We compute quantum character varieties of arbitrary closed surfaces with boundaries and marked points. These are categorical invariants \(\int _S{\mathcal {A}}\) of a surface S, determined by the choice of a braided tensor category \({\mathcal {A}}\), and computed via factorization homology. We identify the algebraic data governing marked points and boundary components with the...

We prove that every 0-shifted symplectic structure on a derived Artin n-stack admits a curved \(A_{\infty }\) deformation quantisation. The classical method of quantising smooth varieties via quantisations of affine space does not apply in this setting, so we develop a new approach. We construct a map from DQ algebroid quantisations of unshifted symplectic structures on a derived...

We study linear Batalin–Vilkovisky (BV) quantization, which is a derived and shifted version of the Weyl quantization of symplectic vector spaces. Using a variety of homotopical machinery, we implement this construction as a symmetric monoidal functor of \(\infty \)-categories. We also show that this construction has a number of pleasant properties: It has a natural extension to...

We compute the image of Enriquez’ elliptic KZB associator in the (maximal) meta-abelian quotient of the fundamental Lie algebra of a once-punctured elliptic curve. Our main result is an explicit formula for this image in terms of Eichler integrals of Eisenstein series, and is analogous to Deligne’s computation of the depth one quotient of the Drinfeld associator. We also show how...

We introduce the notion of a relative pseudomonad, which generalizes the notion of a pseudomonad, and define the Kleisli bicategory associated to a relative pseudomonad. We then present an efficient method to define pseudomonads on the Kleisli bicategory of a relative pseudomonad. The results are applied to define several pseudomonads on the bicategory of profunctors in an...

This paper explores two questions: (1) Which bigraded groups arise as the knot Floer homology of a knot in the three-sphere? (2) Given a knot, how many distinct knots share its Floer homology? Regarding the first, we show there exist bigraded groups satisfying all previously known constraints of knot Floer homology which do not arise as the invariant of a knot. This leads to a...

It was observed by Bump et al. that Ehrhart polynomials in a special family exhibit properties shared by the Riemann \(\zeta \) function. The construction was generalized by Matsui et al. to a larger family of reflexive polytopes coming from graphs. We prove several conjectures confirming when such polynomials have zeros on a certain line in the complex plane. Our main new method...

The space of Bridgeland stability conditions on the bounded derived category of coherent sheaves on \(\mathbf {P}^2\) has a principal connected component \(\hbox {Stab}^\dag (\mathbf{P }^2)\). We show that \(\hbox {Stab}^\dag (\mathbf{P }^2)\) is the union of geometric and algebraic stability conditions. As a consequence, we give a cell decomposition for \(\hbox {Stab}^\dag...

In this paper we study higher Deligne–Lusztig representations of reductive groups over finite quotients of discrete valuation rings. At even levels, we show that these geometrically constructed representations, defined by Lusztig, coincide with certain explicit induced representations defined by Gérardin, thus giving a solution to a problem raised by Lusztig. In particular, we...

We study versions of homological mirror symmetry for hypersurface cusp singularities and the three hypersurface simple elliptic singularities. We show that the Milnor fibres of each of these carries a distinguished Lefschetz fibration; its derived directed Fukaya category is equivalent to the derived category of coherent sheaves on a smooth rational surface \(Y_{p,q,r}\). By...

We present a new method of proving the Diophantine extremality of various dynamically defined measures, vastly expanding the class of measures known to be extremal. This generalizes and improves the celebrated theorem of Kleinbock and Margulis (’98) resolving Sprindžuk’s conjecture, as well as its extension by Kleinbock, Lindenstrauss, and Weiss (’04), hereafter abbreviated KLW...

We construct almost toric fibrations (ATFs) on all del Pezzo surfaces, endowed with a monotone symplectic form. Except for \({\mathbb {C}}P^2\# \overline{{\mathbb {C}}P^2}\), \({\mathbb {C}}P^2\# 2\overline{{\mathbb {C}}P^2}\), we are able to get almost toric base diagrams (ATBDs) of triangular shape and prove the existence of infinitely many symplectomorphism (in particular...

We compute the groups \(H^*(\mathrm {Aut}(F_n); M)\) and \(H^*(\mathrm {Out}(F_n); M)\) in a stable range, where M is obtained by applying a Schur functor to \(H_\mathbb {Q}\) or \(H^*_\mathbb {Q}\), respectively the first rational homology and cohomology of \(F_n\). The answer may be described in terms of stable multiplicities of irreducibles in the plethysm \(\mathrm {Sym}^k...

The main result of this article is a refinement of the well-known subgroup separability results of Hall and Scott for free and surface groups. We show that for any finitely generated subgroup, there is a finite dimensional representation of the free or surface group that separates the subgroup in the induced Zariski topology. As a corollary, we establish a polynomial upper bound...

Let \(\pi \) be a cuspidal, cohomological automorphic representation of \(\mathrm {GL}_n({\mathbb A})\). Venkatesh has suggested that there should exist a natural action of the exterior algebra of a certain motivic cohomology group on the \(\pi \)-part of the Betti cohomology (with rational coefficients) of the \(\mathrm {GL}_n({\mathbb Q})\)-arithmetic locally symmetric space...

In Bayer and Macrì (J Am Math Soc 27(3):707–752, 2014), the first author and Macrì constructed a family of nef divisors on any moduli space of Bridgeland-stable objects on a smooth projective variety X. In this article, we extend this construction to the setting of any separated scheme Y of finite type over a field, where we consider moduli spaces of Bridgeland-stable objects on...

Encoding, transmission and decoding of information are ubiquitous in biology and human history: from DNA transcription to spoken/written languages and languages of sciences. During the last decades, the study of neural networks in brain performing their multiple tasks was providing more and more detailed pictures of (fragments of) this activity. Mathematical models of this...

We prove that any relative character (a.k.a. spherical character) of any admissible representation of a real reductive group with respect to any pair of spherical subgroups is a holonomic distribution on the group. This implies that the restriction of the relative character to an open dense subset is given by an analytic function. The proof is based on an argument from algebraic...

We classify the spectral transfer morphisms (cf. Opdam in Adv Math 286:912–957, 2016) between affine Hecke algebras associated to the unipotent types of the various inner forms of an unramified absolutely simple algebraic group G defined over a non-archimedean local field k. This turns out to characterize Lusztig’s classification (Lusztig in Int Math Res Not 11:517–589, 1995; in...

We study in detail the large N expansion of \({\mathrm {SU}}(N)\) and \({\mathrm {SO}}(N)/{\mathrm {Sp}}(2N)\) Chern–Simons partition function \(Z_N(M)\) of 3-manifolds M that are either rational homology spheres or more generally Seifert fibered spaces. This partition function admits a matrix model-like representation, whose spectral curve can be characterized in terms of a...

After extending the theory of Rankin–Selberg local factors to pairs of \(\ell \)-modular representations of Whittaker type, of general linear groups over a non-Archimedean local field, we study the reduction modulo \(\ell \) of \(\ell \)-adic local factors and their relation to these \(\ell \)-modular local factors. While the \(\ell \)-modular local \(\gamma \)-factor we...

We develop geometry of affine algebraic varieties in \(K^{n}\) over Henselian rank one valued fields K of equicharacteristic zero. Several results are provided including: the projection \(K^{n} \times {\mathbb {P}}^{m}(K) \rightarrow K^{n}\) and blowups of the K-rational points of smooth K-varieties are definably closed maps; a descent property for blowups; curve selection for...