arXiv subjects

Márton Hablicsek

Publications and source records attributed to Márton Hablicsek.

15 recordsLinked to original sources

arXiv2026

Functoriality of logarithmic Hochschild homology of log smooth pairs

The construction of a satisfactory dg category of logarithmic coherent sheaves remains a central open problem in logarithmic geometry. In this paper, we propose an alternative correspondence-theoretic approach based on logarithmic Fourier--Mukai transforms. For smooth proper log pairs, we introduce strong log Fourier--Mukai kernels supported on canonical blow-up compactifications and prove that logarithmic Hochschild homology is functorial with respect to the induced transforms. Unlike the classical setting, logarithmic correspondences do not naturally live on ordinary products, and the standard adjunction formalism fails because of blow-up discrepancies. We overcome these difficulties by constructing explicit unit- and counit-type morphisms that provide the necessary adjunction data without requiring an ambient dg category of logarithmic sheaves. As applications, we construct a dg bicategory of logarithmic correspondences in which logarithmic Hochschild homology and cohomology become categorical invariants. We also define logarithmic Chern characters and a logarithmic Euler pairing compatible with the logarithmic Fourier--Mukai formalism.

math.AG↗

arXiv2026

Motivic (Representation) Stability of Representation Varieties and Character Stacks

In this paper, we introduce the notions of motivic representation stability that is an algebraic counterpart of the notion of representation stability. In the process, we also introduce the notion of motivic decomposition for varieties equipped with an action of a finite group $G$. This motivic decomposition decomposes the virtual class of the variety with respect to irreducible rational representations of $G$. We also formulate conjectures on motivic representation stability in the context of representation varieties and character stacks, and we verify the conjectures for groups whose virtual classes have been extensively studied.

math.AG↗

arXiv2025

Finitely Presentable Higher-Dimensional Automata and the Irrationality of Process Replication

Higher-dimensional automata (HDA) are a formalism to faithfully model the behaviour of concurrent systems. For ordinary automata, there is a correspondence between regular expressions, regular languages and finite automata, which provides a powerful link between algebraic proofs and operational behaviour. It has been shown by Fahrenberg et al. that finite HDA correspond with interfaced interval pomset languages generated by sequential and parallel composition and non-empty iteration, and thereby to a variant of Kleene algebras (KA) with parallel composition. It is known that this correspondence cannot be extended to concurrent KA, which additionally have process replication. An alternative to finite HDA are locally finite HDA, in which every state can only reach finitely many other states, and finitely branching HDA. In this paper, we show that both classes of HDA are closed under process replication and thus models of concurrent KA. To achieve this, we prove that the category of HDA is locally finitely presentable, where the finite HDA generate all other HDA. We then prove that this has the unfortunate side-effect that all HDA are locally finite, which means that the correspondence with concurrent KA trivialises. Similarly, we also show that, even though finitely branching HDA are closed under process replication, the resulting HDA necessarily have infinitely many initial states.

cs.FL↗

arXiv2025

Virtual Classes of Character Stacks

In this paper, we extend the Topological Quantum Field Theory developed by González-Prieto, Logares, and Muñoz for computing virtual classes of $G$-representation varieties of closed orientable surfaces in the Grothendieck ring of varieties to the setting of the character stacks. To this aim, we define a suitable Grothendieck ring of representable stacks, over which this Topological Quantum Field Theory is defined. In this way, we compute the virtual class of the character stack over $BG$, that is, a motivic decomposition of the representation variety with respect to the natural adjoint action. We apply this framework in two cases providing explicit expressions for the virtual classes of the character stacks of closed orientable surfaces of arbitrary genus. First, in the case of the affine linear group of rank $1$, the virtual class of the character stack fully remembers the natural adjoint action, in particular, the virtual class of the character variety can be straightforwardly derived. Second, we consider the non-connected group $\mathbb{G}_m \rtimes \mathbb{Z}/2\mathbb{Z}$, and we show how our theory allows us to compute motivic information of the character stacks where the classical naïve point-counting method fails.

math.AG↗

arXiv2024

Logarithmic Hochschild co/homology via formality of derived intersections

We define log Hochschild co/homology for log schemes that behaves well for simple normal crossing pairs $(X,D)$ or toroidal singularities. We prove a Hochschild-Kostant-Rosenberg isomorphism for log smooth schemes, as well as an equivariant version for log orbifolds. We define cyclic homology and compute it in simple cases. We show that log Hochschild co/homology is invariant under log alterations. Our main technical result in log geometry shows the tropicalization (Artin fan) of a product of log schemes $X \times Y$ is usually the product of the tropicalizations of $X$ and $Y$. This and the machinery of \emph{formality} of derived intersections facilitate a geometric approach to log Hochschild.

math.AG↗

arXiv2023

Arithmetic-Geometric Correspondence of Character Stacks via Topological Quantum Field Theory

In this paper, we introduce Topological Quantum Field Theories (TQFTs) generalizing the arithmetic computations done by Hausel and Rodríguez-Villegas and the geometric construction done by Logares, Muñoz, and Newstead to study cohomological invariants of $G$-representation varieties and $G$-character stacks. We show that these TQFTs are related via a natural transformation that we call the 'arithmetic-geometric correspondence' generalizing the classical formula of Frobenius on the irreducible characters of a finite group. We use this correspondence to extract some information on the character table of finite groups using the geometric TQFT, and vice versa, we greatly simplify the geometric calculations in the case of upper triangular matrices by lifting its irreducible characters to the geometric setting.

math.AG↗

arXiv2022

Virtual Classes of Representation Varieties of Upper Triangular Matrices via Topological Quantum Field Theories

In this paper, we use a geometric technique developed by González-Prieto, Logares, Muñoz, and Newstead to study the $G$-representation variety of surface groups $\mathfrak{X}_G(Σ_g)$ of arbitrary genus for $G$ being the group of upper triangular matrices of fixed rank. Explicitly, we compute the virtual classes in the Grothendieck ring of varieties of the $G$-representation variety and the moduli space of $G$-representations of surface groups for $G$ being the group of complex upper triangular matrices of rank $2$, $3$, and $4$ via constructing a topological quantum field theory. Furthermore, we show that in the case of upper triangular matrices the character map from the moduli space of $G$-representations to the $G$-character variety is not an isomorphism.

math.AG↗

arXiv2021

Derived intersections over the Hochschild cochain complex

The paper generalizes a result of Behrend-Fantechi and Baranovsky-Ginzburg to the 1-shifted cotangent bundle $T^*X[1]$ of a smooth scheme $X$ over the field of complex numbers. We show how one can obtain twisted cotangent bundles as derived intersections of Lagrangians in $T^*X[1]$, moreover and we show that the derived intersection of the quantized Lagrangians coincide with the canonical quantization of the twisted cotangent bundles.

math.AG↗

arXiv2020

Algebraic 3D Graphic Statics: reciprocal constructions

The recently developed 3D graphic statics (3DGS) lacks a rigorous mathematical definition relating the geometrical and topological properties of the reciprocal polyhedral diagrams as well as a precise method for the geometric construction of these diagrams. This paper provides a fundamental algebraic formulation for 3DGS by developing equilibrium equations around the edges of the primal diagram and satisfying the equations by the closeness of the polygons constructed by the edges of the corresponding faces in the dual/reciprocal diagram. The research provides multiple numerical methods for solving the equilibrium equations and explains the advantage of using each technique. The approach of this paper can be used for compression-and-tension combined form-finding and analysis as it allows constructing both the form and force diagram based on the interpretation of the input diagram. Besides, the paper expands on the geometric/static degrees of (in)determinacies of the diagrams using the algebraic formulation and shows how these properties can be used for the constrained manipulation of the polyhedrons in an interactive environment without breaking the reciprocity between the two.

cs.CG↗

arXiv2020

Non-Noetherian representation categories of generalized fields

In this paper, we show that the categories of finitely generated projective $\mathbb{B}$-modules and $\mathbb{F}_\infty$-modules with morphisms being (splittable) injections are not locally Noetherian. This provides another instance of the fact that these generalized fields have strange homological behavior.

math.CO↗

arXiv2016

Explicit computations of Hida families via overconvergent modular symbols

In [Pollack-Stevens 2011], efficient algorithms are given to compute with overconvergent modular symbols. These algorithms then allow for the fast computation of $p$-adic $L$-functions and have further been applied to compute rational points on elliptic curves (e.g. [Darmon-Pollack 2006, Trifković 2006]). In this paper, we generalize these algorithms to the case of families of overconvergent modular symbols. As a consequence, we can compute $p$-adic families of Hecke-eigenvalues, two-variable $p$-adic $L$-functions, $L$-invariants, as well as the shape and structure of ordinary Hida-Hecke algebras.

math.NT↗

arXiv2015

Hodge theorem for the logarithmic de Rham complex via derived intersections

In a beautiful paper Deligne and Illusie proved the degeneration of the Hodge-to-de Rham spectral sequence using positive characteristic methods. In a recent paper Arinkin, Căldăraru and the author of this paper gave a geometric interpretation of the problem of Deligne-Illusie showing that the triviality of a certain line bundle on a derived scheme implies the the Deligne-Illusie result. In the present paper we generalize these ideas to logarithmic schemes and using the theory of twisted derived intersection of logarithmic schemes we obtain the Hodge theorem for the logarithmic de Rham complex.

math.AG↗

arXiv2014

On the joints problem with multiplicities

In this short note we apply a recent theorem of Kollár about the arithmetic genus of curves to give a bound on the number of joints weighted by the multiplicities. This gives an affirmative answer to a conjecture of Carbery in the generic case.

math.CO↗

arXiv2011

Kakeya sets over non-archimedean local rings

In a recent paper of Ellenberg, Oberlin, and Tao, the authors asked whether there are Besicovitch phenomena in F_q[[t]]^n. In this paper, we answer their question in the affirmative by explicitly constructing a Kakeya set in F_q[[t]]^n of measure 0. Furthermore, we prove that any Kakeya set in F_q[[t]]^2 or Z_p^2 is of Minkowski dimension 2.

math.CO↗

arXiv2011

Power map permutations and symmetric differences in finite groups

Let $G$ be a finite group. For all $a \in \Z$, such that $(a,|G|)=1$, the function $ρ_a: G \to G$ sending $g$ to $g^a$ defines a permutation of the elements of $G$. Motivated by a recent generalization of Zolotarev's proof of classic quadratic reciprocity, due to Duke and Hopkins, we study the signature of the permutation $ρ_a$. By introducing the group of conjugacy equivariant maps and the symmetric difference method on groups, we exhibit an integer $d_{G}$ such that $\text{sgn}(ρ_a)=(\frac{d_G}{a})$ for all $G$ in a large class of groups, containing all finite nilpotent and odd order groups.

math.GR↗