Abstract We study fibrations of the projective model for the symmetric space associated with $\operatorname {\mathrm {SL}}(2n,\mathbb {R})$ by codimension $2$ projective subspaces, or pencils of quadrics. In particular we show that if such a smooth fibration is equivariant with respect to a representation of a closed surface group, the representation is quasi-isometrically embedded, and even Anosov if the pencils in the image contain only nondegenerate quadrics. We use this to characterize maximal representations among representations of a closed surface group into $\operatorname {\mathrm {Sp}}(2n,\mathbb {R})$ by the existence of an equivariant continuous fibration of the associated symmetric space, satisfying an additional technical property. These fibrations extend to fibrations of the projective structures associated to maximal representations by bases of pencils of quadrics.

Abstract The moduli space of bundle stable pairs $\overline {M}_C(2,\Lambda )$ on a smooth projective curve C , introduced by Thaddeus, is a smooth Fano variety of Picard rank two. Focusing on the genus two case, we show that its K-moduli space is isomorphic to a GIT moduli of lines in quartic del Pezzo threefolds. Additionally, we construct a natural forgetful morphism from the K-moduli of $\overline {M}_C(2,\Lambda )$ to that of the moduli spaces of stable vector bundles $\overline {N}_C(2,\Lambda )$ . In particular, Thaddeus’ moduli spaces for genus two curves are all K-stable.

Abstract This work concerns generators for the bounded derived category of coherent sheaves over a noetherian scheme X of prime characteristic. The main result is that when the Frobenius map on X is finite, for any compact generator G of $\mathsf {D}(X)$ the Frobenius pushforward $F ^e_*G$ generates the bounded derived category whenever $p^e$ is larger than the codepth of X , an invariant that is a measure of the singularity of X . The conclusion holds for all positive integers e when X is locally complete intersection. The question of when one can take $G=\mathcal {O}_X$ is also investigated. For smooth projective complete intersections it reduces to a question of generation of the Kuznetsov component.

Abstract We show that for any set $A\subset {\mathbb N}$ with positive upper density and any $\ell ,m \in {\mathbb N}$ , there exist an infinite set $B\subset {\mathbb N}$ and some $t\in {\mathbb N}$ so that $\{mb_1 + \ell b_2 \colon b_1,b_2\in B\ \text {and}\ b_1<b_2 \}+t \subset A,$ verifying a conjecture of Kra, Moreira, Richter and Robertson. We also consider the patterns $\{mb_1 + \ell b_2 \colon b_1,b_2\in B\ \text {and}\ b_1 \leq b_2 \}$ , for infinite $B\subset {\mathbb N}$ and prove that any set $A\subset {\mathbb N}$ with lower density $\underline {\!\mathrm {d}}(A)>1/2$ contains such configurations up to a shift. We show that the value $1/2$ is optimal and obtain analogous results for values of upper density and when no shift is allowed.

Abstract We develop a diagrammatic approach to the representation theory of the quantum symmetric pairs corresponding to orthosymplectic Lie superalgebras inside general linear Lie superalgebras. Our approach is based on the disoriented skein category, which we define as a module category over the framed HOMFLYPT skein category. The disoriented skein category admits full incarnation functors to the categories of modules over the iquantum enveloping algebras corresponding to the quantum symmetric pairs, and it can be viewed as an interpolating category for these categories of modules. We define an equivalence of module categories between the disoriented skein category and the iquantum Brauer category (also known as the q -Brauer category), after endowing the latter with the structure of a module category over the framed HOMFLYPT skein category. The disoriented skein category has some advantages over the iquantum Brauer category, possessing duality structure and allowing the incarnation functors to be strict morphisms of module categories. Finally, we construct explicit bases for the morphism spaces of the disoriented skein and iquantum Brauer categories.

Abstract We prove a version of Sylvester’s law of inertia for the Reflection Equation Algebra (=REA). We will only be concerned with the REA constructed from the R -matrix associated to the standard q -deformation of $GL(N,\mathbb {C})$ . For q positive, this particular REA comes equipped with a natural $*$ -structure, by which it can be viewed as a q -deformation of the $*$ -algebra of polynomial functions on the space of self-adjoint N -by- N -matrices. We will show that this REA satisfies a type I -condition, so that its irreducible representations can in principle be classified. Moreover, we will show that, up to the adjoint action of quantum $GL(N,\mathbb {C})$ , any irreducible representation of the REA is determined by its extended signature , which is a classical signature vector extended by a parameter in $\mathbb {R}/\mathbb {Z}$ . It is this latter result that we see as a quantized version of Sylvester’s law of inertia.

Abstract We introduce the index ${\mathcal N}(\omega _1,\omega _2)$ of a pair of pure states on a unital C*-algebra, which is a generalization of the notion of the index of a pair of projections on a Hilbert space. We then show that the Hall conductance associated with an invertible state $\omega $ of a two-dimensional interacting electronic system which is symmetric under $U(1)$ charge transformation may be written as the index $\mathcal {N}(\omega ,\omega _D)$ , where $\omega _D$ is obtained from $\omega $ by inserting a unit of magnetic flux. This exhibits the integrality and continuity properties of the Hall conductance in the context of general topological features of $\mathcal {N}$ .

Abstract A hyperbinary partition of the nonnegative integer n is a partition where every part is a power of $2$ and every power of $2$ appears at most twice. We give three applications of the length generating function for such partitions, denoted by $h_q(n)$ . Morier-Genoud and Ovsienko defined the q -analogue of a rational number $[r/s]_q$ in various ways, most of which depend directly or indirectly on the continued fraction expansion of $r/s$ . As our first application we show that $[r/s]_q=q\,h_q(n-1)/h_q(n)$ where $r/s$ occurs as the n th entry in the Calkin-Wilf enumeration of the non-negative rationals. Next we consider fence posets which are those which can be obtained from a sequence of chains by alternately pasting together maxima and minima. For every n we show there is a fence poset ${\cal F}(n)$ whose lattice of order ideals is isomorphic to the poset of hyperbinary partitions of n ordered by refinement. For our last application, Morier-Genoud and Ovsienko also showed that $[r/s]_q$ can be computed by taking products of certain matrices which are q -analogues of the standard generators for the special linear group $\operatorname {\mathrm {SL}}(2,{\mathbb Z})$ . We express the entries of these products in terms of the polynomials $h_q(n)$ .

Abstract We prove a local-global principle for parametrized $\infty $ -categories: we show that any functor $\mathcal {B} \to \mathcal {C}$ is determined by the following data: the collection of fibers $\mathcal {B}_X$ for X running through the set of equivalence classes of objects of $\mathcal {C}$ endowed with the action of the space of automorphisms $\mathrm {Aut}_X(\mathcal {B})$ on the fiber, the local data, together with a locally cartesian fibration ${\mathcal D} \to \mathcal {C}$ and $\mathrm {Aut}_X(\mathcal {B})$ -linear equivalences ${\mathcal D}_X \simeq {\mathcal P}(\mathcal {B}_X)$ to the $\infty $ -category of presheaves on $\mathcal {B}_X$ , the gluing data. As applications we compute the mapping spaces of the conditionally existing internal hom of $\infty \mathrm {Cat}_{/\mathcal {C}}$ and extend the $\infty $ -categorical Grothendieck-construction by proving that $\infty $ -categories over any $\infty $ -category $\mathcal {C}$ are classified by normal lax 2-functors to a double $\infty $ -category of correspondences.

Abstract We generalize the seminal polynomial partitioning theorems of Guth and Katz [33, 28] to a set of semi-Pfaffian sets. Specifically, given a set $\Gamma \subseteq \mathbb {R}^n$ of k -dimensional semi-Pfaffian sets, where each $\gamma \in \Gamma $ is defined by a fixed number of Pfaffian functions, and each Pfaffian function is in turn defined with respect to a Pfaffian chain $\vec {q}$ of length r , for any $D \ge 1$ , we prove the existence of a polynomial $P \in \mathbb {R}[X_1, \ldots , X_n]$ of degree at most D such that each connected component of $\mathbb {R}^n \setminus Z(P)$ intersects at most $\sim \frac {|\Gamma |}{D^{n - k - r}}$ elements of $\Gamma $ . Also, under some mild conditions on $\vec {q}$ , for any $D \ge 1$ , we prove the existence of a Pfaffian function $P'$ of degree at most D defined with respect to $\vec {q}$ , such that each connected component of $\mathbb {R}^n \setminus Z(P')$ intersects at most $\sim \frac {|\Gamma |}{D^{n-k}}$ elements of $\Gamma $ . To do so, given a k -dimensional semi-Pfaffian set $\mathcal {X} \subseteq \mathbb {R}^n$ , and a polynomial $P \in \mathbb {R}[X_1, \ldots , X_n]$ of degree at most D , we establish a uniform bound on the number of connected components of $\mathbb {R}^n \setminus Z(P)$ that $\mathcal {X}$ intersects; that is, we prove that the number of connected components of $(\mathbb {R}^n \setminus Z(P)) \cap \mathcal {X}$ is at most $\sim D^{k+r}$ . Finally, as applications, we derive Pfaffian versions of Szemerédi-Trotter-type theorems, and also prove bounds on the number of joints between Pfaffian curves.