LSU

Mathematics

Mathematics

The goal of the LSU Mathematics Student Colloquium is to give both undergraduate and graduate students the opportunity to hear and interact with speakers from across the country, providing information and perspective possibly relevant to their graduate and postgraduate careers.

Each invited speaker will spend several days at LSU, giving multiple talks and making himself or herself available to undergraduates.

Talks are not confined to the math department but open to everyone. Those majoring in related fields are encouraged to come.

We are a charted LSU student organization (constitution and bylaws). We are munificently funded by the Student Government Programming, Support, and Initiatives Fund (PSIF), and the LSU Mathematics Department. We are grateful for the generous past funding made by grants from the National Science Foundation (a VIGRE grant) and the Board of Regents.

Fall 2018

**Joshua Sabloff**(October15-17, 2018)

Dr. Sabloff has research interests in Contact and Symplectic Geometry, especially Legendrian and Transversal Knot Theory. He received his PhD from Stanford University. After he spent a year as a postdoctoral lecturer at the University of Pennsylvania, he moved to Haverford. He is now an Associate Professor at Haverford College in Pennsylvania.

**Undergraduate Talk** ( 239 Lockett; Monday, October 15, 3:30-4:20 PM)

Title: How to Tie Your Unicycle in Knots: An Introduction to Legendrian Knot Theory

Abstract: You can describe the configuration of a unicycle on a sidewalk using three coordinates: two position coordinates x and y for where the wheel comes into contact with the ground and one angle coordinate t that describes the angle that the direction the wheel makes with the x axis. How are the instantaneous motions of the unicycle constrained (hint: do you want your tire to scrape sideways)? How can we describe that constraint using generalizations of tools from vector calculus?

The system of constraints at every point in (x,y,t)-space is an example of a "contact structure," and a path that obeys the constraints is a "Legendrian curve." If the curve returns to its starting point, then it is called a "Legendrian knot." A central question in the theory of Legendrian knots is: how can you tell two Legendrian knots apart? How many are there? In other words, how many ways are there to parallel park your unicycle?

There will NOT be a practical demonstration.

**Graduate Talk** ( 239 Lockett; Tuesday, October 16, 3:30- 4:20 PM)

Title: Non-Orientable Lagrangian Fillings of Legendrian Knots

Abstract: Lagrangian fillings of Legendrian knots are interesting objects that are related, on one hand, to the 4-genus of the underlying smooth knot and, on the other hand, to Floer-type invariants of Legendrian knots. Most work on Lagrangian fillings to date has concentrated on orientable fillings. I will present some first steps in constructions of and obstructions to the existence of (decomposable exact) non-orientable Lagrangian fillings. In addition, I will discuss links between the 4-dimensional crosscap number of a knot and the non-orientable Lagrangian fillings of its Legendrian representatives. This is joint work with Linyi Chen, Grant Crider-Philips, Braeden Reinoso, and Natalie Yao.