## “Research in the areas of knot theory, quantum topology and number theory”

**James M Flaherty Visiting Professorship 2018/19 Visit Report**

**Name: ** Dr Robert Osburn

**Home University: ** University College Dublin

**University visited: ** McMaster University, Ontario

**Dates of Visit: ** May – August 2019

**Field of Study:** Mathematics

**Purpose of Visit:**

The purpose of my visit was to conduct research in the areas of knot theory, quantum topology and number theory, build partnerships with mathematicians at McMaster University and co-organize an undergraduate research project at the Fields Institute.

**Visit Details:**

Mathematics provides a universal language for expressing abstractions and plays a central role in solving problems from every imaginable domain of application. Even the most theoretical mathematical ideas, born from natural curiosity and an aesthetic criteria of simplicity, beauty and clarity, have unplanned and unexpected impacts in a vast array of areas.

Knots are objects which appear in nature, science and the arts. They pose particularly beautiful and difficult challenges, e.g., constructing rigorous tests (“invariants”) for determining whether two knots are equivalent. Quantum knot invariants have their origin in the seminal works of two Fields medalists, Vaughan Jones (1984) and Edward Witten (1988) and have fundamental applications in numerous interdisciplinary research areas. A modular form is an analytic object with intrinsic symmetric properties. The study of modular forms has enjoyed long and fruitful interactions with number theory, algebraic geometry, combinatorics, physics and string theory.

Over the past two decades, there have been many striking interactions between quantum knot invariants and modular forms.

For example, quantum invariants of 3-manifolds are typically functions that are defined only at roots of unity and one can ask whether they extend to holomorphic functions on the complex disk with interesting arithmetic properties. A first result in this direction was found in 1999 by Lawrence and Zagier who showed that Ramanujan’s 5th order mock theta functions coincide asymptotically with Witten-Reshetikhin-Turaev (WRT) invariants of Poincare homology spheres. More recently, Zagier has found experimental evidence of modularity properties of a new type for the Kashaev invariants of knots. This “quantum modularity conjecture” implies one of the major outstanding open problems in this field, namely the Volume conjecture. This latter conjecture relates the Kashaev invariant, which is a specifc value of the Nth colored Jones polynomial of a knot (a knot invariant that assigns to each knot a sequence of polynomials indexed by a positive integer N), to its hyperbolic volume and, if true, would give striking relations between hyperbolic geometry, quantum topology and modular forms. Other examples of such intriguing interactions can be found in the construction of new mock theta functions from WRT invariants, the stability of the coefficients of the colored Jones polynomial and generalized quantum modular q-hypergeometric series.

From May 19 to August 9, 2019, I visited McMaster University in Hamilton, Ontario, Canada. There were two main goals. The first goal was to collaborate with Professors Hans Boden and Andrew Nicas and Dr. Will Rushworth (all at McMaster University) in an effort to better understand the underlying theoretical framework behind recent advances in knot theory and quantum topology. The second goal was to co-organize (with Boden and Rushworth) the project entitled “An exploration of quantum invariants of knots and modularity” as part of the Fields Institute for Research and Mathematical Sciences Undergraduate Summer Research Program, see http://www.fields.utoronto.ca/activities/19-20/2019-fusrp.

The aim of this program is to provide a high-quality and enriching mathematics research experience for undergraduates. We supervised the following outstanding students over a period of nine weeks: Colin Bijaoui (McMaster University), Beckham Myers (Harvard University), Aaron Tronsgard (University of Alberta) and Shaoyang Zhou (Vanderbilt University). The main objective of the project was to perform knot theoretic calculations to explicitly construct new infinite families of q-hypergeometric series arising from computations of the colored Jones polynomial. We also focused on number theoretic properties of closely related q-hypergeometic series with a view towards proving congruences, duality and quantum modularity. As part of this project, I gave four one and a half hour lectures on the necessary background material.

In addition, I gave the following talks:

(1) On May 28 and 30th, two one hour research talks entitled “q-series, knots and modular forms” and “Generalized Kontsevich-Zagier series via knots” in the Geometry and Topology seminar at McMaster University,

(2) On June 19, a one hour talk entitled “Generalized Kontsevich-Zagier series via knots” in the Number Theory seminar at Queen’s University (Kingston, Ontario). A collaborative research project concerning topics in algebraic number theory was discussed with my host, Professor Ram Murty,

(3) On July 24, an invited one hour Colloquium talk entitled “Generalized Kontsevich-Zagier series via knots” at Oregon State University (Corvallis, USA). A joint research project on q-series and quantum modular forms was also formulated with my host, Professor Holly Swisher,

(4) On August 6, a one hour talk entitled “Ramanujan, partitions and knots” for undergraduates at McMaster University.

During my stay in Canada, I was also invited to visit the prestigious Korea Institute for Advanced Study in Seoul, South Korea. This visit took place from August 13-17, 2019, immediately after my stay in Canada. A joint research project on knots, q-series and quantum modularity is in preparation with my host, Professor Byungchan Kim (SeoulTech). The visit culminated in the one-day workshop on August 16 entitled “2019 2nd q-day: q-series and related topics”, see http://events.kias.re.kr/h/qDay2/?pageNo=3874.

The aim of this workshop was to provide a forum to discuss recent developments in q-series and related topics as well as to encourage synergy among q-series researchers in various backgrounds with a concentration on interactions between physics, quantum topology, geometry and number theory. It was an honor to represent the ICUF as the plenary speaker of this workshop.

**Future Continuing Collaboration:**

I will return to McMaster University to give the James M. Flaherty Visiting Professorship public lecture on September 27, 2019. The talk is entitled “Knots, modularity and beyond”. During this visit, I will continue discussions with Boden and Rushworth concerning the preparation of the outputs of the research conducted with the undergraduates at the Fields Institute. We fully expect this work to lead to high-quality research publications. In addition, future research projects and plans to visit Ireland are underway.

**Conclusion:**

My research visit to McMaster University was extremely productive. I am very grateful to the Ireland Canada University Foundation for this opportunity.