### 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) |