Research

Overview

My work in quantum and low-dimensional topology is focused on establishing connections between quantum invariants (such as the Reshetikhin-Turaev and Turaev-Viro invariants) of 3-dimensional manifolds and their geometric properties (such as hyperbolic volume).  Much of this work has been done in collaboration with Dr. Sanjay Kumar.

Low-dimensional manifolds are spaces of dimension 4 or lower that locally look like our normal world but may have different global shape and structure, and the study of low-dimensional manifolds is an active area of research in mathematics. The primary way we try to understand and distinguish manifolds is to associate to them algebraic and geometric properties called topological invariants to encode their shape and structure. In dimension 3, for example, the volume of a manifold is often an invariant encoding its geometric properties. Another example is the family of Turaev-Viro invariants (TV) of 3-dimensional manifold, which are motivated in some sense by quantum physics. Roughly speaking, the Turaev-Viro invariants of  a 3-dimensional manifold are a sequence of real numbers computed using combinatorial data from the manifold called a triangulation. 

At first glance, there is no reason to think that the combinatorial/quantum TV invariants  encode any particular geometric information about a manifold, but recent work (see CY15, DKY18, O18, WY20, BDKY18, B20, vdV08, W19, K20) established for several manifolds and families of manifolds that the sequence of TV invariants increases at an exponential rate proportional to the volume of the manifold. This deep connection between the combinatorial/quantum and geometric invariants of 3-dimensional manifolds is believed to hold for all 3-dimensional manifolds (see CY15) , and my research primarily involves making progress on this volume conjecture. 

Asymptotics of the TV Invariants for an Infinite Family

In joint work with Dr. Kumar (see KM21), we constructed infinite families of 3-dimensional manifolds using elementary building blocks pictured below which satisfy this volume conjecture and establish methods for creating new manifolds satisfying the conjecture. Our family of 3-dimensional manifolds can be represented by 2-dimensional diagrams, as shown by the following images. 

Left: Examples of an S-piece (left) and an A-piece (right), the elementary building blocks of our family of 3-dimensional manifolds.

Right: Example of a manifold constructed using S- and A-pieces that satisfies the volume conjecture. Slice this surface along the blue curves to obtain copies of the elementary pieces.

TV Invariants and Cable Spaces

In other joint work with Dr. Kumar (see KM22 ), we verify the volume conjecture for another infinite family of manifolds constructed through certain twisting and gluing operations on manifolds. Unfortunately, these operations do not lend themselves as well to diagrams for visualization. The primary takeaway from this paper is that the exponential growth rate of the TV invariants of a 3-dimensional space (which is conjectured to capture geometric properties of the space) does not change when that space undergoes a certain type of "geometry-preserving" gluing operation.

q-hyperbolic Manifolds

In work with my advisor, Dr. Kalfagianni (see KM23), we study the asymptotics of the TV invariants of infinite families of knots in the 3-sphere and some of their interesting geometric properties. We construct the first infinite families of hyperbolic knots in the 3-sphere where their TV invariants are known to grow at least exponentially. This exponential growth property is called q-hyperbolicity, and our constructions, in conjunction with experimentation by Futer-Purcell-Schleimer (see FPS23), allowed us to identify many low-crossing and low-volume knots from the various manifold censuses (see SnapPy) with this property.  Many of these knots have the following form:

Dehn surgery sequences taking a framed figure-eight knot to a framed double twist knot, which establishes that the TV invariants of these two families of knots (for m, n  nonzero integers) and their complements in the 3-sphere grow exponentially. These are the first infinite families of knots which have been shown to be q-hyperbolic. 

In addition, we apply our results to an open conjecture of Andersen-Masbaum-Ueno (see AMU06) asserting connections between certain algebraic structures associated to surfaces, called quantum representations, and the geometry of hyperbolic 3-manifolds constructed from those surfaces.

Genus g surface with mapping class corresponding to a composition of Dehn twists about each of the labeled curves satisfying the AMU Conjecture.

Overview

I am also interested in using topological tools such as persistent (co)homology and fiber bundle theory to analyze data. These tools have been used in a wide variety of settings by numerous mathematicians and data scientists. Much of the original theory is nicely contained in the book EH10, and ZC05,  C09,  G14, and P19 are also excellent resources  for learning about the theory of persistence and how it was originally developed for use in data science. 

Persistent (co)homology is an algebraic and computational tool which recovers topological features of data such as loops, holes, voids, etc. Of greatest interest to me has been its applications to the study of data from images (see PC14, CIDSZ08).  Efficient computation of persistence is important when analyzing large and/or high-dimensional datasets, and standard tools, such as Ripser and GUDHI, have been implemented in Python to aid in computation. 

Persistent Lifts of Persistent Cohomology Classes

My research in this field has largely been in collaboration with Dr. Hitesh Gakhar and Dr. Jose Perea. Following the constructions and results from P20 and PP20, we develop algorithms which refine (in a sense) the topological information traditionally obtained from computing persistence. We then apply this refinement procedure to algorithmically re-coordinatize potentially high-dimensional datasets into spaces that better reflect the shape of the dataset.