Related papers: Comparing Dualities in the $K(n)$-local Category

The author has previously associated to each commutative ring with unit $\Bbbk$ and \'etale groupoid $\mathscr G$ with locally compact, Hausdorff, totally disconnected unit space a $\Bbbk$-algebra $\Bbbk\mathscr G$. The algebra…

In this paper we take some classical ideas from commutative algebra, mostly ideas involving duality, and apply them in algebraic topology. To accomplish this we interpret properties of ordinary commutative rings in such a way that they can…

We develop the $p$-adic representation theory of $p$-adic Lie groups on solid vector spaces over a complete non-archimedean extension of $\mathbb{Q}_p$. More precisely, we define and study categories of solid, solid locally analytic and…

Let $G$ be a connected reductive algebraic group over an algebraically closed field of positive characteristic, $\mathfrak{g}$ be its Lie algebra, and $B$ be a Borel subgroup. We prove a formula for the dimensions of extension groups, in…

In this mostly expository article, we provide a new account of our proof with Minsky and Sisto that mapping class groups and Teichm\"uller spaces admit bicombings. More generally, we explain how the hierarchical hull of a pair of points in…

In this paper, we show an isomorphism of homological knot invariants categorifying the Reshetikhin-Turaev invariants for $\mathfrak{sl}_n$. Over the past decade, such invariants have been constructed in a variety of different ways, using…

We construct a natural transformation from the Bousfield-Kuhn functor evaluated on a space to the Topological Andre-Quillen cohomology of the K(n)-local Spanier-Whitehead dual of the space, and show that the map is an equivalence in the…

The Krein--Tannaka duality for compact groups was a generalization the Pontryagin--Van Kampen duality for locally compact abelian groups and a remote predecessor of the theory of tensor categories. It is less known that it found…

We generalize Blumberg-Mandell's K-theoretic Poitou-Tate duality to arithmetic schemes of arbitrary dimension, smooth and proper over S-integers. As in our earlier papers on the subject, we discuss how to model the compactly supported side…

We construct a new effective orbifold $\widehat{\Y}$ with an $S^1$-gerbe $c$ to study an $S^1$-gerbe $\mathfrak{t}$ on a $G$-gerbe $\Y$ over an orbifold $\B$. We view the former as the relative dual, relative to $\B$, of the latter. We show…

Let X be Drinfeld's upper half space of dimension d over a finite extension K of Q_p. We construct for every homogeneous vector bundle F on the projective space P^d a GL_{d+1}(K)-equivariant filtration by closed K-Frechet spaces on F(X).…

We study N = 2 superconformal theories with gauge group SU(N) and 2N fundamental flavours in a locus of the Coulomb branch with a Z_N symmetry. In this special vacuum, we calculate the prepotential, the dual periods and the period matrix…

We study sheaves of Lie-Rinehart algebras over locally ringed spaces. We introduce morphisms and comorphisms of such sheaves and prove factorization theorems for each kind of morphism. Using this notion of morphism, we obtain (higher)…

We define an inner product (suitably interpreted) on the K(n)-local spectrum LG := L_{K(n)}BG_+, where G is a finite group or groupoid. This gives an inner product on E^*BG_+ for suitable K(n)-local ring spectra E. We relate this to the…

We discuss the proof of Kazhdan and Lusztig of the equivalence of the Drinfeld category D(g,h) of g-modules and the category of finite dimensional Uq(g)-modules, q=exp(\pi ih), for h\in C\Q*. Aiming at operator algebraists the result is…

We give a new proof of the equivalence between two of the main models for $(\infty,n)$-categories, namely the $n$-fold Segal spaces of Barwick and the $\Theta_{n}$-spaces of Rezk, by proving that these are algebras for the same monad on the…

We show that every involutive Hopf monoid in a complete and finitely cocomplete symmetric monoidal category gives rise to invariants of oriented surfaces defined in terms of ribbon graphs. For every ribbon graph this yields an object in the…

Let $\mathbb{F}_q$ be the field of $q$ elements and let $A=\mathbb{F}_q[t]$ be the polynomial ring over $\mathbb{F}_q$. Let $\mathfrak{n}\in A\setminus \mathbb{F}_q$ be a monic polynomial with a prime factor of degree prime to $q-1$. Let…

This paper is the first of a pair that aims to classify a large number of the type $II$ quantum subgroups of the categories $\mathcal{C}(\mathfrak{sl}_{r+1},k)$. In this work we classify the braided auto-equivalences of the categories of…

Using degree N isogenies of elliptic curves, we produce a spectrum Q(N). This spectrum is built out of spectra related to tmf. At p=3 we show that the K(2)-local sphere is built out of Q(2) and its K(2)-local Spanier-Whitehead dual. This…