Summer Research in Mathematics

Math Research Experience 2018

The mathematics department will be hosting an internal research experience for undergraduates and early graduate students in Summer 2018. If your idea of fun is thinking about mathematics, then this may be your chance for getting paid for having fun.

Projects

  • Simplicial Complexes and Connectivity. An n-simplex is an n-dimensional triangle. For example, a 0-simplex is a point, a 1-simplex is an edge, a 2-simplex is a filled in triangle and a 3-simplex is a solid tetrahedron, etc. A simplicial complex is a space built in a very rigid way from n-simplices. We will explore the differences between various simplicial complex arising in the study of an algebraic phenomenon called homological stability. Will ask questions about their connectivity. This will involve learning some concepts from category theory and algebraic topology.

    Mentor: Agnes Beaudry. Prerequisite. MATH 3140 or MATH 2135.

  • Graphs, quadratic forms, and period maps. Period maps provide a method of parameterizing certain algebraic and geometric data arising from the Hodge theory of complex projective manifolds. In extending period maps to the boundary of the domains of definition, one is led to study cones of quadratic forms. Specifically, one wants to know whether a given monodromy cone is contained in a "standard" cone of quadratic forms. In certain cases, these monodromy cones can be described from the combinatorics of graphs. In this project, we will investigate this type of situation for the Torelli map and for the Prym map, where the monodromy cones arise from graphs, and where there are several open questions. Our approach will primarily be computational, working out examples, and writing some computer code to investigate. The point is that from previous work there is an expectation as to what graphs will lead to monodromy cones that do not lie in a "standard" cone, and we would like to confirm this computationally in some further examples.

    Mentor: Sebatian Casalaina-Martin. Prerequisite. Ideally MATH 3140, but MATH 2135 may be sufficient. Some experience writing computer code would be beneficial.

  • Invertible random matrices. A random matrix is a matrix whose entries are random variables. One goal of Random Matrix Theory is to study the "typical" (or "average") behavior of matrices. For example, the eigenvalues of a random matrix describe the typical eigenvalue behavior of all matrices. This project aims to study how often random matrices are invertible and to understand properties of the inverse. We will consider questions such as: What is the probability that a random matrix is invertible? When it is invertible, how large (or small) can the entries of the inverse be? Part of the project will involve numerical simulations using software such as MATLAB, but no prior programming knowledge is required.

    Mentor: Sean O'Rourke. Prerequisite. Probability theory and linear algebra.

  • Liber Mathematicae and Mathematics presentation on the web. Properly presenting Mathematics on the web is still a challenging task. The proposed REU addresses this problem by extending the Liber Mathematicae project and writing appropriate code in ruby on rails. Mathematically it is planned to work on the visualization of singularities within Liber Mathematicae.

    Mentor: Markus Pflaum. Prerequisite. Both acquaintance with fundamental algebra and programming.

  • Power bases for rings of integers. The rings of integers of algebraic number fields are one of the fundamental objects of study in algebraic number theory. Questions about these rings encompass many of the problems that motivate current research.

    For this project, we are interested in when the integral basis of this ring admits a very simple description. Specifically, our question is: When does the ring of integers admit a power basis, a basis consisting of 1 and powers of some other element? Number fields whose ring of integers has this property are called monogenic. The problem of classifying such number fields was posed in the 1960's by Helmut Hasse and is often called Hasse's Problem.

    Hasse's problem is far from solved. Most attacks on Hasse's problem have involved classifying monogenic or non-monogenic families of a given degree. Our project will follow in this tradition. Using a tool called the Montes algorithm; we hope to classify infinite families of irreducible monic integer polynomials of degree 5 and greater that generate monogenic fields. Not only will this work help solve Hasse's problem, but the rings of integers we classify will supply other mathematicians with examples that are efficient for computational purposes.

    This project may continue during the school year, depending on the interest of the participants.

    Mentor: Kate Stange and Hanson Smith. Prerequisite. A strong grade in MATH 3140 required. MATH 4140 even better.

  • Tinkering with canonical forms. Equivalence relations on sets can give a mathematical way to encode when we believe elements behave the same. Similarity of matrices is a fundamental such relation that preserves eigenvalues, order, invertibility, and many other important properties. If we restrict our attention to invertible n by n matrices with eigenvalue 1, then similarity classes of matrices are in fact indexed by integer partitions of n. This project seeks to better understand the relationship between similarity and another equivalence relation that has classes indexed by set partitions of {1,2,..., n}. At this point we know that the two relations are completely incompatible, but we'd like a more nuanced description. That is, we'd like to convert the basic linear algebraic problem into a nice combinatorial description with broader representation theoretic implications.

    Mentor: Nat Thiem. Prerequisite. Ideally MATH 3140, but linear algebra plus one or more MATH 3000+ level courses is probably sufficient.

Dates

  • Starting date

  • May

  • June

  • July