# Schedule

## Schedule

### Summer 2022

The summer 2022 program will be organized by Dr. Wayne Tarrant.

### Summer 2021

Below are the schedule of activities for the summer of 2021. All activities will be held in Microsoft Teams for June 7-18 and July 19-30 and in Crapo Hall room G219 for June 21-July 16. All times are EDT. The 2021 program was organized by Dr. Kenji Kozai.

Date Time (EDT) Description
June 7 10:00am Introduction and orientation
June 8 11:00am Workshop: Literature search
June 10 11:00am Workshop: Reading Mathematics Critically
June 15 11:00am Seminar: Bianca Thompson (Westminster College)

Title: The connection between Diophantine equations and binary trees

Abstract: Binary trees are used to encode a lot of different types of information and can be used to make decisions by following different paths along the tree. It turns out you can also use these trees to represent the different possible 2-adic valuations for sequences like x^2+D.Further, these valuation trees allow us to look at Diophantine equations of the form x^2+D= 2^cy, y odd, and determine the possible solutions. The goal of this talk is to share how to construct binary trees to create valuation trees and then use those trees to determine the possible integer solutions of specific Diophantine equations.
June 17 11:00am Week 2 progress report presentations
June 21 9:00am In-person meet and greet
June 22 11:00am Workshop: Mathematical Writing in Overleaf/LaTeX
June 24 11:00am Seminar: Tracy Weyand (Rose-Hulman Institute of Technology)

Title: Applications of Graphs, Eigenvalues, and Eigenfunctions

Abstract: Eigenvalues are not only important mathematically; eigenvalues and eigenfunctions can tell us important physical information in applications. You have looked at graphs of functions for years; how can a graph model something?

A discrete graph is a collection of vertices (dots), some of which are connected by edges (lines). Graphs can be used to model many things including the internet, subway routes, and materials. In these situations, each vertex represents and object and each line represents a relation between the two vertices it connects. The benefit is that these graph models usually have less complexity than traditional models. “Action” on such a system can be modeled with a matrix. We will look at some concrete examples where eigenvalues and eigenfunctions tell us important information about a physical system.
June 29 11:00am Workshop: Giving a mathematics talk
July 1 11:00am Seminar: Ranjan Rohatgi (Saint Mary's College)

Title: "Gerrymandering: how can we measure it?"

Abstract: Every ten years after the Census is complete, each state redraws its districts for the US House and its own state legislature. We’ll discuss how some states make it easy for politicians to gerrymander – draw districts to favor or disfavor one party or group – and how we can detect it when it occurs. We’ll also look at some proposed fixes, including political and independent commissions. Additionally, I’ll explain some of the work I’m doing as a member of the Indiana Citizens Redistricting Commission and how you can help us!
July 6 11:00am Workshop: Computing with Sage/Mathematica/Maple
July 8 10:30am Week 5 progress report presentations
July 13 11:00am Workshop: Coping with imposter syndrome
July 15 11:00am Seminar: Victoria Noquez (Indiana University)

Title: Fractals as final coalgebras

Abstract: In this talk we will explore the development of a surprising connection between category theory and fractal sets. Final coalgebras are a category theoretic construction which provide a useful way to capture continuous information (such as infinite streams). The connection between fractal sets and final coalgebras was first explored by Freyd, who showed that the unit interval is a final coalgebra of a certain endofuctor on the category of bipointed sets, and this work was further developed by Leinster. Bhattacharya, Moss, Ratnayake, and Rose took this in a different direction by considering categories whose objects are metric spaces, and showed that the Sierpinski Gasket is Bilipschitz equivalent to a final coalgabra of an appropriately chosen endofunctor. Most recently, we have extended this work to consider the Sierpinski Carpet, which requires a substantially different technical framework to accomodate a fractal which occurs via gluing copies of a set along line segments (rather than the Sierpinski Gasket, which can be constructed by gluing at points). We will introduce final coalgebras and outline the development of machinery to consider different fractal sets.
July 20 11:00am Workshop: Graduate school and career preparation, with guests Havi Ellers (Ph.D. University of Michigan 2025) and Casey Garner (Ph.D. University of Minnesota 2024)
July 27 11:00am Workshop: Research plans and personal statements
July 28 1:00pm Final presentation @ Indiana Undergraduate Math Research Conference (Link)
July 29 11:00am Seminar: Stephen Oloo (Kalamazoo College)

Title: You've been doing geometry wrong!?

Abstract: Join me in traveling to infinity. Literally. We will do this by getting acquainted with projective space and learning how it is a better setting for doing geometry than our more familiar euclidean space. Along the way we will encounter weird facts like how circles and parabolas are actually the same thing, and learn the meanings of fancy mathematical notions like 'compactification' and 'moduli space'.

### Summer 2020

Below are the schedule of activities for the summer of 2020. All activities will be held in Microsoft Teams unless otherwise noted. The 2020 program was organized by Dr. Kenji Kozai.

Date Time (EDT) Description
June 1 10:00am Introduction and orientation
June 2 11:00am Workshop: Literature search and reading
June 4 11:30am Technical overview and get-together
June 9 11:00am Workshop: Mathematical writing in LaTeX/Overleaf
June 11 11:30am Week 2 progress report presentations
June 16 11:00am Workshop: Computing with Sage/Mathematica/Maple
June 18 11:30am Seminar: Aamir Rasheed (Rose-Hulman Institute of Technology)
Title: A brief introduction to Morse theory
Abstract: Morse theory is a very beautiful area of topology. Roughly speaking, topology is the study of shape of a geometric object. By shape, we mean such properties of geometric objects that remain unchanged if the object is deformed continuously. A basic goal of topology, is to distinguish geometric objects. Sometimes this is easy. We all know the difference between a donut and a sphere; one has a hole, and the other doesn’t. But to distinguish more complicated objects, subtler tools are needed. A lot of such tools are provided by Morse theory. The basic idea in Morse theory is that the shape of a space can be studied by studying the functions defined on that space. In this talk, we will discuss and illustrate this idea by means of some examples.
June 23 11:00am Workshop: Giving a mathematics talk
June 25 11:30am Seminar: Ivan Ventura (Cal Poly Pomona)
Title: Inverse Problems: An Introduction
Abstract: In this talk we be introduced into the wide world of inverse problems. Inverse problems is a general categorization problems that require you to do the “opposite” of what is typically done. We will focus inverse spectral problems, which attempt to recover geometric or other properties from the spectrum (read “eigenvalues”) of a linear transformation. Specifically I will introduce the famous “Can you hear the shape of the drum?” problem.
June 30 11:00am Workshop: Forming a research plan/proposal
July 2 11:30am Week 5 progress report presentations
July 7 11:00am Workshop: Graduate school and career preparation, with guests Max Hlavacek (Ph.D. UC Berkeley 2022) and Alvin Moon (Ph.D. UC Davis 2020)
July 9 11:30am Seminar: Katie Ansaldi (Wabash College)
Title: Rainbow numbers of Zn for Linear Equations in Three Variables
Abstract: An exact r-coloring of a set S is a surjective function c : S → [r]. The rainbow number of a set S for an equation eq is the smallest integer r such that every exact r-coloring of S contains a rainbow solution to eq, that is, a solution in which no two elements have the same color. In this presentation, we discuss the rainbow numbers of Zn, for the equation a1 x1 + a2 x2 + a3 x3 = b.
July 14 11:00am Workshop: Coping with imposter syndrome
July 16 11:30am Seminar: Lauren Lazarus (Wentworth Institute of Technology)
Title: Modeling Oscillations with Delayed Feedback
Abstract: In many physical systems, a delay occurs between events and their effects: information takes time to travel, or a system needs time to process information and react accordingly. Unfortunately -- and fortunately! -- when we model these systems with differential equations, the delay causes the solution space to have infinite dimensions and exhibit a deeper set of possible behaviors. We'll discuss some intuitive reasoning and results around a simplistic delay differential equation that shows cyclic behavior.
July 21 11:00am Workshop: Writing papers and the publication process
July 23 11:30am Final presentation session (date/time subject to change)