Research

I'm fascinated by how quantum physics gives rise to beautiful and unexpected patterns in mathematics.
My research explores geometry, topology, and algebra inspired by quantum field theory and string theory.
More specifically, my research interests include:

A detailed summary of some of my projects can be found in my Research Statement.

Papers

My papers on arXiv
My Google Scholar page
My MathSciNet profile

In preparation

Publications and preprints


  1.   preprint [arXiv:2601.18042]
    We study BPS $q$-series associated to 3-manifolds decorated by a line defect along an embedded link. We prove that these $q$-series depend only on the class of the link in the skein module, thereby defining a homomorphism from the skein module to the space of $q$-series. The image of this homomorphism is conjectured to be holomorphically quantum modular, which suggests a new approach to Langlands duality for skein modules through $q$-series.


  2. Generalized Global Symmetries of T[M] Theories: Part II (by Sergei Gukov, Po-Shen Hsin, and Du Pei)
      preprint [arXiv:2511.13696]
    In this short appendix titled "\(\widehat{Z}\) for knot complements and the volume conjecture", I formulate the volume conjecture for the \(\widehat{Z}\)-invariant for knot complements and prove (the exponential part of) it for the figure-eight knot.


  3. (with Tobias Ekholm, Pietro Longhi, and Vivek Shende)
      preprint [arXiv:2510.19041]
    Given a 3-manifold \(M\), and a branched cover arising from the projection of a Lagrangian 3-manifold \(L\) in the cotangent bundle of \(M\) to the zero-section, we define a map from the skein of \(M\) to the skein of \(L\) via the skein-valued counting of holomorphic curves. When \(M\) and \(L\) are products of surfaces and intervals, we show that wall crossings in the space of the branched covers obey (a skein-valued lift of) the Kontsevich-Soibelman wall-crossing formula. Holomorphic curves in cotangent bundles correspond to Morse flow graphs; in the case of branched double covers, this allows us to give an explicit formula for the skein trace. After specializing to the case where \(M\) is a surface times an interval, and additionally specializing the HOMFLYPT skein to the \(\mathfrak{gl}(2)\) skein on \(M\) and the \(\mathfrak{gl}(1)\) skein on \(L\), we recover an existing prescription of Neitzke and Yan.


  4. (with Samuel Panitch)
      preprint [arXiv:2509.09100]
    This paper studies the connection between the quantum trace map -- which maps the \(\mathfrak{sl}_2\)-skein module to the quantum Teichmüller space for surfaces and to the quantum gluing module for 3-manifolds -- and the quantum UV-IR map -- which maps the \(\mathfrak{gl}_2\)-skein module to the \(\mathfrak{gl}_1\)-skein module of the branched double cover. We show that the two maps are compatible in a precise sense, and that the compatibility map is natural under changes of triangulation; for surfaces, this resolves a conjecture of Neitzke and Yan. As a corollary, under a mild hypothesis on the 3-manifold, the quantum trace map can be recovered from the quantum UV-IR map, hence providing yet another construction of the recently introduced 3d quantum trace map.


  5. (with Piotr Kucharski, Pietro Longhi, Dmitry Noshchenko, and Piotr Sułkowski)
      preprint [arXiv:2508.09729]
    We propose a symmetrization relation between BPS quivers encoding 4d \(\mathcal{N}=2\) theories and symmetric quivers associated to 3d \(\mathcal{N}=2\) theories. We analyse in detail the symmetrization of BPS quivers for a series of \(A_m\) Argyres-Douglas theories by engineering 3d-4d systems in geometric backgrounds involving appropriate 3-manifolds and Riemann surfaces. We discuss properties of these geometric backgrounds and derive the corresponding quiver partition functions from the perspective of skein modules, which forms the foundation of the symmetrization map for the minimal chamber. We also prove that the structure of wall-crossing in 4d \(A_m\) Argyres-Douglas theories is isomorphic to the structure of unlinking of symmetric quivers encoding their partner 3d theories, which allows for a proper definition of the symmetrization map outside the minimal chamber. Finally, we show that the Schur indices of 4d theories are captured by symmetric quivers that include symmetrization of 4d BPS quivers.


  6. (with Rostislav Akhmechet and Peter K. Johnson)
      Quantum Topology (2025), published online first [journal | arXiv:2403.14461]
    We introduce an invariant of negative definite plumbed knot complements unifying knot lattice homology, due to Ozsváth, Stipsicz, and Szabó, and the BPS \(q\)-series of Gukov and Manolescu. This invariant is a natural extension of weighted graded roots of negative definite plumbed 3-manifolds introduced earlier by the first two authors and Krushkal. We prove a surgery formula relating our invariant with the weighted graded root of the surgered 3-manifold.


  7. (with Samuel Panitch)
      Algebraic & Geometric Topology, to appear [to appear | arXiv:2403.12850]
    We construct the 3d quantum trace map, a homomorphism from the Kauffman bracket skein module of an ideally triangulated 3-manifold to its (square root) quantum gluing module, thereby giving a precise relationship between the two quantizations of the character variety of ideally triangulated 3-manifolds. This map, whose existence was conjectured earlier by Agarwal, Gang, Lee, and Romo, is a natural 3-dimensional analog of the 2d quantum trace map of Bonahon and Wong. Our construction is based on the study of stated skein modules and their behavior under splitting, especially into face suspensions.


  8. (with Tobias Ekholm, Angus Gruen, Sergei Gukov, Piotr Kucharski, Marko Stošić, and Piotr Sułkowski)
      Journal of Geometry and Physics 177 (2022), 104520 [journal | arXiv:2110.13768]
    We generalize the \(F_K\) invariant, i.e. \(\widehat{Z}\) for the complement of a knot \(K\) in the 3-sphere, the knots-quivers correspondence, and \(A\)-polynomials of knots, and find several interconnections between them. We associate an \(F_K\) invariant to any branch of the \(A\)-polynomial of \(K\) and we work out explicit expressions for several simple knots. We show that these \(F_K\) invariants can be written in the form of a quiver generating series, in analogy with the knots-quivers correspondence. We discuss various methods to obtain such quiver representations, among others using \(R\)-matrices. We generalize the quantum \(a\)-deformed \(A\)-polynomial to an ideal that contains the recursion relation in the group rank, i.e. in the parameter \(a\), and describe its classical limit in terms of the Coulomb branch of a 3d-5d theory. We also provide \(t\)-deformed versions. Furthermore, we study how the quiver formulation for closed 3-manifolds obtained by surgery leads to the superpotential of 3d \(\mathcal{N}=2\) theory \(T[M_3]\) and to the data of the associated modular tensor category \(\text{MTC} [M_3]\).



  9.   preprint [arXiv:2106.03942]
    S. Gukov and C. Manolescu conjectured that the Melvin-Morton-Rozansky expansion of the colored Jones polynomials can be re-summed into a two-variable series \(F_K(x,q)\), which is the knot complement version of the 3-manifold invariant \(\hat{Z}\) whose existence was predicted earlier by S. Gukov, P. Putrov and C. Vafa. In this paper we use an inverted version of the R-matrix state sum to prove this conjecture for a big class of links that includes all homogeneous braid links as well as all fibered knots up to 10 crossings. We also study an inverted version of Habiro's cyclotomic series that leads to a new perspective on \(F_K\) and discovery of some regularized surgery formulas relating \(F_K\) with \(\hat{Z}\). These regularized surgery formulas are then used to deduce expressions of \(\hat{Z}\) for some plumbed 3-manifolds in terms of indefinite theta functions.


  10. (with Sergei Gukov and Pavel Putrov)
      Annales Henri Poincaré 22 (2021), 4173-4203 [journal | arXiv:2009.11874]
    Many BPS partition functions depend on a choice of additional structure: fluxes, Spin or Spin\(^c\) structures, etc. In a context where the BPS generating series depends on a choice of Spin\(^c\) structure we show how different limits with respect to the expansion variable \(q\) and different ways of summing over Spin\(^c\) structures produce different invariants of homology cobordisms out of the BPS \(q\)-series.


  11. (with Tobias Ekholm, Angus Gruen, Sergei Gukov, Piotr Kucharski, and Piotr Sułkowski)
      Communications in Mathematical Physics 396 (2022), 143-186 [journal | arXiv:2005.13349]
    Reducing a 6d fivebrane theory on a 3-manifold \(Y\) gives a \(q\)-series 3-manifold invariant \(\widehat{Z}(Y)\). We analyse the large-\(N\) behaviour of \(F_K = \widehat{Z}(M_K)\), where \(M_K\) is the complement of a knot \(K\) in the 3-sphere, and explore the relationship between an \(a\)-deformed (\(a=q^N\)) version of \(F_K\) and HOMFLY-PT polynomials. On the one hand, in combination with counts of holomorphic annuli on knot complements, this gives an enumerative interpretation of \(F_K\) in terms of counts of open holomorphic curves. On the other, it leads to closed form expressions for \(a\)-deformed \(F_K\) for \((2,2p+1)\)-torus knots. They suggest a further \(t\)-deformation based on superpolynomials, which can be used to obtain a \(t\)-deformation of ADO polynomials, expected to be related to categorification. Moreover, studying how \(F_K\) transforms under natural geometric operations on \(K\) indicates relations to quantum modularity in a new setting.


  12. (with Sergei Gukov, Po-Shen Hsin, Hiraku Nakajima, Du Pei, and Nikita Sopenko)
      Journal of Geometry and Physics 168 (2021), 104311 [journal | arXiv:2005.05347]
    By studying Rozansky-Witten theory with non-compact target spaces we find new connections with knot invariants whose physical interpretation was not known. This opens up several new avenues, which include a new formulation of \(q\)-series invariants of 3-manifolds in terms of affine Grassmannians and a generalization of Akutsu-Deguchi-Ohtsuki knot invariants.



  13.   Journal of Knot Theory and Its Ramifications 29 (2020), no. 14, 2050097 [journal | arXiv:2004.02087]
    The Gukov-Manolescu series, denoted by \(F_K\), is a conjectural invariant of knot complements that, in a sense, analytically continues the colored Jones polynomials. In this paper we use the large color \(R\)-matrix to study \(F_K\) for some simple links. Specifically, we give a definition of \(F_K\) for positive braid knots, and compute \(F_K\) for various knots and links. As a corollary, we present a class of `strange identities' for positive braid knots.


  14. (with Sungbong Chun, Sergei Gukov, and Nikita Sopenko)
      Journal of High Energy Physics 09 (2020), 152 [journal | arXiv:1911.08456]
    One of the main challenges in 3d-3d correspondence is that no existent approach offers a complete description of 3d \(N=2\) SCFT \(T[M_3]\) --- or, rather, a "collection of SCFTs" as we refer to it in the paper --- for all types of 3-manifolds that include, for example, a 3-torus, Brieskorn spheres, and hyperbolic surgeries on knots. The goal of this paper is to overcome this challenge by a more systematic study of 3d-3d correspondence that, first of all, does not rely heavily on any geometric structure on \(M_3\) and, secondly, is not limited to a particular supersymmetric partition function of \(T[M_3]\). In particular, we propose to describe such "collection of SCFTs" in terms of 3d \(N=2\) gauge theories with "non-linear matter'' fields valued in complex group manifolds. As a result, we are able to recover familiar 3-manifold invariants, such as Turaev torsion and WRT invariants, from twisted indices and half-indices of \(T[M_3]\), and propose new tools to compute more recent \(q\)-series invariants \(\hat Z (M_3)\) in the case of manifolds with \(b_1 > 0\). Although we use genus-1 mapping tori as our "case study," many results and techniques readily apply to more general 3-manifolds, as we illustrate throughout the paper.



  15.   Symmetry, Integrability and Geometry: Methods and Applications (SIGMA) 16 (2020), 044, 17 pages [journal | arXiv:1909.13002]
    We study \(q\)-series-valued invariants of 3-manifolds that depend on the choice of a root system \(G\). This is a natural generalization of the earlier works by Gukov-Pei-Putrov-Vafa [arXiv:1701.06567] and Gukov-Manolescu [arXiv:1904.06057] where they focused on \(G={\rm SU}(2)\) case. Although a full mathematical definition for these ''invariants'' is lacking yet, we define \(\hat{Z}^G\) for negative definite plumbed 3-manifolds and \(F_K^G\) for torus knot complements. As in the \(G={\rm SU}(2)\) case by Gukov and Manolescu, there is a surgery formula relating \(F_K^G\) to \(\hat{Z}^G\) of a Dehn surgery on the knot \(K\). Furthermore, specializing to symmetric representations, \(F_K^G\) satisfies a recurrence relation given by the quantum \(A\)-polynomial for symmetric representations, which hints that there might be HOMFLY-PT analogues of these 3-manifold invariants.


Expository articles



PhD Thesis


A piece of recreational mathematics


Mathematica files