I am an RTG postdoc in the Mathematics Department at UCLA, working with Ko Honda.
From 2021-2024 I was a postdoc in the Mathematics Department at Michigan State University. My faculty mentor was Efstratia Kalfagianni. In 2021 I finished my PhD in Mathematics at UC Santa Barbara. My advisor was Stephen Bigelow.
Email: higginsv[at]math.ucla.edu
Research
My research interests focus on interactions between low-dimensional topology and algebra, in a field known as quantum topology. 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
Summer 2025 REU (RTG Geometry/Topology REU at UCLA)
In Summer 2025 I supervised several projects for a Research Experience for Undergraduates (REU) as part of the Geometry and Topology RTG group at UCLA. The title of the program was "Quantum Topology of Knots and Links." The program was overseen by faculty member Ko Honda. Additional mentors included graduate students Roman Krutowski, Jung Joo Suh, and Yan Tao. There were 8 undergraduate students in the REU program: Cole Ellison (UC Santa Barbara), Mark Sandey (UC Riverside), Bryan Silva (UC Irvine), Amy Somers (UC Santa Barbara), David Stephens (UC San Diego), Srijan Sundar (Oxford), Andrew Sylvester (UC Santa Barbara), and Parth Wokhlu (U Chicago). In Fall 2025, Amy and Parth presented a poster at the Undergraduate Mathematics Symposium at UIC and Amy, Andrew, and Cole presented a talk in the UCSB Topology Seminar. In January 2026, at the Joint Mathematics Meeting in Washington D.C., students presented a talk and 3 posters.
Summer 2023 REU (SURIEM REU at MSU)
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 was published in the journal Algebraic & Geometric Topology. The students gave presentations about their work at JMM 2024 (poster) in San Francisco, the SUMMR Conference at University of Michigan-Dearborn, the UnKnot V Conference at Seattle University, and the Young Mathematicians Conference at The Ohio State University.
Conference and Seminar Organization:
With Rhea Palak Bakshi, Jozef Przytycki, and Ka Ho Wong, I organized an AMS Special Session on Recent Progress in Quantum Topology at the Joint Mathematics Meeting (JMM) in Washington, D.C. Jan 4-7, 2026.
With Rhea Palak Bakshi, Dionne Ibarra, and Sujoy Mukherjee, I organized an AMS Special Session on Knot Theory and Quantum Topology at the AMS Spring Western Sectional Meeting at Cal Poly, San Luis Obispo, CA May 3-4, 2025.
With Elijah Bodish, Artem Kalmykov, Kang Lu, and Haihan Wu, I organized an AMS Special Session on Recent Progress on Categorification and Quantum Groups at the AMS Spring Eastern Sectional Meeting at UConn, Hartford, CT April 5-6, 2025.
From 2021-2024 I was one of the organizers of the MSU Topology Seminar and the MSU RTG Seminar.
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.