Papers
Published
Purity and 2-Calabi-Yau categories (to appear, Invent. Math)
In this paper I show that when a stack of objects in a 2-Calabi-Yau category has a good moduli space, the direct image of the constant mixed Hodge module complex along the morphism to the good moduli space satisfies the decomposition theorem, i.e. it is pure. This has quite a few applications, which I discuss in the second half of the paper, including Halpern-Leistner's conjecture on the purity of the Borel-Moore homology of the stack of coherent sheaves on a K3 surface, purity of the Borel-Moore homology of stacks of semistable Higgs bundles and global nilpotent cones, and a generalisation of the perverse filtration arising from the Hitchin system, from the case of coprime (r,d) to general (r,d).
The integrality conjecture and the cohomology of preprojective stacks (Crelle's Journal (2023) volume 804, pages 105-154)
In this paper I prove that the Borel-Moore homology of the stack of representations of a preprojective algebra is pure. This turns out to have a number of consequences: the paper also contains a proof of the Schiffmann-Vasserot conjecture that the (fully equivariant) CoHA embed in a noncommutative deformation of the shuffle algebra, the Bozec-Schiffmann-Vasserot conjecture on the positivity of modified Kac polynomials, and results on the vanishing cycle cohomology of Hilbert scheme of C^3.
Refined invariants of finite-dimensional Jacobi algebras (To appear, Algebraic Geometry)
In this paper I proved that the refined/motivic/cohomological BPS/GV invariants of finite-dimensional Jacobi algebras are always positive. The original goal of this project was to show exactly the opposite, and thereby disprove the conjecture of Brown and Wemyss that all such Jacobi algebras are the contraction algebras of flopping curves. So this paper, unwittingly, provides evidence for their conjecture.
Nonabelian Hodge theory for stacks and a stacky P=W conjecture (Adv. Math. (2023) Volume 415, 108889)
This paper presents a proposal for constructing a nonabelian Hodge isomorphism at the level of moduli stacks of Higgs bundles and local systems on a smooth compact curve. Furthermore, a version of the P=W conjecture is presented at the level of stacks, which is conjectured to be equivalent to both the original P=W conjecture and also the PI=WI conjecture. Everything is proved here in genus 0 and 1 (genus 1 is the first place where things start to get quite tricky), with higher genus being the starting point for further work (see above).
A boson-fermion correspondence in cohomological Donaldson-Thomas theory (Glasgow Math. J., Special edition for the British Mathematical Colloquium 2022, pages 1-25)
Here I used some of the deformed dimensional reduction techniques worked out in my earlier paper with Tudor Pădurariu to develop a "fermionization" procedure for the preprojective CoHA. Along the way I calculate the BPS cohomology of the central extensions of preprojective algebras introduced by Etingof and Rains, as well as the deformed preprojective algebras introduced by Crawley-Boevey and Van den Bergh.
Deformed dimensional reduction (With Tudor Pădurariu) (Geometry & Topology 2022, volume 26 number 2, pages 721-776)
This paper is about a generalization of dimensional reduction for vanishing cycle cohomology. In Donaldson-Thomas theory, the point of this generalization is to answer the question: what happens to refined DT invariants as we deform the potential. We use the general answer that we arrive at to prove a conjecture of Cazzaniga, Morrison, Pym, and Szendrői, as well as to connect work of Dobrovolska, Ginzburg and Travkin to the study of BPS cohomology.
Strong positivity for quantum theta bases of quantum cluster algebras (With Travis Mandel) (Invent. Math. 2021 volume 226 number 3, pages 725-843)
This paper is about quantum theta bases for quantum cluster algebras. We show that these deform the GHKK theta basis, and moreover satisfy quantum versions of positivity and strong positivity, and are even atomic with respect to the quantum scattering atlas, thus proving the strong positivity conjecture for quantum cluster algebras.
The local motivic DT/PT correspondence (With Andrea Ricolfi) (J. London Math. Soc. 2021 volume 104 number 3, pages 1384-1432)
In this paper we carry out an analysis of the virtual motive of the quot scheme of a line in affine three space, showing that in the relative+motivic setting, the virtual motive fits into a wall crossing formula that is a local refinement of the DT/PT correspondence
Cohomological Donaldson-Thomas theory of a quiver with potential and quantum enveloping algebras (with Sven Meinhardt) (Invent. Math. (2020) 221, pages 777–871)
In this paper we discuss and prove the categorical lift of the integrality conjecture in motivic DT theory. The equality in the usual integrality conjecture involving plethystic exponentials is replaced by an isomorphism involving symmetric algebras. The resulting theorem states that the critical cohomological Hall algebra of semistable modules of fixed slope possesses a (surprising) perverse filtration, which is respected by the product and the localised coproduct. The resulting associated graded object is free supercommutative. We furthermore upgrade the product form of the wall crossing formula found in the work of Kontsevich and Soibelman to a tensor product description for the entire cohomological Hall algebra.
Enumerating coloured partitions in 2 and 3 dimensions (With Jared Ongaro and Balázs Szendrői) (Mathematical Proceedings of the Cambridge Philosophical Society (2020) 169(3), 479-505)
This paper is part review, part conjecture, part partial proof of said conjecture, regarding coloured partitions in 2 and 3 dimensions.
Positivity for quantum cluster algebras (Annals of Math. 157-219 Volume 187 (2018), Issue 1)
Using the cohomological wall crossing formula, and building on the beautiful work of Kontsevich, Soibelman, Nagao and Efimov, in this paper I proved the quantum cluster positivity conjecture for skew-symmetric quantum cluster algebras.
Purity of critical cohomology and Kac's conjecture (Math. Res. Lett. (2018) Volume 25(2) 469 – 488)
A new proof of Kac's positivity conjecture using cohomological DT theory.
Cohomological Hall algebras and character varieties (Int. J. Math. (2016) Volume 27, Issue 07)
This paper was written for the proceedings of VBAC 2014 in Berlin. The paper demonstrates that the compactly supported cohomology of the genus g character stack has a Hall algebra structure coming from cohomological DT theory, via the theory of brane tilings of Riemann surfaces.
The Critical CoHA of a quiver with potential (Quarterly J. Math. (2017) Vol 68 (2) pp. 635-703)
In this paper I proved a couple of structural results on critical cohomological Hall algebras associated to quivers with potential, as defined by Kontsevich and Soibelman. The main result is the construction of a localised coproduct - this turned out later (see above) to be a key gadget in proving that the critical CoHA for a quiver with potential is a deformed universal enveloping algebra, and much later turned out to be an important ingredient in proving Okounkov's conjecture regarding Kac polynomials. Secondly, in the appendix, I upgrade the motivic dimensional reduction theorem of Behrend, Bryan and Szendrői to cohomology. This turns out to be an essential part in my reproof of the Kac conjecture, as well as providing the bridge to the cohomological Hall algebras considered by Schiffmann and Vasserot.
Purity for graded potentials and quantum cluster positivity (With Davesh Maulik, Jörg Schürmann and Balázs Szendrői) Compositio Math. (2015) Vol 151, No. 10, pp. 1913–1944
Here we prove the positivity conjecture for quantum cluster variables for quivers admitting nondegenerate quasihomogeneous potentials.
Motivic Donaldson-Thomas theory and the role of orientation data: Glasgow J. Math. (2016) Vol. 58, No. 1, pp. 229-262
This paper is a general introduction to the problem of orientation data, via an example for which the integration map built without OD fails to preserve associative products. This is the friendliest paper on integration maps here, and is an expanded version of a chapter of my thesis.
The motivic Donaldson-Thomas invariants of (-2) curves (with Sven Meinhardt) (Algebra and Number Theory (2017) Vol. 11, No. 6, 1243–1286)
We put some of the below theory into practice, calculating the DT invariants of everyone's second favourite minimal 3-fold flopping contraction.
Motivic DT-invariants for the one loop quiver with potential (with Sven Meinhardt) Geometry & Topology (2015), Vol. 19, pp. 2535–2555, (2015)
This paper calculates the motivic DT invariants for the Jacobi quiver with potential. It turns out this involves a lot of thinking about Looijenga's exotic product, and monodromy, so a lot of this kind of theory is spelt out in here. The answer turns out to be pretty simple, confirming a kind of motivic analogue of the integrality phenomenon in "usual" DT theory.
Superpotential algebras and manifolds (Adv. Math. (2012) Volume 231, Issue 2, Pages 879--912):
This paper is all about the question of whether fundamental group algebras of compact orientable manifolds without boundary are superpotential algebras. It's a kind of folklore result that they are Calabi-Yau algebras, so it's natural to suppose that the answer is yes, but it turns out the answer is no. Using the interplay between the Connes long exact sequence and the Gysin long exact sequence for an S^1 fibration it turns out this algebraic statement has a neat topological interpretation.
Consistency conditions for brane tilings (J. Algebra 338 (2011), pp. 1-23)
This paper concerns a construction that occurs as a sort of noncommutative analogue of toric geometry, called a brane tiling algebra. There's a way of thinking of these algebras in terms of simple homotopy theory, which seems to be quite powerful. In particular there's a proof in here that the resulting algebras (assuming the eponymous consistency conditions) are 3-Calabi-Yau.
Preprints
Okounkov's conjecture via BPS Lie algebras (with Tommaso Maria Botta)
In this paper we prove that for an arbitrary quiver Q the Maulik-Okounkov Lie algebra, which generates their version of the Yangian associated to Q via stable envelopes and Nakajima quiver varieties, is isomorphic to the BPS Lie algebra for the associated tripled quiver with its canonical cubic potential. As a consequence, we prove Okounkov's conjecture, stating that the graded dimensions of this Lie algebra are given by the coefficients of the Kac polynomials for Q.
BPS algebras and generalised Kac-Moody algebras from 2-Calabi-Yau categories (with Lucien Hennecart and Sebastian Schlegel Mejia)
For a very general type of 2-Calabi-Yau category with good moduli space, in our earlier paper we defined the BPS algebra, via the study of relative Hall algebras and perverse filtrations. In this paper we give a complete description of this algebra: it is one Half of the generalised Kac-Moody algebra defined by putting certain intersection cohomology spaces as spaces of Chevalley roots; in particular, spaces of generators at hyperbolic imaginary roots are identified with intersection cohomology of intersection cohomology of components of the coarse moduli space. Applications include the proof of the cohomological integrality theorem for 2CY categories, a Lie theoretic description of all coefficients of Kac polynomials, a proof of the Bozec-Schiffmann conjecture regarding positivity of cuspidal polynomials, and a new description of the total cohomology of Nakajima quiver varieties in terms of lowest weight modules for the BPS Lie algebra
BPS Lie algebras for totally negative 2-Calabi-Yau categories and nonabelian Hodge theory for stacks (with Lucien Hennecart and Sebastian Schlegel Mejia) (submitted)
This paper starts off by developing the general theory of cohomological Hall algebras and in categories of homological dimension at most 2, lifting the standard construction in terms of virtual pullbacks to an algebra object in categories of mixed Hodge modules over an arbitrary base monoid, and we introduce the BPS algebra of (suitably geometric) 2-Calabi-Yau categories. With this rather dry-sounding work out of the way, we move to the main examples of interest (in this paper), which we call totally negative 2-Calabi-Yau categories. We prove that for such categories, the BPS algebra is freely generated by intersection cohomology, and prove a PBW theorem for the whole CoHA, again expressed in terms of intersection cohomology. The two main applications are a proof of (a class of cases of) the Bozec-Schiffmann positivity conjecture (a strengthening of the Kac positivity conjecture), and a construction of a nonabelian Hodge isomorphism in Borel-Moore homology for moduli stacks associated to curves of genus at least two.
Affine BPS algebras, W algebras, and the cohomological Hall algebra of A^2 (submitted)
In this paper I introduce deformations and affinizations of BPS Lie algebras associated to preprojective CoHAs. Building on lots of previous work, I calculate the cohomological Hall algebra of zero-dimensional support sheaves, along with its 1 and 2-parameter deformations, expressed in terms of affinized BPS Lie algebras, deformed affinized BPS Lie algebras, and (half) Yangians
BPS Lie algebras and the less perverse filtration on the preprojective CoHA (submitted)
Here I used some recent results from cohomological DT theory to prove a relative purity result for the Borel-Moore homology of the preprojective stack, which implies that the Hall algebra of raising operators carries a perverse filtration. The zeroth cohomology turns out to be isomorphic to a certain BPS Lie algebra, which is equal to the Kac-Moody algebra of the underlying quiver in cohomological degree zero, but typically also has generators in nonzero cohomological degree, which I (conjecturally) completely classify using the new perverse filtration.
Donaldson-Thomas theory for categories of homological dimension one with potential (with Sven Meinhardt)
In this paper we rigorously write down the foundational background to motivic DT theory. In particular, we prove the integrality conjecture.
This is actually my PhD thesis, and as such is just a place holder for a paper that I have to write. In it you'll find an explanation of what orientation data is, and a worked example showing why it matters. There's a lot of other stuff in here too, as there should be given how hopelessly long it is. For example there's a section explaining that the "naive" motivic weight on moduli spaces of representations of modules for Jacobi algebras is the same one that Kontsevich and Soibelman use, with the aid of orientation data. There is also a proof of one of their conjectures, regarding the invariance of orientation data under derived equivalence.