arXiv subjects

Atabey Kaygun

Publications and source records attributed to Atabey Kaygun.

At least 19 recordsLinked to original sources

arXiv2026

Local Smith Profiles of Twisted Calabi--Yau Algebras

The matrix Hilbert series of a locally finite elementary twisted Calabi--Yau algebra is the inverse of a matrix polynomial. The Smith normal form of this polynomial over the power series ring at $x=1$ produces a finite list of local exponents refining the Gelfand--Kirillov dimension, which records only the largest of them. We show that the Calabi--Yau symmetry makes this local data rigid: the algebra decomposes into ring factors according to the average Artin--Schelter index along Nakayama cycles, and after a local normalization the symmetry induces a nonsingular linking form on the Smith cokernel together with a finite-order Nakayama action on its layers, forcing reciprocal-eigenvalue and parity constraints on the multiplicities of the exponents. We compute the complete local data for cyclic skew-group algebras, derive a parity sieve for quiver classifications in dimension three, and realize, as an iterated smash product of a graded down-up algebra, a four-vertex type that a recent classification had left open.

math.RA↗

arXiv2026

Combinatorial Models for Linear Homotopy Theories

For a field $k$ of characteristic $0$, we compare $k$-linear chain complexes, semisimplicial vector spaces, augmented semisimplicial vector spaces, semicubical vector spaces, and arboreal vector spaces through small differential categorical algebras. We prove that semisimplicial modules and augmented semisimplicial modules are equivalent to appropriate chain-complex homotopy theories, both at the Gabriel--Zisman localization and the Quillen model-categorical level. The semicubical sign embedding gives a natural comparison from semicubical modules to augmented semisimplicial modules and induces a Quillen adjunction, but not a Quillen equivalence on the full semicubical category since there is an obstruction in augmented homology at degree $-1$.

math.AT↗

arXiv2026

Arboreal Objects and Their Homotopy Theory

We construct a category $\OrdFor$ as an arboreal extension of $\Delta_{\mathrm{epi}}\subseteq\Delta$, whose morphisms are ordered forests composed by grafting. We define a full functor $\pi\colon \OrdFor\to\Delta_{\mathrm{epi}}^{op}$ extracting the semisimplicial shadow. For every complete category $\mathcal C$, this induces a fully faithful functor from semisimplicial objects in $\mathcal C$ to $\mathcal C$-valued presheaves on $\OrdFor$, with right adjoint given by right Kan extension. We show that if weak equivalences of arboreal objects are detected by this right adjoint, then their Gabriel--Zisman localization is equivalent to that of semisimplicial objects. For bicomplete cofibrantly generated model categories, under the usual acyclicity hypothesis for right-induced transfer, the corresponding model structure on arboreal objects is Quillen equivalent to the Reedy model structure on semisimplicial objects.

math.AT↗

arXiv2026

Geometric Rigidity in Moduli Stacks of Algebras

We study quadratic moduli schemes $X$ of algebra laws on a fixed vector space $W$ under the transport-of-structure action of $GL(W)$ on $Hom(W^{\otimes 2},W)$. We construct an intrinsic three-term deformation complex on $X$ whose fibers encode transverse first-order classes and primary obstructions, and whose cohomology agrees on the operadic loci with the standard low-degree deformation cohomology (\`a la Gerstenhaber and Nijenhuis--Richardson). We then define a canonical quadratic map $\kappa^{inc}_{2,\mu}\colon H^2_{inc}(\mu)\to H^3_{inc}(\mu)$ that controls second-order lifts modulo isotriviality. If $\mu$ is smooth point in a reduced component and $(\kappa^{inc}_{2,\mu})^{-1}(0)=\{0\}$, then the $G$-orbit of $\mu$ is Zariski open in that component. This provides a coordinate-free explanation of Richardson-type geometric rigidity even when the second deformation cohomology does not vanish.

math.AG↗

arXiv2025

Geometry of Deformations via Incidence Varieties

We provide a unified geometric realization of the classical deformation complexes. We construct GL-equivariant bilinear incidence varieties whose diagonal slices recover the varieties of associative, commutative, Leibniz, and Lie algebra structures on a finite-dimensional vector space. We prove that the fiber of the incidence map at a given algebra law is canonically isomorphic to the space of 2-cocycles in the corresponding cohomology theory (Hochschild, Harrison, Leibniz, or Chevalley--Eilenberg). Furthermore, we introduce invariant bilinear forms to define open strata of separable and semisimple algebras, and demonstrate that these strata consist of open GL-orbits, establishing the rigidity of generic points in the coarse moduli spaces for all four geometries.

math.RA↗

arXiv2024

The Leibniz PROP is a crossed presimplicial algebra

We prove that the Leibniz PROP is isomorphic (as $\Bbbk$-linear categories) to the symmetric crossed presimplicial algebra $\Bbbk[(\Delta^+)^{op} \mathbb{S}]$ where $\Delta^+$ is the skeletal category of finite well-ordered sets with surjections, but the distributive law between $(\Delta^+)^{op}$ and the symmetric groups $\mathbb{S} = \bigsqcup_{n\geq 1} S_n$ is not the standard one.

math.CT↗

arXiv2024

A Model Categoric Equivalence for Crossed Simplicial Modules

We construct a model categorical equivalence between the category of simplicial vector spaces and the category of representations of a crossed simplicial group $\Delta G$ when each $G_n$ is finite and the characteristic of the ground field is 0.

math.AT↗

arXiv2022

Persistent Homology, Matroids and Cobordisms

The homological information about a filtered simplicial complex over the poset of positive real numbers is often presented by a barcode which depicts the evolution of the associated Betti numbers. However, there is a wonderfully complex combinatorics associated with the homology classes of a filtered complex, and one can do more than just counting them over the index poset. Here, we show that this combinatorial information can be encoded by filtered matroids, or even better, by rooted forests. We also show that these rooted forests can be realized as cobordisms.

math.AT↗

arXiv2022

Classification of Stochastic Processes with Topological Data Analysis

In this study, we examine if engineered topological features can distinguish time series sampled from different stochastic processes with different noise characteristics, in both balanced and unbalanced sampling schemes. We compare our classification results against the results of the same classification tasks built on statistical and raw features. We conclude that in classification tasks of time series, different machine learning models built on engineered topological features perform consistently better than those built on standard statistical and raw features.

stat.ML↗

arXiv2022

Quantum van Est Isomorphism

Motivated by the fact that the Hopf-cyclic (co)homologies of function algebras over Lie groups and universal enveloping algebras over Lie algebras capture the Lie group and Lie algebra (co)homologies, we hereby upgrade the classical van Est isomorphism to ones between the Hopf-cyclic (co)homologies of quantized algebras of functions and quantized universal enveloping algebras, both in $h$-adic and $q$-deformation frameworks.

math.QA↗

arXiv2021

Time Series Classification via Topological Data Analysis

In this paper, we develop topological data analysis methods for classification tasks on univariate time series. As an application, we perform binary and ternary classification tasks on two public datasets that consist of physiological signals collected under stress and non-stress conditions. We accomplish our goal by using persistent homology to engineer stable topological features after we use a time delay embedding of the signals and perform a subwindowing instead of using windows of fixed length. The combination of methods we use can be applied to any univariate time series and in this application allows us to reduce noise and use long window sizes without incurring an extra computational cost. We then use machine learning models on the features we algorithmically engineered to obtain higher accuracies with fewer features.

stat.ML↗

arXiv2021

Birational Equivalences and Kac-Moody Algebras

We show that every Kac-Moody algebra is birationally equivalent to a smash biproduct of two copies of a Weyl algebra together with a polynomial algebra. We also show that the same is true for quantized Kac-Moody algebras where one replaces Weyl algebras with their quantum analogues.

math.QA↗

arXiv2021

Enumerating Labeled Graphs that Realize a Fixed Degree Sequence

A finite non-increasing sequence of positive integers $d = (d_1\geq \cdots\geq d_n)$ is called a degree sequence if there is a graph $G = (V,E)$ with $V = \{v_1,\ldots,v_n\}$ and $deg(v_i)=d_i$ for $i=1,\ldots,n$. In that case we say that the graph $G$ realizes the degree sequence $d$. We show that the exact number of labeled graphs that realize a fixed degree sequence satisfies a simple recurrence relation. Using this relation, we then obtain a recursive algorithm for the exact count. We also show that in the case of regular graphs the complexity of our algorithm is better than the complexity of the same enumeration that uses generating functions.

math.CO↗

arXiv2020

A New Non-archimedean Metric on Persistent Homology

In this article, we define a new non-archimedean metric structure, called cophenetic metric, on persistent homology classes of all degrees. We then show that zeroth persistent homology together with the cophenetic metric and hierarchical clustering algorithms with a number of different metrics do deliver statistically verifiable commensurate topological information based on experimental results we obtained on different datasets. We also observe that the resulting clusters coming from cophenetic distance do shine in terms of different evaluation measures such as silhouette score and the Rand index. Moreover, since the cophenetic metric is defined for all homology degrees, one can now display the inter-relations of persistent homology classes in all degrees via rooted trees.

math.AT↗

arXiv2020

Birational Equivalences and Generalized Weyl Algebras

We calculate suitably localized Hochschild homologies of various quantum groups and Podle\'s spheres after realizing them as generalized Weyl algebras (GWAs). We use the fact that every GWA is birationally equivalent to a smash product with a 1-torus. We also address and solve the birational equivalence problem, and the birational smoothness problem for GWAs.

math.KT↗

arXiv2020

Noncommutative Fibrations

We show that faithfully flat smooth extensions are reduced flat, and therefore, fit into the Jacobi-Zariski exact sequence in Hochschild homology and cyclic (co)homology even when the algebras are noncommutative or infinite dimensional. We observe that such extensions correspond to étale maps of affine schemes, and we propose a definition for generic noncommutative fibrations using distributive laws and homological properties of the induction and restriction functors. Then we show that Galois fibrations do produce the right exact sequence in homology. We then demonstrate the versatility of our model on a geometro-combinatorial example. For a connected unramified covering of a connected graph $G'\to G$, we construct a smooth Galois fibration $\mathcal{A}_{G}\subseteq\mathcal{A}_{G'}$ and calculate the homology of the corresponding local coefficient system.

math.KT↗

arXiv2019

Homology of quantum linear groups

For every $n\geq 1$, we calculate the Hochschild homology of the quantum monoids $M_q(n)$, and the quantum groups $GL_q(n)$ and $SL_q(n)$ with coefficients in a 1-dimensional module coming from a modular pair in involution.

math.KT↗

arXiv2018

Jacobi-Zariski Exact Sequence for Hochschild Homology and Cyclic (Co)Homology

We prove that for an inclusion of unital associative but not necessarily commutative algebras $B\subseteq A$ we have long exact sequences in Hochschild homology and cyclic (co)homology akin to the Jacobi-Zariski sequence in André-Quillen homology, provided that the quotient $B$-bimodule $A/B$ is flat. We also prove that for an arbitrary r-flat morphism $f:B\to A$ with an H-unital kernel, one can express the Wodzicki excision sequence and the corresponding Jacobi-Zariski sequence in Hochschild homology and cyclic (co)homology as a single long exact sequence.

math.KT↗