CyberSec Research

Multiparameter persistence module can capture more topological differences across data instances compared to using a single parameter, where the well-studied matching distance investigates the distance along a straight line in the multiparameter space that gives the biggest difference. We propose to generalize the straight line to a monotone path filtration and offer software implementations.

It is known that, for all n, there exist compact differentiable orientable n-manifolds with dual Stiefel-Whitney class wbar_{n-ahat(n)} nonzero, and this is best possible, but the proof is nonconstructive. Here ahat(n) equals the number of 1's in the binary expansion of n if n equiv 1 mod 4 and exceeds this by 1 otherwise. We find, for all n nonzero mod 4, examples of real Bott manifolds with this property.

Hessenberg varieties are a family of subvarieties of full flag varieties. This family contains well-known varieties such as Springer fibers, Peterson varieties, and permutohedral varieties. It was introduced by De Mari-Procesi-Shayman in 1992 and has been actively studied in this decade. In particular, unexpected relations to hyperplane arrangements and the Stanley-Stembridge conjecture in graph theory have been discovered. Hessenberg varieties can be defined in partial flag varieties. In this paper, we study their cohomology by relating them to the cohomology of Hessenberg varieties in the full flag varieties.

This paper constructs numerous examples of highly connected Poincar\'{e} complexes, each homotopy equivalent to a topological manifold yet not homotopy equivalent to any smooth manifold. Furthermore, we determine the homotopy type of any closed $2k$-connected framed $(4k+2)$-manifold with Kervaire invariant one for $k=7,15,31$.

For logarithmic conformal field theories whose monodromy data is given by a not necessarily semisimple modular category, we solve the problem of constructing and classifying the consistent systems of correlators. The correlator construction given in this article generalizes the well-known one for rational conformal field theories given by Fuchs-Runkel-Schweigert roughly twenty years ago and solves conjectures of Fuchs, Gannon, Schaumann and Schweigert. The strategy is, even in the rational special case, entirely different. The correlators are constructed using the extension procedures that can be devised by means of the modular microcosm principle. It is shown that, as in the rational case, the correlators admit a holographic description, with the main difference that the holographic principle is phrased in terms of factorization homology. The latter description is used to prove that the coefficients of the torus partition function are non-negative integers. Moreover, we show that the derived algebra of local operators associated to a consistent system of correlators carries a Batalin-Vilkovisky structure. We prove that it is equivalent to the Batalin-Vilkovisky structure on the Hochschild cohomology of the pivotal module category of boundary conditions, for the notion of pivotality due to Schaumann and Shimizu. This proves several expectations formulated by Kapustin-Rozansky and Fuchs-Schweigert for general conformal field theories.

Monoidal product, braiding, balancing and weak duality are pieces of algebraic information that are well-known to have their origin in oriented genus zero surfaces and their mapping classes. More precisely, each of them correspond to operations of the cyclic framed $E_2$-operad. We extend this correspondence to include another algebraic piece of data, namely the modified trace, by showing that it amounts to a homotopy fixed point structure with respect to the homotopy involution that reverses the orientation of surfaces and dualizes the state spaces. We call such a homotopy fixed point structure reflection equivariance. As an application, we describe the effect of orientation reversal on spaces of conformal blocks and skein modules in the non-semisimple setting, throughout relying on their factorization homology description. This has important consequences: For a modular functor that is reflection equivariant relative to a rigid duality, i) the circle category is modular, and the resulting mapping class group representations are automatically the ones built by Lyubashenko, and ii) the resulting internal skein algebras have one simple representation, carrying a unique projective mapping class group representation making the action equivariant. While i) is a new topological characterization of not necessarily semisimple modular categories, ii) generalizes the implicit description of spaces of conformal blocks purely through the representation theory of moduli algebras given by Alekseev-Grosse-Schomerus from rational conformal field theories admitting a Hopf algebra description to finite rigid logarithmic conformal field theories. This also generalizes several results of Faitg from ribbon factorizable Hopf algebras to arbitrary modular categories.

The persistent homology transform (PHT) of a subset $M \subset \mathbb{R}^d$ is a map $\text{PHT}(M):\mathbb{S}^{d-1} \to \mathbf{Dgm}$ from the unit sphere to the space of persistence diagrams. This map assigns to each direction $v\in \mathbb{S}^{d-1}$ the persistent homology of the filtration of $M$ in direction $v$. In practice, one can only sample the map $\text{PHT}(M)$ at a finite set of directions $A \subset \mathbb{S}^{d-1}$. This suggests two natural questions: (1) Can we interpolate the PHT from this finite sample of directions to the entire sphere? If so, (2) can we prove that the resulting interpolation is close to the true PHT? In this paper we show that if we can sample the PHT at the module level, where we have information about how homology from each direction interacts, a ready-made interpolation theory due to Bubenik, de Silva, and Nanda using Kan extensions can answer both of these questions in the affirmative. A close inspection of those techniques shows that we can infer the PHT from a finite sample of heights from each direction as well. Our paper presents the first known results for approximating the PHT from finite directional and scalar data.

We study finite abelian group actions on weakly Lefschetz cohomologically symplectic (WLS) manifolds, a collection of manifolds that includes all compact connected Kaehler manifolds. We prove that for any WLS manifold $X$ there exists a number $C$ such that, for any integer $m\geq C$, if $({\mathbf Z}/m)^k$ acts freely on $X$, then $\sum_j b_j(X;{\mathbf Q})\geq 2^k$. We also prove a structure theorem for effective actions on WLS manifolds of $({\mathbf Z}/p)^r$, where $p$ is a big enough prime, analogous to some results for tori of Lupton and Oprea, and we find bounds on the discrete degree of symmetry of WLS manifolds. Our technique, which may be of independent interest, is based on studying the cohomology of abelian covers of WLS manifolds $X$ associated to certain maps $\pi:X\to T^k$. We prove that, in the presence of actions of arbitrarily big finite abelian groups, some of these abelian covers have finitely generated cohomology, and the spectral sequence associated to $\pi$ degenerates at the second page over the rationals.

We translate the operations of polarization and depolarization from monomial ideals in a polynomial ring to abstract simplicial complexes. As a result, we explicitly describe the relation between the Koszul simplicial complex of a monomial ideal and that of its polarization. Using the simplicial translation of depolarization we propose a way to reduce a simplicial complex to a smaller one with the same homology. This type of reduction, that can be interpreted as non-elementary collapse, can be used as a pre-process step for algorithms on simplicial complexes. We apply this methodology to the efficient computation of the Alexander dual of abstract simplicial complexes.

We give an intuitive combinatorial proof of Ky Fan's covering lemma based on the Borsuk-Ulam theorem. We then show how this approach can be generalized to Ky Fan's covering lemma for several linear orders.

In this work, we explore the structure of the embedding space of a transformer model trained for playing a particular reinforcement learning (RL) game. Specifically, we investigate how a transformer-based Proximal Policy Optimization (PPO) model embeds visual inputs in a simple environment where an agent must collect "coins" while avoiding dynamic obstacles consisting of "spotlights." By adapting Robinson et al.'s study of the volume growth transform for LLMs to the RL setting, we find that the token embedding space for our visual coin collecting game is also not a manifold, and is better modeled as a stratified space, where local dimension can vary from point to point. We further strengthen Robinson's method by proving that fairly general volume growth curves can be realized by stratified spaces. Finally, we carry out an analysis that suggests that as an RL agent acts, its latent representation alternates between periods of low local dimension, while following a fixed sub-strategy, and bursts of high local dimension, where the agent achieves a sub-goal (e.g., collecting an object) or where the environmental complexity increases (e.g., more obstacles appear). Consequently, our work suggests that the distribution of dimensions in a stratified latent space may provide a new geometric indicator of complexity for RL games.

We study Swan modules, which are a special class of projective modules over integral group rings, and their consequences for the homotopy classification of CW-complexes. We show that there exists a non-free stably free Swan module, thus resolving Problem A4 in the 1979 Problem List of C. T. C. Wall. As an application we show that, in all dimensions $n \equiv 3$ mod $4$, there exist finite $n$-complexes which are homotopy equivalent after stabilising with multiple copies of $S^n$, but not after a single stabilisation. This answers a question of M. N. Dyer. We also resolve a question of S. Plotnick concerning Swan modules associated to group automorphisms and, as an application, obtain a short and direct proof that there exists a group with $k$-periodic cohomology which does not have free period $k$. In contrast to the original proof our R. J. Milgram, our proof circumvents the need to compute the Swan finiteness obstruction.

Understanding the behavior and evolution of a dynamical many-body system by analyzing patterns in their experimentally captured images is a promising method relevant for a variety of living and non-living self-assembled systems. The arrays of moving liquid crystal skyrmions studied here are a representative example of hierarchically organized materials that exhibit complex spatiotemporal dynamics driven by multiscale processes. Joint geometric and topological data analysis (TDA) offers a powerful framework for investigating such systems by capturing the underlying structure of the data at multiple scales. In the TDA approach, we introduce the $\Psi$-function, a robust numerical topological descriptor related to both the spatiotemporal changes in the size and shape of individual topological solitons and the emergence of regions with their different spatial organization. The geometric method based on the analysis of vector fields generated from images of skyrmion ensembles offers insights into the nonlinear physical mechanisms of the system's response to external stimuli and provides a basis for comparison with theoretical predictions. The methodology presented here is very general and can provide a characterization of system behavior both at the level of individual pattern-forming agents and as a whole, allowing one to relate the results of image data analysis to processes occurring in a physical, chemical, or biological system in the real world.

The Enomoto-Satoh (ES) trace detects the Johnson cokernel, and its 1-cocycle property is important for the proof that the Johnson image is annihilated by the ES trace. Via the natural map from the ribbon graph complex introduced by Merkulov and Willwacher to the Chevalley-Eilenberg complex of the Lie algebra of symplectic derivations, where the Johnson image lives, the ES trace is essentially obtained from the 1-cocycle given by the unique ribbon graph with one vertex and one edge. In this perspective, this ribbon graph is the "universal" version of the ES trace. The main result of this paper is that there are no other (linearly independent) 1-cocycles in the ribbon graph complex, showing that nothing can be found there for the detection of the Johnson cokernel. The proof is done by applying the result of Church-Farb-Putman and Morita-Sakasai-Suzuki, which states that the virtual top-dimensional cohomology of the moduli space of marked Riemann surfaces vanishes.

We prove a rigidity result for certain $p$-complete \'etale $\mathbf{A}^{1}$-invariant sheaves of anima over a qcqs finite-dimensional base scheme $S$ of bounded \'etale cohomological dimension with $p$ invertible on $S$. This generalizes results of Suslin--Voevodsky, Ayoub, Cisinski--D\'eglise, and Bachmann to the unstable setting. Over a perfect field we exhibit a large class of sheaves to which our main theorem applies, in particular the $p$-completion of the \'etale sheafification of any $2$-effective $2$-connective motivic space, as well as the $p$-completion of any $4$-connective $\mathbf{A}^{1}$-invariant \'etale sheaf. We use this rigidity result to prove (a weaker version of) an \'etale analog of Morel's theorem stating that for a Nisnevich sheaf of abelian groups, strong $\mathbf{A}^{1}$-invariance implies strict $\mathbf{A}^{1}$-invariance. Moreover, this allows us to construct an unstable \'etale realization functor on $2$-effective $2$-connective motivic spaces.

Let $X$ be a compact Riemann surface $X$ of genus $\geqslant 2$ and let $\sigma:X \to X$ be an anti-holomorphic involution. Using real and quaternionic systems of Hodge bundles, we study the topology of the real locus $\mathbb{R} \mathbf{M}_{\mathrm{Dol}}(r,d)$ of the moduli space of semistable Higgs bundles of rank $r$ and degree $d$ on $X$, for the induced real structure $(E,\phi) \to (\sigma^*(\overline{E}),\sigma^*(\overline{\phi}))$. We show in particular that, when $\mathrm{gcd}(r,d)=1$, the number of connected components of $\mathbb{R} \mathbf{M}_{\mathrm{Dol}}(r,d)$ coincides with that of $\mathbb{R} \mathrm{Pic}_d(X)$, which is well-known.

We show that, for a finite spectrum $X$, the Spanier-Whitehead duality isomorphism induces an isomorphism between the cohomological and homological Atiyah-Hirzebruch spectral sequences. As an application, it follows that Poincar\'e duality for a Poincar\'e duality complex, which is oriented over a ring spectrum $E$, induces an isomorphism between the two spectral sequences.

Let $p$ be an odd prime, and let $n\in \N$ be an integer. We show that the $n^{\text{th}}$ mod-$p$ cohomology of a solvable saturable pro-$p$ group is isomorphic to the $n^{\text{th}}$ mod-$p$ cohomology of its associated $\Z_p$-Lie algebra $\g$ as a $\F_p$-vector space. Addittonally, we obtain that the $n^{\text{th}}$ mod-$p$ cohomology of $\g$ and of $\g/p\g$ are isomorphic as $\F_p$-vector spaces.