurston proved that branched covers of the plane (these are generalizations of polynomials) that satisfy some finiteness properties can either be realized as a polynomial or have a certain kind of topological obstruction.
The goal of this project is to compose braids and polynomials to create and make sense of higher degree braid diagrams. In doing so, we’ll search for compositions and braid diagrams that are always/never/sometimes realized as polynomials.
Prerequisites: Background (either coursework or previous research experience) in one of: topology (understanding of continuous maps), programming (ideally Python/Sage or Mathematica), or complex analysis/dynamics.
Final Report: Click here.
TA: Sayantan Khan
Participants:
Katalin Berlow
Patrick Chan
Urshita Pal
Aris Lemmenes
Tori Braun
Organizers:
Malavika Mukundan
Swaraj Pande
Nancy Wang
Michael Mueller
Ben Riley
Chris Stith
Schinella D'Souza
Shelby Cox
Scott Neville
Sanal Shivaprasad
MREG organizing committee email: mreg-committee@umich.edu
Click here to go to the website for MREG 2021.