Algebra, Geometry, Topology & Applications

Abstract: One of the areas of active research in the field of categorical algebra is the study of so-called exactness properties of categories. The idea is to capture axiomatically various common features of universes (categories) of mathematical structures. The most famous examples of classes of categories defined by exactness properties are abelian categories and toposes.

Matrix properties are exactness properties that can be expressed as particular types of Horn clauses. They describe geometric features of internal relations in categories. In the field of universal algebra, they correspond to a special kind of linear Mal’tsev conditions. Both abelian categories and duals of toposes have a number of matrix properties in common. In this talk, after a non-technical introduction to the subject, we will present some results on the infinite poset of matrix properties of finitely complete categories, which have been obtained with some help from the computer in [1] and [2]. This talk will be an expanded version of a similar, but shorter talk [3] given in 2021.

[1] M. Hoefnagel, P.-A. Jacqmin, and Z. Janelidze, The matrix taxonomy of finitely complete categories, submitted for publication, 2021 (https://www.overleaf.com/read/rddpksbbjjbv)

[2] M. Hoefnagel, P.-A. Jacqmin, Z. Janelidze, and E. van der Walt, On binary matrix properties, submitted for publication, 2021 (https://www.overleaf.com/read/ttczxgsxhzgg)

[3] Z. Janelidze, A surprising story of how a computer was taught to prove some theorems in finitely complete categories, talk given at the 2021 Congress of South African Mathematical Society (https://www.zurab.online/2021/11/a-surprising-story-of-how-computer-was.html)

Abstract: Given any category C where one can do stable homotopy theory (such as a stable infinity category or a stable model category), together with a nice homology theory, we give conditions which guarantee that certain truncations of C are algebraic. This solves Franke’s algebraicity conjecture from 1996. As an application we give algebraic models for modules over certain ring spectra and the chromatic stable homotopy category for large primes. First half of the talk will be a general introduction in the subject. This is all joint with Piotr Pstrągowski.

Abstract: Graph and Network Visualization is a research area concerning automatic creation of pictorial representations of graphs. A node-link diagram (or simply a drawing of a graph) is one of the most intuitive of these representations: the vertices are represented as 2 or 3-dimensional objects and edges as (poly-) lines or curves connecting the adjacent vertices. Graph drawings are used in a number of fields, including social science, bioinformatics, neuroscience, electronics, software engineering, business informatics and humanities.

Many of the formally defined Graph Visualization problems are geometric in their nature. One of them is the problem of designing straight-line graph drawing in the scenario where the positions of the vertices are limited to a pre-specified set of points. This setting leads to a long-standing open problem [1] in the Graph Visualization and Computational Geometry fields. The problem asks whether there exists a set of O(n) points on the plane that can be used as vertex locations for a planar drawing for every planar graph with n vertices. This problem has been actively studied since late 80s and still remains widely open. In this talk, I will present an overview of the results on this problem and our recent progress on it.

[1] Small universal point sets for planar graphs

Abstract: This talk will contain an introduction to persistent homology, a fairly recent data analysis method which uses techniques from pure mathematics to obtain a short numeric summary for a large class of complex data sets. This numeric summary, called a persistence diagram or sometimes barcode, satisfies strong stability results with respect to a custom metric.

In the second part of this talk, we will introduce a novel analytical tool based on persistent homology that extracts quantitative features from chest CT scans to describe the geometric structure of the airways inside the lungs. We show that our new radiomic features stratify COPD patients in agreement with the GOLD guidelines for COPD and can distinguish between inspiratory and expiratory scans. These CT measurements are very different to those currently in use -- which do not give a complete picture of COPD -- and we demonstrate that they convey significant medical information. The results of this study are a proof of concept that topological methods can enhance the standard methodology to create a finer classification of COPD and increase the possibilities of more personalized treatment.

Abstract: Leibniz algebras were first introduced in [1] as non-skew-symmetric analogues of Lie algebras, but they became very popular after Loday rediscovered them in [5], mainly due to the development of a new, Leibniz (co)homology theory for Lie algebras. Since then, many authors have studied them and obtained very relevant algebraic results, and also in relation with physics and geometry. In particular, many results of Lie algebras have been extended to the Leibniz case.

The notions of non-abelian tensor and exterior products were introduced for groups by Brown and Loday [2] as tools for homotopy theory, but they can also give interesting information about central extensions and (co)homology. They were extended to the Lie setting in [3]. Later, the notion of the non-abelian tensor product was generalized to Leibniz algebras in [4]. But the exterior product for Leibniz algebras was left as an open question, as we believe, due to technical difficulties.

In this talk, we present the construction of the non-abelian exterior product of Leibniz algebras, with applications in low dimensional Leibniz homology and the study of capability property of Leibniz algebras. These results were obtained in [6, 7].

[1] A. Bloh, A generalization of the concept of Lie algebra. Sov. Math. Dokl. 6, 1450-1452 (1965).

[2] R. Brown, J. L. Loday, Van Kampen theorems for diagrams of spaces. Topology 26(3), 311-335 (1987).

[3] G.J. Ellis, A non-abelian tensor product of Lie algebras. Glasgow Math. J. 33(1), 101-120 (1991).

[4] A.V. Gnedbaye, A non-abelian tensor product of Leibniz algebras. Ann. Inst. Fourier (Grenoble) 49(4), 1149-1177 (1999).

[5] J.-L. Loday, Une version non commutative des algèebres de Lie: les algèbres de Leibniz. Enseign. Math. 39(3-4), 269-293 (1993).

[6] G. Donadze, X. Garcia-Martinez, E. Khmaladze, A non-abelian exterior product and homology of Leibniz algebras, Revista Mat. Complutense 31(1) (2018), 217-236.

[7] E. Khmaladze, R. Kurdiani, M. Ladra, On capability of Leibniz algebras, Georgian Math. J. 28 (2) (2021), 271-279.

Abstract: Finding a simple description of a convex hull of a set K in n-dimensional Euclidean space is a basic problem in mathematics.

When K has some additional geometric structures one may hope to give an explicit construction of its convex hull. A good starting point is when K is a space curve. In this talk I will describe convex hulls of space curves which have a "very" positive torsion. In particular, we obtain parametric representation of the boundary of the convex hull, different formulas for their Euclidean volumes, and the solution to a general moment problem corresponding to such curves .

Joint work with Jaume de Dios Pont and José Madrid.

Abstract: The aim of this talk is to give an axiomatic definition of the fundamental groupoid using the language of 2-categories. As it turns out, the fundamental groupoid satisfies a universal property: this is true for both the topological one (generalising the Poincare group) and the étale one (which generalises the Galois group).

Other than for reasons of a more conceptual nature, this result has also nice calculatory applications, even for the standard fundamental group.

We will give a few examples of this, in the second part of our talk. Indeed, there is a whole class of such examples: the real(-valued) toric varieties (and more generally, monoid schemes over the real numbers).

Abstract: In our joint work [1] with Alex Martsinkovsky we established some properties of the stable category of a left hereditary ring. In particular, we showed that if, in addition, a ring is left perfect and right coherent, then the stable category is complete. We also showed that the stable category (of an arbitrary ring) has all (small) coproducts. However, the existence of cokernels was proved only in the case where a ring is left hereditary and its injective envelope, viewed as a left module over itself, is projective (the latter condition is equivalent to the condition that the stable category is Abelian, as the main result of the above-mentioned paper asserts).

In the present work the above-mentioned result on cokernels is generalized. Namely, it is proved that if a ring is left hereditary, left perfect and right coherent, then the stable category has cokernels, and their explicit construction is given. For a left hereditary ring, the issue when the converse statement is true is studied. In particular, it is shown that the stable category of a Dedekind domain has cokernels if and only if the domain is left perfect. Several new necessary and sufficient conditions for a left hereditary ring to be left perfect and right coherent are found. One of them requires that the full subcategory of projective modules be reflective in the category of modules.

[1] A. Martsinkovsky, D. Zangurashvili, The stable category of a left hereditary ring, J. Pure Appl. Algebra, 219(2015), 4061-4089.

Abstract: In [1], Bazzoni and Glaz consider the problem of determining the possible values for the weak global dimension of a Gaussian ring. At the end of their paper they conjectured that the weak global dimension of a Gaussian ring is either 0, 1 or infinity. In [2], the author shows that the weak global dimension of a coherent Gaussian ring is either infinity or at most one. She also shows that the weak global dimension of a Gaussian ring is at most one if and only if it is reduced. So to prove the conjecture it is enough to show that the weak global dimension of an arbitrary non-reduced Gaussian ring is infinity. The latter was proved in [3].

In this talk we will present an overview of the subject.

[1] S. Bazzoni, S. Glaz, Gaussian properties of total rings of quotients, J. Algebra 310 (2007) 180-193

[2] S. Glaz, Weak dimension of Gaussian rings, Proc. Amer. Math. Soc. 133 (2005) 2507-2513

[3] G. Donadze, V. Z.Thomas, Bazzoni-Glaz Conjecture, J. Algebra 420 (2014) 141-160

Anstract: Bourn protomodular categories and S-protomodular categories in the sense of Bourn, Martins-Ferreira, Montoli, and Sobral will be defined; some examples will be given. The aim of the talk is to describe a new approach to the notion of S-protomodular category.

Abstract: linked pdf