# Mentoring

## For undergraduate students

We organize mentoring for undergraduate students. Suppose you want to discuss your REU project with a graduate student, or you wish to attend graduate school in the future but have no idea what it looks like or how to apply.

For the Spring 2024 semester we are running a reading groups program, where undergrads can learn about a topic by reading books or papers under the guidance of a grad student. **Reading groups are open to all undergraduates and non-math majors are encouraged to join.** These reading group topics can possibly turn into an honors thesis by expanding upon the work done with a faculty advisor.

**New reading group: Around Roth’s and Szemerédi’s Theorems. Contact Donovan if you would like to join!**

**All other groups are now started for Spring 2024. If you would still like to try to join a reading group, contact Donovan.**

If you have any comments or questions, contact Donovan Snyder.

# Reading Group Topics for Spring 2024

### Around Roth’s and Szemerédi’s Theorems

**Grad student organizer:**Pablo Bhowmik and Firdavs Rakhmonov**Description:**This reading group will delve into two fundamental results in additive combinatorics: Roth’s and Szemerédi’s Theorems. Of particular interest is Szemerédi’s Theorem, which asserts that any subset of the natural numbers with a positive upper density contains an arithmetic progression of length \(k\) for every \(k\geq 1\). The special case where \(k=3\) is famously known as Roth’s Theorem. The proof of Roth’s Theorem is quite technical, and we intend to analyze it thoroughly during our meetings. Additionally, we will introduce the concept of discrete Fourier analysis, a powerful tool in combinatorics that plays a pivotal role in proving Roth’s Theorem.**Prerequisites:**Participants are expected to possess a level of mathematical maturity and a familiarity with undergraduate-level combinatorics (MATH 238 or equivalent). Basic knowledge of abstract algebra (MATH 236) and mathematical analysis (MATH 170’s, 265) will be assumed at certain points.**Resources to be used:***Galois Cohomology and Class Field Theory*by Harari,*Algebraic Number Theory*by Cassels-Frohlich,*Algebraic Number Theory*by Neukirch

### Class Field Theory

**Grad student organizer:**John Lin**Description:**This course is on class field theory, one of the greatest achievements of 20th century number theory. It is a topic in number theory which describes the abelian extensions of certain fields, including number fields. We will start with local and global fields, which are generalizations of number fields. We will then go over group cohomology, which is an important technique. Then we will state and prove the main theorems of class field theory, which are the reciprocity theorem and existence theorem. If time permits, we will go over applications such as the Hilbert class field, Grunwald-Wang theorem, Chebotarev density, and the Kronecker-Weber theorem.**Prerequisites:**Algebraic number theory MATH 430**Resources to be used:***Galois Cohomology and Class Field Theory*by Harari,*Algebraic Number Theory*by Cassels-Frohlich,*Algebraic Number Theory*by Neukirch

### Fractal projections and intersections problems in Geometric Measure Theory (Limited spots available)

**Grad student organizer:**Quy Pham**Description:**Seventy years ago, John Marstrand published a paper that relates the Hausdorff dimensions of fractal sets in the plane to the dimensions of their orthogonal projections onto lines. We will consider two questions: How do orthogonal projections affect the Hausdorff dimension of a given set?

What can we say about the dimensions if we intersect a subset \(A\subset \mathbb{R}^n\) with a subset \(B\subset \mathbb{R}^n\).

For example, in \(\mathbb{R}^3\), generally, if we intersect a plane with a line, then the intersection is just one point. However, when \(A\) and \(B\) are fractal subsets, the situation is much more complicated.

Our goal is to understand the classical results and techniques in Geometric Measure theory, in particular related to fractal projections and intersection problems.**Prerequisites:**Analysis of 170’s, 265H, or 471. The topics of the reading course will be adjusted to the level of the students.**Resources to be used:***Fractal Geometry: Mathematical Foundations and Applications*by Kenneth Falconer,*Geometry of Sets*and*Fourier analysis and Hausdorff dimension*by Pertti Mattila

### Iterated Function Systems

**Grad student organizer:**Donovan Snyder**Description:**In an attempt to move towards Quantum Iterated Function Systems, we will start with the basics in topology and dynamical systems in order to study Iterated Function Systems. These act on spaces in interesting ways (often producing fractals), and we will study their properties rigorously.**Prerequisites:**A sequence of calculus (140’s,160’s, 170’s) is necessary. The more knowledge of multi-dimensional calculus, linear algebra, and topology, the easier the ramp up will be.**Resources to be used:***Fractals Everywhere*by Michael Barnsley and other texts.

### Witness Sets in Fractal Settings using VC-Dimension

**Grad student organizer:**Donovan Snyder**Description:**Given a set \(X\) and a collection of subsets, a witness set has at least one point present in every subset. We use the notion of the VC-dimension to find such witness sets in both the finite and infinite settings. One of the main focus for this project will be the setting of fractal sets, though other settings of interested are welcome!**Prerequisites:**No strict prerequisites, but knowledge of linear algebra (MATH 173, 235, or 165), analysis (MATH 170’s, 265), or some probability (MATH 201) is useful.**Resources to be used:**A few sets of notes written by people in the research group like this one. We would start at a more foundational level.

## For graduate students

Mentoring can help you build valuable skills which you will find useful throughout your career, whether you go into academia or industry.

If you want to join this program as a mentor for Spring 2024, contact Donovan.