Vijay Higgins

I am a postdoctoral research associate in the Mathematics Department at Michigan State University. My faculty mentor is Efstratia Kalfagianni. In 2021 I finished my PhD in Mathematics at UC Santa Barbara. My advisor was Stephen Bigelow.

Research

My research interests focus on interactions between low-dimensional topology and algebra.  I am interested in link invariants that arise from quantum groups, and generalizations of these invariants. I study skein modules of 3-manifolds with a particular focus on the algebra structure associated to the skein module of a thickened surface. These produce invariants of 3-manifolds as well as invariants of links in thickened surfaces or, more generally, invariants of ribbon graphs in thickened surfaces. The topological features of these skein modules are intimately related to the algebraic structure of quantum groups. I am interested in obtaining topological definitions of quantum groups and their representation categories, canonical bases, and topological realizations of important algebraic objects, such as Lusztig's Frobenius map.

On this site you can find my papers and CV.

Activities

Seminars:

 I am currently one of the organizers of the MSU Topology Seminar and the MSU RTG Seminar.

Summer 2023 REU

In Summer 2023 Joseph Melby and I supervised a Research Experience for Undergraduates (REU) as part of the SURIEM REU program hosted by the Lyman Briggs College at Michigan State University. The project members were Bodie Beaumont-Gould (Lewis & Clark College), Erik Brodsky (Michigan State University), Alaina Hogan (Grand Valley State University), and Joshua Piazza (Wheaton College). A paper resulting from the project is now available on the arXiv. You might find the project members giving presentations about their work at conferences such as JMM 2024 (poster) in San Francisco, the SUMMR Conference at University of Michigan-Dearborn, or the Young Mathematicians Conference at The Ohio State University.

Previous Organization:

In 2022, I organized with Rhea Palak Bakshi and Wade Bloomquist an AMS Special Session on Skein Theory and Quantum Algebra for the Joint Mathematics Meeting (JMM 2022).

In Summer 2021 I organized with Wade Bloomquist and Sanjay Kumar the Remote Rendezvous for Quantum Topologists (RRQT) , which was an asynchronous conference held during the week of August 9-13, 2021.

During the 2020-2021 academic year I organized UCSB's Quantum Algebra and Topology Seminar.