Related papers: $K$-theoretic pullbacks for Lagrangians on derived…

In this paper we introduce and study the so-called continuous $K$-theory for a certain class of "large" stable $\infty$-categories, more precisely, for dualizable presentable categories. For compactly generated categories, the continuous…

We show that certain tilting results for quivers are formal consequences of stability, and as such are part of a formal calculus available in any abstract stable homotopy theory. Thus these results are for example valid over arbitrary…

Here we propose a different perspective of the deep factorisation in Kyprianou (2015) based on determining potentials. Indeed, we factorise the inverse of the MAP-exponent associated to a stable process via the Lamperti-Kiu transform. Here…

The "fundamental theorem" for algebraic $K$-theory expresses the $K$-groups of a Laurent polynomial ring $L[t,t^{-1}]$ as a direct sum of two copies of the $K$-groups of $L$ (with a degree shift in one copy), and certain "nil" groups of…

Let $V$ be the set of $n\times n$ complex or real general matrices, Hermitian matrices, symmetric matrices, positive definite (resp. semi-definite) matrices, diagonal matrices, or upper triangular matrices. Fix $k\in \mathbb{Z}\setminus…

We formulate a constructive theory of noncommutative Landau-Ginzburg models mirror to symplectic manifolds based on Lagrangian Floer theory. The construction comes with a natural functor from the Fukaya category to the category of matrix…

We introduce a real-parameter refinement of the classical integer hierarchies underlying Schmidt number, block-positivity, and $k$-positivity for maps between matrix algebras. Starting from a compact family of $\alpha$-admissible unit…

We construct a geometric realization of categories of representations of affine Hecke algebras and split reductive $p$-adic groups via a $K$-motivic Springer theory. We suggest a connection to the coherent Springer theory of Ben-Zvi, Chen,…

Let X be a non-compact Calabi-Yau manifold and f be a holomorphic function on X with compact critical locus. We introduce the notion of f-twisted Sobolev spaces for the pair (X,f) and prove the corresponding Hodge-to-de Rham degeneration…

We cast Kasparov's equivariant KK-theory in the framework of model categories. We obtain a stable model structure on a certain category of locally multiplicative convex $G$-$C^*$-algebras, which naturally contains the stable…

We describe the mod $p^r$ pro $K$-groups $\{K_n(A/I^s)/p^r\}_s$ of a regular local $\mathbb F_p$-algebra $A$ modulo powers of a suitable ideal $I$, in terms of logarithmic Hodge-Witt groups, by proving pro analogues of the theorems of…

Let $G$ be a finite subgroup of $\mathrm{SU}(4)$ whose elements have age not larger than one. In the first part of this paper, we define $K$-theoretic stable pair invariants on the crepant resolution of the affine quotient $\mathbb{C}^4/G$,…

This paper combines several new constructions in mathematics and physics. Mathematically, we study framed flat PGL(K,C)-connections on a large class of 3-manifolds M with boundary. We define a space L_K(M) of framed flat connections on the…

For any holomorphic mapping $f\colon X\to Y$ between a complex manifold $X$ and a complex Hermitian manifold $Y$ we extend the pullback $f^*$ from smooth forms to a class of currents. We provide a basic calculus for this pullback and show…

We study some arithmetic properties of the mirror maps and the quantum Yukawa coupling for some 1-parameter deformations of Calabi-Yau manifolds. First we use the Schwarzian differential equation, which we derived previously, to…

The derived Maurer-Cartan locus $\text{MC}^\bullet(L)$ is a functor from differential graded Lie algebras to cosimplicial schemes. If L is differential graded Lie algebra, let $L_+$ be the truncation of $L$ in positive degrees $i>0$. We…

This work investigates the Frobenius morphism on derived categories associated with algebraic stacks in positive characteristic. Particularly, we show that in many cases sufficiently many Frobenius pushforwards of a compact generator…

We argue that Lagrangian correspondences are the correct framework to study functoriality of virtual fundamental classes arising from a $-2$-symplectic derived structure.

Given a compact Lie group $G$ acting on a space $X$, the classical Atiyah-Segal completion theorem identifies topological $K$-theory of the homotopy quotient $X/G$ with an explicit completion of $G$-equivariant topological $K$-theory of…

This paper is the first in a series in which we offer a new framework for hermitian K-theory in the realm of stable $\infty$-categories. Our perspective yields solutions to a variety of classical problems involving Grothendieck-Witt groups…