Quoc P. Ho

Abstract: We prove a conjecture of Gorsky, Hogancamp, Mellit, and Nakagane in the Weyl group case. Namely, we show that the left and right adjoints of the parabolic induction functor between the associated Hecke categories of Soergel bimodules differ by the relative full twist.

Abstract: We provide a uniform construction of “mixed versions” or “graded lifts” in the sense of Beilinson–Ginzburg–Soergel which works for arbitrary Artin stacks. In particular, we obtain a general construction of graded lifts of many categories arising in geometric representation theory and categorified knot invariants. Our new theory associates to each Artin stack of finite type $\mathcal{Y}$ over $\Fqbar$ a symmetric monoidal $\DG$-category $\Shv_{\gr, \compct}(\mathcal{Y})$ of constructible graded sheaves on $\mathcal{Y}$ along with the six-functor formalism, a perverse $t$-structure, and a weight (or co-$t$-)structure in the sense of Bondarko and Pauksztello, compatible with the six-functor formalism, perverse $t$-structures, and Frobenius weights on the category of (mixed) $\ell$-adic sheaves.

Classically, mixed versions were only constructed in very special cases due to non-semisimplicity of Frobenius. Our construction sidesteps this issue by semi-simplifying the Frobenius action itself. However, the category $\Shv_{\gr, \compct}(\mathcal{Y})$ agrees with those previously constructed when they are available. For example, for any reductive group $G$ with a fixed pair $T\subset B$ of a maximal torus and a Borel subgroup, we have an equivalence of monoidal $\DG$ weight categories $\Shv_{\gr, \compct}(B\backslash G/B) \simeq \Ch^{\bounded}(\SBim_W)$, where $\Ch^{\bounded}(\SBim_W)$ is the monoidal $\DG$-category of bounded chain complexes of Soergel bimodules and $W$ is the Weyl group of $G$.

Abstract: Using a geometric argument building on our new theory of graded sheaves, we compute the categorical trace and Drinfel'd center of the (graded) finite Hecke category $\Hecke_W^\gr = \Ch^{\bounded}(\SBim_W)$ in terms of the category of (graded) unipotent character sheaves, upgrading results of Ben-Zvi–Nadler and Bezrukavnikov–Finkelberg–Ostrik. In type $A$, we relate the categorical trace to the category of $2$-periodic coherent sheaves on the Hilbert schemes $\Hilb_n(\CC^2)$ of points on $\CC^2$ (equivariant with respect to the natural $\CC^* \times \CC^*$ action), yielding a proof of (a $2$-periodized version of) a conjecture of Gorsky–Neguț–Rasmussen which relates HOMFLY-PT link homology and the spaces of global sections of certain coherent sheaves on $\Hilb_n(\CC^2)$. As an important computational input, we also establish a conjecture of Gorsky–Hogancamp–Wedrich on the formality of the Hochschild homology of $\Hecke_W^\gr$.

Abstract: In the topological setting, we lift homological densities for (generalized) configuration spaces in the sense of Farb–Wolfson–Wood to the level of spectra.

Abstract: Using factorization homology with coefficients in twisted commutative algebras (TCAs), we prove two flavors of higher representation stability for the cohomology of (generalized) configuration spaces of a scheme/topological space \(\mathsfit{X}\). First, we provide an iterative procedure to study higher representation stability using actions coming from the cohomology of \(\mathsfit{X}\) and prove that all the modules involved are finitely generated over the corresponding TCAs. More quantitatively, we compute explicit bounds for the derived indecomposables in the sense of Galatius–Kupers–Randal-Williams. Secondly, we prove that when certain \(\mathsf{C}_\infty\)-operations on the cohomology of \(\mathsfit{X}\) vanish, the cohomology of its configuration spaces form a free module over a twisted commutative algebra built out of the configuration spaces of the affine space. This generalizes a result of Church–Ellenberg–Farb on the freeness of \(\mathsf{FI}\)-modules arising from the cohomology of configuration spaces of open manifolds and, moreover, resolves the various conjectures of Miller–Wilson in this case.

Abstract: Motivated by spectral gluing patterns in the Betti Langlands program, we show that for any reductive group $G$, a parabolic subgroup $P$, and a topological surface $M$, the (enhanced) spectral Eisenstein series category of $M$ is the factorization homology over $M$ of the $\En_2$-Hecke category $\Hecke_{G, P} = \IndCoh(\LocSys_{G, P}(\Dsk^2, \Sphere^1))$, where $\LocSys_{G, P}(\Dsk^2, \Sphere^1)$ denotes the moduli stack of $G$-local systems on a disk together with a $P$-reduction on the boundary circle.

More generally, for any pair of stacks $\mathcal{Y}\to \mathcal{Z}$ satisfying some mild conditions and any map between topological spaces $N\to M$, we define $(\mathcal{Y}, \mathcal{Z})^{N, M} = \mathcal{Y}^N \times_{\mathcal{Z}^N} \mathcal{Z}^M$ to be the space of maps from $M$ to $\mathcal{Z}$ along with a lift to $\mathcal{Y}$ of its restriction to $N$. Using the pair of pants construction, we define an $\En_n$-category $\Hecke_n(\mathcal{Y}, \mathcal{Z}) = \IndCoh_0\left(\left((\mathcal{Y}, \mathcal{Z})^{\Sphere^{n-1}, \Dsk^n}\right)^\wedge_{\mathcal{Y}}\right)$ and compute its factorization homology on any $d$-dimensional manifold $M$ with $d\leq n$, \[ \int_M \Hecke_n(\mathcal{Y}, \mathcal{Z}) \simeq \IndCoh_0\left(\left((\mathcal{Y}, \mathcal{Z})^{\partial (M\times \Dsk^{n-d}), M}\right)^\wedge_{\mathcal{Y}^M}\right), \] where $\IndCoh_0$ is the sheaf theory introduced by Arinkin–Gaitsgory and Beraldo. Our result naturally extends previous known computations of Ben-Zvi–Francis–Nadler and Beraldo.

Abstract: The \(\otimes^\star\)- structure on the category of sheaves on the Ran space is not pro-nilpotent in the sense of Francis–Gaitsgory. However, under some connectivity assumptions, we prove that Koszul duality induces an equivalence of categories and that this equivalence behaves nicely with respect to Verdier duality on the Ran space and integrating along the Ran space, i.e. taking factorization homology. Based on ideas sketched by Gaitsgory, we show that these results also offer a simpler alternative to one of the two main steps in the proof of the Atiyah–Bott formula.

Abstract: We prove that the factorization homologies of a scheme with coefficients in truncated polynomial algebras compute the cohomologies of its generalized configuration spaces. Using Koszul duality between commutative algebras and Lie algebras, we obtain new expressions for the cohomologies of the latter. As a consequence, we obtain a uniform and conceptual approach for treating homological stability, homological densities, and arithmetic densities of generalized configuration spaces. Our results categorify, generalize, and in fact provide a conceptual understanding of, the coincidences appearing in the work of Farb–Wolfson–Wood. Our computation of the stable homological densities also yields rational homotopy types which answer a question posed by Vakil–Wood. Our approach hinges on the study of homological stability of cohomological Chevalley complexes, which is of independent interest.

Abstract: We provide a construction of free factorization algebras in algebraic geometry and link factorization homology of a scheme with coefficients in a free factorization algebra to the homology of its (unordered) configuration spaces. As an application, this construction allows for a purely algebro-geometric proof of homological stability of configuration spaces.

Abstract: Employing a geometric setting inspired by the proof of the Fundamental Lemma, we study some counting problems related to the average size of 2-Selmer groups and hence obtain an estimate for it.

Abstract: We present a low-scaling diagrammatic Monte Carlo approach to molecular correlation energies. Using combinatorial graph theory to encode many-body Hugenholtz diagrams, we sample the Møller-Plesset (MPn) perturbation series, obtaining accurate correlation energies up to $\mathsf{n=5}$, with quadratic scaling in the number of basis functions. Our technique reduces the computational complexity of the molecular many-fermion correlation problem, opening up the possibility of low-scaling, accurate stochastic computations for a wide class of many-body systems described by Hugenholtz diagrams.