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 2019 I am teaching three sections of Math 111 - Calculus I; I am currently scheduled to teach two more sections of Math 111 and one section of Math 221Q in the spring semester. I am also overseeing one Oxford Research Scholar throughout the academic year, for 2 credit hours per term.
In past academic years at Oxford I have taught Math 111 - Calculus I (5 sections overall), Math 110 - Transition to Calculus (3 sections overall), Math 212 - Differential equations (3 sections overall) and Math 221Q - Linear algebra (1 section).
Publications
- Virtual Strings and Free knots, to appear in "A concise encyclopedia of knot theory", under review.
- The Multi-variable affine index polynomial, submitted for publication. Preprint available on the arXiv.
- The Wriggle polynomial for virtual tangles, submitted for publication. Preprint available on the arXiv.
- Index Polynomials for virtual tangles, published in Journal of Knot Theory and its Ramifications, 27 (12), 2018.
- Finite-type Invariants of long and framed virtual knots, published in Journal of Knot Theory and its Ramifications, Jun 2019.
Emory students can access the online version of the Journal of Knot Theory and its Ramifications at this link (requires library login).
Presentations
Upcoming presentations
Joint Mathematics Meeting: Contributed paper session, Denver, CO, January 2020
Past presentations
- Knots in Washington XLVIII, Washington DC, May 2019
- Spring AMS Sectional Meeting: Special Session on Invariants of Knots, Links and Low-dimensional Manifolds, Hartford, CT, April 2019
- Joint Mathematics Meeting: Special Session on Algebraic Structures motivated by Knot Theory, Baltimore, MD, January 2019
- 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, and also given a couple of generalizations of the Affine Index Polynomial.
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