# Undergraduate Mathematics SymposiumSaturday, October 24, 2015University of Illinois at Chicago

Organized by Alex Austin, David Dumas, Steven Hurder, and Kevin Tucker

The Undergraduate Mathematics Symposium at UIC is an annual one-day meeting focusing on undergraduate mathematical research and education. The meeting features invited lectures by mathematical researchers, as well as contributed lectures and posters by undergraduates on their own research projects.

The 2015 UMS will be held at UIC on Saturday, October 24.

### Plenary lectures

• Chelsea Walton (Temple University)
Hamilton's quaternions
Abstract: In this talk I will discuss the last great achievement of Sir William Rowan Hamilton- the discovery of the quaternion number system. This discovery was very controversial for its time and nearly drove Hamilton mad! The talk will be full of drama, intrigue, and wonderful mathematics. Some familiarity with the complex number system would help, but is not needed.
• Henry Segerman (Oklahoma State University)
How to make sculptures of 4-dimensional things
Abstract: Our brains have evolved in a three-dimensional environment, and so we are very good at visualising two- and three-dimensional objects. But what about four-dimensional objects? The best we can really do is to look at three-dimensional "shadows". Just as a shadow of a three-dimensional object squishes it into the two-dimensional plane, we can squish a four-dimensional shape into three-dimensional space, where we can then make a sculpture of it. If the four-dimensional object isn't too complicated and we choose a good way to squish it, then we can get a very good sense of what it is like. We will explore the sphere in four-dimensional space, the four-dimensional polytopes (which are the four-dimensional versions of the three-dimensional polyhedra), and various sculptures, puzzles, and virtual reality experiences that have come from thinking about these things.
Density in Apollonian packings
Abstract: We describe how to compute different notions of density in Apollonian circle and sphere packings, and an unexpected connection to the Riemann Hypothesis. This is work that grew out of an IGL project with Jason Hempstead, Kaiyue Hou, Danni Sun, and Joe Vandehey, and also encompasses joint work with Cristian Cobeli and Alexandru Zaharescu.

### Attending the symposium

Everyone is welcome to attend the symposium. To help with our planning, we ask that attendees complete the registration form.

### Location

All symposium events will take place in the Science and Engineering Offices building (SEO) on UIC's East Campus. Registration, lunch and coffee will be provided in room 300, while the lectures will take place in room 636.

### Schedule of events

Plenary lectures are 50 minutes and student lectures 20 minutes; breaks of 10 minutes between talks allow for questions and discussion. A catered lunch of sandwiches and salads is provided for all symposium participants.

 8:15 - 8:50am Sign-in and coffee in SEO 300 Morning session — Plenary Lectures — SEO 636 8:55am Opening remarks 9:00am Jayadev Athreya (UIUC) — Density in Apollonian packings 10:00am Chelsea Walton (Temple University) — Hamilton's quaternions 11:00am Henry Segerman (Oklahoma State University) — How to make sculptures of 4-dimensional things 12:00pm Lunch in SEO 300 Afternoon Session 1 — Student lectures — SEO 636 1:00pm Yao-Rui Yeo (University of Minnesota) — The Hurwitz Existence Problem 1:30pm John Spaw (Millikin University) — Locating Numbers and Sets for Disconnected Graphs 2:00pm Alex Page (UIUC) — The Broken Stick Problem in Higher Dimensions: From a Classic Puzzle to Modern Distance Geometry 2:30pm Coffee break in SEO 300 Afternoon session 2 — Student lectures and Poster Session — SEO 636 and SEO 300 3:00pm Elena Malloy (Yale University) — Quasi-synchronization on a fractal network of logistic maps 3:30pm Ben Lowe (University of Chicago) — Configuration Spaces of Hinge Structures and Singularities 4:00 - 5:00pm Poster Session in SEO 300