Aug. 7-11, 2017

Georgia Institute of Technology, Atlanta, GA

Photo credit: Joe Rabinoff

This is a week-long workshop for undergraduate math majors featuring two mini-courses, problem sessions, faculty talks, as well as graduate school and career panels. Participants will also have the opportunity to give a short talk on their research and / or a poster presentation. Funding is available to help cover travel and local accommodations.

All activities will take place on the ground floor of Skiles Classroom Building in room 005 and the adjacent atrium.

## Mini-courses:

### Geometric approaches to data science (Allison Gilmore, One Medical and Joseph Ross, SignalFx)

This mini-course will describe an approach to data science that combines a geometric perspective on traditional machine learning algorithms with methods arising directly from geometric and topological data analysis. No background in data science or programming is required.

Supporting documents are available at https://github.com/alligilmore-sum/sum_data_science.

### Bridge trisections and knotted surfaces in four-dimensional space (Jeff Meier, University of Georgia; TA Marla Williams, University of Nebraska, Lincoln)

Is it possible to tie a sphere in a knot? Can a knotted circle be the boundary of a disk? The answers to these questions are "No" if we restrict our attention to our familiar three-dimensional setting. However, the answers become "Yes" once we allow ourselves the extra room provided by four-dimensional space. This raises new questions: What does a knotted sphere look like? How do we visualize it? Which knotted circles bounds disks? What else happens in four-dimensional space?

This mini-course will give an introduction to the theory of knotted surfaces in four-space, and the main tool employed to help us understand these counter-intuitive objects will be the recently discovered bridge trisection theory. A bridge trisection is a decomposition of a knotted surface into three simple pieces. This decomposition makes it much easier for us to visualize and study these complicated objects. In fact, we will see that bridge trisections allow us to represent any knotted surface using diagrams that can easily be manipulated and studied, drawing connections will well studied areas of classical knot theory such as bridge splittings and braids.

We'll begin by classifying surfaces, with an eye towards bridge trisections. Then, we'll discuss bridge splittings and braidings for knots and links. After that, we'll give an overview of knotted surface theory and explore lots of examples, with a focus on closed surface knots. Finally, we'll discuss the case of properly embedded surfaces in four-dimensional space and introduce an adaptation of bridge trisections in this setting. Throughout, we'll draw connections to the study of classical knots and braids.

## Faculty Talks:

Speaker: Sabetta Matsumoto (Georgia Tech)

Title: Non-euclidean virtual reality

Joint work with Vi Hart, Andrea Hawksley, and Henry Segerman

Abstract: The properties of euclidean space seem natural and obvious to us, to the point that it took mathematicians over two thousand years to see an alternative to Euclid’s parallel postulate. The eventual discovery of hyperbolic geometry in the 19th century shook our assumptions, revealing just how strongly our native experience of the world blinded us from consistent alternatives, even in a field that many see as purely theoretical. Non-euclidean spaces are still seen as unintuitive and exotic, but with direct immersive experiences we can get a better intuitive feel for them. The latest wave of virtual reality hardware, in particular the HTC Vive, tracks both the orientation and the position of the headset within a room-sized volume, allowing for such an experience. We use this nacent technology to explore the three-dimensional geometries of the Thurston/Perelman geometrization theorem. This talk focuses on our simulations of H³ and H²×E.

Speaker: Candice Price (University of San Diego)

Title: The Tangle Model: An Application of Topology to Biology

Abstract: The tangle model was developed in the 1980’s by professors DeWitt Sumner and Claus Ernst. This model uses the mathematics of tangles to model protein-DNA binding. An n-string tangle is a pair (B,t) where B is a 3-dimensional ball and t is a collection of n non-intersecting curves properly embedded in B. N-string tangles are formed by placing 2n points on the boundary of B, and attaching n non-intersecting curves inside B. Tangles, like knots and links, are studied through their diagrams. In the tangle model for DNA site-specific recombination, one is required to solve simultaneous equations for unknown tangles which are summands of observed DNA knots and links. This discussion will give a review of the tangle model including definitions.

Speaker: Margaret Symington (Mercer University)

Title: Experiencing geometries in 4-space

Abstract: Mathematical definitions and the theorems that rely on them are by their nature precise and abstract. A deeper understanding of mathematical objects can come as one tries to get up close and experience the objects, asking questions, making observations, even using analogies. In this talk I will attempt to give a ``feel" for 4-space equipped with each of three geometries--complex, symplectic and Euclidean. That ``feel" will be gained by exploring consequences and implications of the abstract definitions using examples, thought ​experiments and visualization.

## Career Panel:

Allison Gilmore (One Medical)

Dan Margalit (Georgia Tech)

Candice Price (University of San Diego)

Joseph Ross (SignalFx)

Margaret Symington (Mercer University)

Mohammad Ghomi (Graduate Chair at Georgia Tech): Overview of Georgia Tech PhD Program

Yuchen (Roy) He (Georgia Tech)

Peter Lambert-Cole (PhD from Louisiana State University; postdoc at Georgia Tech)

Samantha Petti (Georgia Tech)

Marla Williams (University of Nebraska, Lincoln)

## Organizers:

Jennifer Hom (Georgia Tech)

Caitlin Leverson (Georgia Tech)

Tye Lidman (North Carolina State University)

Supported by NSF CAREER Grant DMS-1552285