Nicolas Petit

Visiting Assistant Professor and Teaching Fellow of Mathematics

Welcome to my directory page! I'm a Visiting Professor of Mathematics here at Oxford College, where I've been since 2017.

For more information about my research and teaching, feel free to check out my CV, my Research Statement and my Teaching statement.

If you're an undergraduate student who successfully took Math 221 (linear algebra), and you're interested in learning more about knots from a mathematical standpoint (possibly for a summer research project), feel free to contact me.


Education

PhD| Dartmouth College| 2016

MA| Dartmouth College| 2013

MA| Università degli Studi di Torino| 2011

BA| Università degli Studi di Torino| 2009

Courses Taught

In the fall of 2018 I will be teahing two sections of Math 110 - Calculus with Precalculus, as well as one section of Math 221Q - Linear Algebra.

In the Academic Year 2016-17 I taught 5 sections of Math 111 - Calculus I (3 in the fall, 2 in the spring) and one section of Math 212 - Differential Equations.

Publications

Virtual Strings and Free knots, to appear in "A concise encyclopedia of knot theory", under review.

Index Polynomials for virtual tangles. Submitted for publication. Preprint available at arXiv:1805:08178 

Finite-type Invariants of long and framed virtual knots. Submitted for publication. Preprint available at arXiv:1610.03825 

Presentations

Upcoming presentations

Joint Mathematics Meeting: Special session on Not KNerds, Baltimore, MD, January 2019

Past presentations 

Knots in Washington XLVI, Washington D.C., May 2018

Spring AMS Sectional Meeting: Special Session on Algebraic, Combinatorial and Quantum invariants of Knots and Manifolds, Columbus, OH, March 2018

Fall AMS Sectional Meeting: Special Session on Algebraic and Combinatorial Structures in Knot Theory, Riverside, CA, November 2017

Spring AMS Sectional Meeting: Special Session on Knot Theory and its applications, Charleston, SC, March 2017

Spring AMS Sectional Meeting: Special Session on Algebraic Structures in Knot Theory, Athens, GA, March 2016 

Joint Mathematics Meeting: Special Session on Knots in Washington (State), Seattle, WA, January 2016

Fall AMS Sectional Meeting: Special Session on Algebraic and Combinatorial Invariants of Knots, Chicago, IL, November 2015 


Research Interests

I am a knot theorist, mainly interested in combinatorial invariants of knots. My thesis work revolved around generalizing three Vassiliev invariants of order one for virtual knots, first defined by A. Henrich, to the case of long and framed virtual knots; since then, I've generalized the polynomial invariant to the case of virtual tangles.

I am currently studying the properties of the new invariants I've defined, especially with regards to connected sum of tangles. I am also investigating GPV invariants of virtual knots, and their relation to Vassiliev invariants: it is known that every GPV invariant gives rise to a Vassiliev invariant, but little is known about the space of such extensions. With some collaborators, I'm also exploring the definition of GPV invariants for virtual string links (a special type of virtual tangles), as well as the relation between band-pass invariance and knot concordance for GPV invariants. 

I get intuition for my research using Mathematica code; I am currently developing code (based on Bar-Natan's code) that evaluates a Gauss Diagram formula (another way of defining GPV invariants) on a given virtual knot, and that computes the dimension of the space of Vassiliev extensions of GPV invariants. If you're interested in the code feel free to contact me at nicolas.petit@emory.edu