Michigan Research Experience for Graduates 2022

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

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

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

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

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

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

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