Mentor: Eliot Bitting

Area: Algebra

Overview of the topic: Explore basic mathematical cryptography via Hoffstein, Pipher, and Silverman. 

Pre-requisites: Is enrolled or have taken math 3000 

Text: An introduction to mathematical Cryptography by Hoffstein, Pipher, Silverman 

Mentor: Stephen Landsittel

Area: Topology

Overview of the topic: This program is very flexible depending on the student’s background. Starting with basic general topology via Munkres, possibly skipping first few chapters if the student has that background, work through selected exercises, present an interesting problem of my choosing at the end. Goal (if the student has not seen any ring theory) is to understand basic theorems about paracompactness and metric spaces and present related results.

Pre-requisites: Really should have at least seen some set theory if not already comfortable at that level (this is all I ask as a bare minimum). If the student knows basic ring theory then I can bring in some interesting ideas about topological rings, but this isn’t totally necessary. If the student has actually already taken a topology course then I can also adapt this program slightly more-so to mostly concern topological rings. If the student has background in both topology and ring theory, then we can briefly observe some products and convergence in Munkres, and then mainly focus on topological rings and number theory.

Text: Probably primarily selected chapters of Munkres, potentially Attiyah-Macdonald, Notes that I provide from more advanced texts in algebra (& possibly number theory), I will possibly bring in elements from Altman-Kleinman 

Mentor: Brent Koogler

Area: Analysis

Overview of the topic: Depending on interest, either one or a combination of the following: (1) introduction to fundamental numerical algorithms, with an emphasis on array oriented python implementations, or (2) an introduction to theoretical numerical analysis (advanced).

Pre-requisites: (1) Know how to read and write proofs, competency with python, and a general interest in numerics, or (2) a course in advanced calculus and mathematical maturity.  

Text: (1) "Elementary Numerical Analysis" by Atkinson and Han, or (2) "Theoretical Numerical Analysis: A Functional Analysis Framework" by Atkinson and Han

Mentor: Luis Flores

Area: Analysis

Overview of the topic: Convexity is a simple, but very powerful mathematical concept that is often forgotten. It also has a wide variety of applications in industry as well as in academia. We will explore how far we can push this concept from an analytical and geometrical point of view. For example, we will observe that the convexity of a function is intimately related to how smooth (differentiable) a function is. We will also see observe that we can obtain information about the convexity of a function by analyzing the convexity of an appropriate set. Convexity can also be exploited in terms of optimization. For example, we will see that a convex continuous function on a convex and compact set attains its minimum and maximum at corresponding extreme points. This is why if you want to optimize a convex and continuous function of two variables on a closed square, you only have to evaluate the function at the four corners of the square. 

Pre-requisites: Students interested in this subject should have a background in linear algebra and basic analysis. We will not delve into abstract algebra or measure theory. Analysis will be kept at a level similar to that of advanced calculus (Think supremums, infimums, limit superiors, limit inferiors, compactness, etc). Background in topology and multivariable advanced calculus is a huge plus but not required.  

Text: Convex Analysis by Rockafellar, and Convex Optimization Theory by Bertsekas 

Mentor: Srishti Singh

Area: Algebra

Overview of the topic: We will learn the basics of Macaulay2 and Sage, software designed to perform computations in algebraic geometry and commutative algebra. We will also solve some combinatorial algebra problems (basically permutations and combinations but level 9000). 

Pre-requisites: Basic proof techniques; have taken at least one abstract algebra course (or are taking it).   

Text: N/A 

Mentor: Saloni Sinha

Area: Number Theory

Overview of the topic: The goal is to study some well-known topics in analytic number theory at a slightly advanced level. The topics will include study of arithmetic functions, prime numbers, characters and some classic theory of L-functions.  

Pre-requisites: Calculus 1-3 sequence, some introductory complex analysis  

Text: Analytic Number Theory by H. Iwaniec and E. Kowalski 

Mentor: James Warta

Area: Analysis 

Overview of the topic: Introduction to measure theory and the Hausdorff/Lebesgue Measures 

Pre-requisites: Undergraduate analysis at minimum, advanced calculus preferred   

Text: A selection from Folland, Wheedon-Zygmund, Evans-Gariepy