CyberSec Research

In this article we provide a model-independent definition of the concept of lax $2$-functors from $(\infty,2)$-category theory and show that it agrees with the existing and widely used combinatorial model for those in terms of inert-cocartesian functors, which is utilized for example in the foundational work of Gaitsgory and Rozenblyum on Derived Algebraic Geometry to talk about the lax Gray tensor product.

We construct differential models for twisted $\mathrm{Spin}^c$-bordism and for its Anderson dual, and employ the latter to define a twisted anomaly map whose source is the differential twisted $K$-theory. Our differential model for the twisted Anderson dual follows the formalism developed in [YY23]. To connect these constructions with the geometric framework of the Atiyah-Singer index theory, we further present a gerbe-theoretic formulation of our models in terms of bundle gerbes and gerbe modules [Mur96] [BCMMS02]. Within this geometric setting, we define the twisted anomaly map \[ \widehat{\Phi}_{\widehat{\mathcal{G}}}\colon \widehat{K}^{0}(X,\widehat{\mathcal{G}}^{-1}) \longrightarrow \bigl(\widehat{I\Omega^{\mathrm{Spin}^c}_{\mathrm{dR}}}\bigr)^{n}(X,\widehat{\mathcal{G}}), \] whose construction naturally involves the reduced eta-invariant of Dirac operators acting on Clifford modules determined by the twisted data. Conceptually, this map is expected to encode the anomalies of twisted $1|1$-dimensional supersymmetric field theories, in accordance with the perspectives developed in [ST11] and [FH21].

It is a classical problem in algebraic topology to decide whether a given graded $\mathbb{Z}$-algebra can be realized as the cohomology ring of a space. In this paper, we introduce families of Stanley-Reisner algebras depending on graphs, and relate their realizability to the span coloring of the graph.

Working in a generic derived algebro-geometric context, we lay the foundations for the general study of affineness and local descendability. When applied to $\mathbf{E}_\infty$ rings equipped with the fpqc topology, these foundations give an $\infty$-category of spectral stacks, a viable functor-of-points alternative to Lurie's approach to nonconnective spectral algebraic geometry. Specializing further to spectral stacks over the moduli stack of oriented formal groups, we use chromatic homotopy theory to obtain a large class of $0$-affine stacks, generalizing Mathew--Meier's famous $0$-affineness result. We introduce a spectral refinement of Hopkins' stack construction of an $\mathbf{E}_\infty$ ring, and study when it provides an inverse to the global sections of a spectral stack. We use this to show that a large class of stacks, which we call reconstructible, are naturally determined by their global sections, including moduli stacks of oriented formal groups of bounded height and the moduli stack of oriented elliptic curves.

We prove the statement in the title, solving in this way a conjecture stated by Ginot for manifolds with corners. Along the way, we establish a derived Swiss-cheese additivity theorem and an alternative proof for the hyperdescent of factorization algebras over those manifolds.

In this, the first of three papers about $C_2$-equivariant complex quadrics, we calculate the equivariant ordinary cohomology of smooth antisymmetric quadrics. One of these quadrics coincides with a $C_2$-equivariant Grassmannian, and we use this calculation to prove an equivariant refinement of the result that there are 27 lines on a cubic surface in $\mathbb{P}^3$.

These are lecture notes for a course in Winter 2022/23, updated and completed in October 2025. The goal of the lectures is to present some recent developments around six-functor formalisms, in particular: the abstract theory of 6-functor formalisms; the 2-category of cohomological correspondences, and resulting simplifications in the proofs of Poincar\'e--Verdier duality results; the relation between 6-functor formalisms and ``geometric rings''; many examples of 6-functor formalisms, both old and new.

We investigate how the notions of pairings of operads of May and compatible pairs of indexing systems of Blumberg--Hill relate via the correspondence between indexing systems and $N_{\infty}$-operads. We show that a pairing of operads induces a pairing on the associated indexing systems. Conversely, we show that in many cases, compatible pairs of indexing systems can be realized by a pairing of $N_{\infty}$-operads.

Let $G$ be a finite group. A $G$-Tambara functor $T$ consists of collection of commutative rings $T(G/H)$ (one for each subgroup $H$ of $G$), together with certain structure maps, satisfying certain axioms. In this note, we show that, for any integer $k$, any $G$-Tambara functor $T$, and any subgroups $H_1, H_2 \leq G$, $k$ is a unit in $T(G/H_1)$ if and only if $k$ is a unit in $T(G/H_2)$.

We prove an induction theorem for the higher algebraic K-groups of group algebras $kG$ of finite groups $G$ over characteristic $p$ finite fields $k$. For a certain class of finite groups, which we call $p$-isolated, this reduces calculations to calculations for their $p$-subgroups. We do so by showing that the stable module categories of $kH$ as $H$ ranges over subgroups of $G$ assemble into a categorical Green functor, which results in a spectral Green functor structure on K-theory. By general induction theory, this reduces proving a spectrum-level induction statement to proving an induction statement on $\pi_0$ Green functors, which we accomplish using modular representation theory. For $p$-isolated groups with Sylow $p$-subgroups of order $p$, we produce explicit new calculations of K-groups.

I formalize the ontology of apocalyptic events as synchronized morphogenetic manifolds within the framework of Thom's catastrophe theory. Local catastrophes (folds, cusps, umbilici) are extended to higher-order systemic collapses through the synchronization of multiple morphogenetic manifolds. The resulting construct is the Apocalypsis: a topological meta-singularity generated by the alignment of local singularities into a global structure of collapse. The mathematical formalization of Apocalyptic events and Apocalypsis integrates dynamical systems theory, topological stability, and probabilistic dependence structures using Archimedean copulas that capture nonlinear interrelations among coupled subsystems. The Inevitability Theorem demonstrates the existence, genericity, and almost-sure occurrence of Apocalypsis under stochastic coupling.

We study a family of pseudodifferential operators (quantum Hamiltonians) on $L^{2}(\mathbb{R}^{n};\mathbb{C}^{d})$ whose spectrum exhibits two energy bands exchanging a finite number of eigenvalues. We show that this number coincides with the Chern index of a vector bundle associated to the principal symbol (the classical Hamiltonian). This result provides a simple yet illustrative instance of the Atiyah Singer index formula, with applications in areas such as molecular physics, plasma physics or geophysics. We also discuss the phenomenon of topological contact without exchange between energy bands, a feature that cannot be detected by the Chern index or K theory, but rather reflects subtle torsion effects in the homotopy groups of spheres.

We give an internal description of constructible objects in an $\infty$-topos. More precisely, $P$-consctructible objects are locally constant objects internal to Fun($P$,An), for any noetherian poset $P$.

We reprove the generalized Nandakumar-Ramana Rao conjecture for the prime case using representation ring-graded Bredon cohomology. Our approach relies solely on the $RO(C_p)$-graded cohomology of configuration spaces, viewed as a module over the $RO(C_p)$-graded Bredon cohomology of a point.

This paper introduces a novel paradigm for the analysis and verification of concurrent programs -- the Singularity Theory. We model the execution space of a concurrent program as a branched topological space, where program states are points and state transitions are paths. Within this framework, we characterize deadlocks as attractors and livelocks as non-contractible loops in the execution space. By employing tools from algebraic topology, particularly homotopy and homology groups, we define a series of concurrent topological invariants to systematically detect and classify these concurrent "singularities" without exhaustively traversing all states. This work aims to establish a geometric and topological foundation for concurrent program verification, transcending the limitations of traditional model checking.

Let $M$ be the disk or a compact, connected surface without boundary different from the sphere $S^2$ and the real projective plane $\mathbb{R}P^2$, and let $N$ be a compact, connected surface (possibly with boundary). It is known that the pure braid groups $P_n(M)$ of $M$ are bi-orderable, and, for $n\geq 3$, that the full braid groups $B_n(M)$ of $M$ are not bi-orderable. The main purpose of this article is to show that for all $n \geq 3$, any subgroup $H$ of $B_n(N)$ that satisfies $P_n(N) \subsetneq H \subset B_n(N)$ is not bi-orderable.

We develop a systematic framework for constructing (3+1)-dimensional topological quantum field theories (TQFTs) that realize specified anomalies of finite symmetries, as encountered in gauge theories with fermions or fermionic lattice systems. Our approach generalizes the Wang-Wen-Witten symmetry-extension construction to the fermionic setting, building on two recent advances in the study of fermionic TQFTs and related homotopy theory. The first is the categorical classification of anomalous TQFTs in (3+1)d. The second, which we develop further in a planned sequel to this paper, is a hastened Adams spectral sequence for computing supercohomology groups, closely paralleling techniques from cobordism theory. By integrating supercohomology and cobordism methods within the recently developed categorical framework of fusion 2-categories, we provide a concrete and systematic route to constructing fermionic TQFTs with specified anomalies, thereby establishing a conceptual bridge between anomaly realization, cobordism, and higher-categorical structures.

The goal of this paper is to put the theory of approximate fibrations into the framework of higher topos theory. We define the notion of an approximate fibration for a general geometric morphism of $\infty$-topoi, give several characterizations in terms of shape theory and compare it to the original definition for maps of topological spaces of Coram and Duvall. Furthermore, we revisit the notion of cell-like maps between topoi, and generalize Lurie's shape-theoretic characterization by giving a purely topos-theoretical proof.