Related papers: On some spectral spaces associated to tensor trian…

In infinite dimensions and on the level of trace-class operators $C$ rather than matrices, we show that the closure of the $C$-numerical range $W_C(T)$ is always star-shaped with respect to the set $\operatorname{tr}(C)W_e(T)$, where…

This lecture series is based on joint work in progress with Shaul Barkan, as well as work in progress of the author. The five sections of these notes correspond to the five lectures, but more details have been added. $2$-dimensional…

We generalize the construction of tensor categories of endomorphisms of a type III factor $M$ associated with a $G$-kernel, from the case of a discrete group $G$ to that of a compact second countable group. Our approach is based on the…

We study moduli spaces of flat metrics on closed Riemannian orbifolds admitting such metrics. We show that for such orbifolds $\mathcal{O}$, the Teichm\"uller space of flat metrics $\mathcal{T}_{\text{flat}}(\mathcal{O})$ serves as a…

We use categorification of monoid actions to study algebraic geometry over symmetric monoidal categories. This brings together the relative algebraic geometry over symmetric monoidal categories developed by To\"{e}n and Vaqui\'{e}, along…

A p-periodic 3-manifold is a 3-manifold that admits a Z_{p}-action whose fixed point set is a circle. We give a congruence relates the quantum invariant of a p-periodic 3-manifold associated to any modular category over an integrally closed…

We develop a rigidity criterion to show that in simplicial model categories with a compatible symmetric monoidal structure, operad structures can be automatically lifted along certain maps. This is applied to obtain an unpublished result of…

Let R be a Cohen-Macaulay local ring. Denote by mod R the category of finitely generated R-modules. In this paper, we consider the classification problem of resolving subcategories of mod R in terms of specialization-closed subsets of Spec…

Recall that the definition of the $K$-theory of an object C (e.g., a ring or a space) has the following pattern. One first associates to the object C a category A_C that has a suitable structure (exact, Waldhausen, symmetric monoidal, ...).…

We investigate properties of essential spectra of disjointness preserving operators acting on Banach $C(K)$-modules. In particular, we prove that under some very mild conditions the upper semi-Fredholm spectrum of such an operator is…

A finite bosonic or fermionic symmetry can be described uniquely by a symmetric fusion category $\mathcal{E}$. In this work, we propose that 2+1D topological/SPT orders with a fixed finite symmetry $\mathcal{E}$ are classified, up to $E_8$…

This is a continuation of the paper "Modular tensor categories and orbifold theories", arXiv:math.QA/0104242. It discusses orbifold models of conformal filed theory, or, in mathematical language, question of constructing the category of…

Let S be an oriented surface of genus g with m punctures. If 3g-3+m is at least 4 then we construct for every compact subset K of moduli space a closed Teichmueller geodesic not intersecting K.

The space of differential operators acting on skewsymmetric tensor fields or on smooth forms of a smooth manifold are representations of its Lie algebra of vector fields. We compute the first cohomology spaces of these representations and…

We introduce the notion of solid monoid and rigid monoid in monoidal categories and study the formal properties of these objects in this framework. We show that there is a one to one correspondence between solid monoids, smashing…

We show that every logmodular subalgebra of $M_n(\mathbb{C})$ is unitary equivalent to an algebra of block upper triangular matrices, which was conjectured in \cite{VM}. In particular, this shows that every unital contractive representation…

We study the topology of a class of proper submodules and some of its distinguished subclasses and call them structure spaces. We give several criteria for the quasi-compactness of these structure spaces. We study $T_0$ and $T_1$ separation…

We treat spectral problems by twisted groupoid methods. To Hausdorff locally compact groupoids endowed with a continuous $2$-cocycle one associates the reduced twisted groupoid $C^*$-algebra. Elements (or multipliers) of this algebra admit…

The category of strict polynomial functors inherits an internal tensor product from the category of divided powers. To investigate this monoidal structure, we consider the category of representations of the symmetric group which admits a…

Kneser-Haken Finiteness asserts that for each compact 3-manifold M there is an integer c(M) such that any collection of k>c(M) closed, essential, 2-sided surfaces in M must contain parallel elements. We show here that if M is closed then…