arXiv subjects

Mariam Pirashvili

Publications and source records attributed to Mariam Pirashvili.

13 recordsLinked to original sources

arXiv2026

Homology-based Morphometry of Brain Atrophy: Methods and Applications

Understanding the structure of the brain, and how it changes with time and disease, is a core goal of structural neuroimaging. Contemporary approaches to structural brain analysis are dominated by voxel-wise, mass-univariate methods such as voxel-based morphometry (VBM). However, these techniques require images to be normalized to a standard template, which can obscure subject-specific geometric features. Normalization to a common stereotactic space can also be problematic when comparing groups with substantial brain pathology, lesions, or other anatomical abnormalities. Here, we introduce two complementary pipelines based on persistent homology (PH), a tool from topological data analysis, to quantify multiscale geometric features of structural T1-weighted MRI scans. Pipeline 1 quantifies regional thinning by applying the Euclidean distance transform to tissue masks in a slice-wise manner. Pipeline 2 uses $α$-filtrations to measure structural similarity between pairs of scans, capturing sulcal widening and ventricular enlargement. Synthetic experiments with controlled induced lesions showed that Pipeline 1 is best suited to between-subject analyses, whereas Pipeline 2 is better suited to within-subject designs. Applied to real-world data from the Alzheimer's Disease Neuroimaging Initiative (ADNI), Pipeline 1 separated Alzheimer's disease (AD) from cognitively normal (CN) participants using single-modality T1-weighted MRI without nonlinear registration (ROC-AUC = 0.895), with peak effects localized to medial temporal regions. Pipeline 2 captured disease-related longitudinal change, with follow-up scans remaining closest to their own baselines and AD subjects showing greater short-interval change than CN subjects. Together, these pipelines provide interpretable topological biomarkers for cross-sectional group comparisons and longitudinal tracking.

math.AT↗

arXiv2025

An isometry theorem for persistent homology of circle-valued functions

This paper explores persistence modules for circle-valued functions, presenting a new extension of the interleaving and bottleneck distances in this setting. We propose a natural generalisation of barcodes in terms of arcs on a geometric model associated to the derived category of quiver representations. The main result is an isometry theorem that establishes an equivalence between the interleaving distance and the bottleneck distance for circle-valued persistence modules.

math.AT↗

arXiv2025

On the centre of crossed modules of Lie algebras

This paper studies the relationship between crossed modules of Lie algebras and their centres. We show that any crossed module $\partial : L_1\to L_0$ of Lie algebras fits in an exact sequence involving cohomology of the homotopy Lie algebras $π_0(L_*)$ and $π_1(L_*)$.

math.CT↗

arXiv2025

On the Gottlieb group, Drinfeld centre and the centre of a crossed module

This new version includes a connection of the main construction to the Gottlieb group, which was absent in the previous versions. However, the first version included material about Lie algebras which will become available soon as a separate paper. The aim of this paper is to introduce the concept of the centre of a crossed module $\G_* = (\G_1\to \G_0)$. This centre is closely related to the Gottlieb group of the classifying space of a crossed module and also to the Drinfeld centre of a monoidal category introduced independently by Drinfeld and Joyal and Street. Our definition of the centre is based on certain crossed homomorphisms $\G_0\to \G_1$, which makes it easy to relate it to group cohomology. This connection is used to relate the Gottlieb group of a 2-type to its Whitehead centre.

math.KT↗

arXiv2021

Symmetric cohomology of groups and Poincaré duality

Let $G$ be a finite group of order $n$ and let $M$ be a $G$-module. We construct groups $H_*^\varkappa(G,M)$ for which $H_k^\varkappa (G,M^{tw}) \cong H^{n-k-1}_λ(G,M),$ where $M^{tw}$ is a twisting of a $G$-module $M$ defined in Section $5$ and $H^{*}_λ(G,M)$ is a variation of the group cohomology introduced by Zarelua, which in many cases is isomorphic to the symmetric cohomology of groups defined by Staic. The groups $H_*^\varkappa(G,M)$ come together with transformations from Tate cohomology. We find conditions under which these transformations are isomorphisms.

math.GR↗

arXiv2020

On the nonabelian cohomology with coefficients in a crossed module

This paper is concerned with the nonabelian cohomology of groups with coefficients in crossed modules. These objects were introduced by Dedecker and studied by Breen, Borovoi, Noohi and many others. In this paper we study several important objects that are classified by these cohomologies and for them we develop a theory similar to the Schreier obstruction theory of group extensions.

math.GR↗

arXiv2020

On the complexity of zero-dimensional multiparameter persistence

Multiparameter persistence is a natural extension of the well-known persistent homology, which has attracted a lot of interest. However, there are major theoretical obstacles preventing the full development of this promising theory. In this paper we consider the interesting special case of multiparameter persistence in zero dimensions which can be regarded as a form of multiparameter clustering. In particular, we consider the multiparameter persistence modules of the zero-dimensional homology of filtered topological spaces when they are finitely generated. Under certain assumptions, we characterize such modules and study their decompositions. In particular we identify a natural class of representations that decompose and can be extended back to form zero-dimensional multiparameter persistence modules. Our study of this set of representations concludes that despite the restrictions, there are still infinitely many classes of indecomposables in this set.

math.AT↗

arXiv2020

An outline of obstruction theories of extensions via track categories

Abelian track categories can be classified via the third Baues-Wirsching cohomology of small categories. This approach is used in this paper to compare and classify different generalisations of the obstruction theory of non-abelian group extensions, due to Cegarra, Garzón and Grandjean, Cegarra, Garzón and Ortega, and Chen, Du and Wang.

math.CT↗

arXiv2019

Topology and geometry of molecular conformational spaces and energy landscapes

Understanding the geometry and topology of configuration or conformational spaces of molecules has relevant applications in chemistry and biology such as the proteins folding problem, drug design and the structure activity relationship problem. Despite their relevance, configuration spaces of molecules are only partially understood. In this paper we discuss both theoretical and computational approaches to the configuration spaces of molecules and their associated energy landscapes. Our mathematical approach shows that when symmetries of the molecules are taken into account, configuration spaces of molecules give rise to certain principal bundles and orbifolds. We also make use of a variety of geometric and topological tools for data analysis to study the topology and geometry of these spaces.

q-bio.QM↗

arXiv2019

Crossed modules and symmetric cohomology of groups

This paper links the third symmetric cohomology (introduced by Staic and Zarelua ) to crossed modules with certain properties. The equivalent result in the language of 2-groups states that an extension of 2-groups corresponds to an element of $HS^3$ iff it possesses a section which preserves inverses in the 2-categorical sense. This ties in with Staic's (and Zarelua's) result regarding $HS^2$ and abelian extensions of groups.

math.KT↗

arXiv2017

Symmetric cohomology of groups

We investigate the relationship between the symmetric, exterior and classical cohomologies of groups. The first two theories were introduced respectively by Staic and Zarelua. We show in particular, that there is a map from exterior cohomology to symmetric cohomology which is a split monomorphism in general and an isomorphism in many cases, but not always. We introduce two spectral sequences which help to explain the realtionship between these cohomology groups. As a sample application we obtain that symmetric and classical cohomologies are isomorphic for torsion free groups.

math.GR↗

arXiv2015

Endomorphisms in short exact sequences

We sudy the behaviour of endomorphisms and automorphisms of groups involved in abelian group extensions. The main result can be stated as follows: Let $0\to N\to G\to Q \to 1$ be an abelian group extension. Then one has the following exact sequence of groups: $$0\to End^{N,Q}(G)\to End^Q_N(G)\to End_Q(N)\to H^2(Q,N)\to H^2(G,N)$$ where $End^{N,Q}(G)$ denotes the set of all endomorphisms of $G$ which centralise $N$ and induce identity on $Q$, $End^Q_N(G)$ denotes the set of all endomorphisms of $G$ which normalise $N$ and induce identity on $Q$ and $End_Q(N)$ denotes the set of endomorphisms of $N$ which are compatible with the action of $Q$ on $N$. This exact sequence is obtained using the five-term exact sequence in group cohomology. An interesting fact we discovered is that the first three terms involved have ring structure and the maps between them are ring homomorphisms. The ring structure on $End_Q(N)$ is well-known, however the ring structure of the second term is a little more exotic. Restricted on quasi-regular elements, this gives the exact sequence proved recently in \cite{passi} by Passi, Singh and Yadav.

math.GR↗

arXiv2015

Schreier Theory of Track Categories

This paper is a continuation of our study of non-abelian Baues-Wirsching cohomologies. In our previous paper, we defined second non-abelian cohomology H2(C;D) of a small category C with coefficients in a so-called centralised natural system D. We proved that H2(C;D) classifies linear extensions of C by D, generalising the corresponding result for abelian natural systems. For an abelian natural system D, the third cohomology classifies certain abelian track categories. A track category is a 2-category where all 2-morphisms are isomorphisms. A track category is called abelian if for every 1-morphism f, the group Aut(f) is abelian. In a similar fashion to the above, we want to generalise this result for non-abelian track categories. In this paper we solve this problem for the following important case: Given categories K and C and a functor ? p: K -> C, which is identity on objects and surjective on morphisms, and G, a centralised natural system of groups on K, we describe the equivalence classes of all track categories T for which K is the underlying category and C is the homotopy category and G_f = Aut(f).

math.CT↗