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The theory of flag algebras, introduced by Razborov in 2007, has opened the way to a systematic approach to the development of computer-assisted proofs in extremal combinatorics. It makes it possible to derive bounds for parameters in…
We associate with the Grassmann algebra a topological algebra of distributions, which allows the study of processes analogous to the corresponding free stochastic processes with stationary increments, as well as their derivatives.
We answer some questions related to multiplicity formulas by Rosenthal and Zelevinsky and by Lakshmibai and Weyman for points on Schubert varieties in Grassmannians. In particular, we give combinatorial interpretations in terms of…
Existing approaches to solving combinatorial optimization problems on graphs suffer from the need to engineer each problem algorithmically, with practical problems recurring in many instances. The practical side of theoretical computer…
Linear algebraic expressions are the essence of many computationally intensive problems, including scientific simulations and machine learning applications. However, translating high-level formulations of these expressions to efficient…
We examine ordered graphs, defined as graphs with linearly ordered vertices, from the perspective of homomorphisms (and colorings) and their complexities. We demonstrate the corresponding computational and parameterized complexities, along…
We survey recent applications of topology and singularity theory in the study of the algebraic complexity of concrete optimization problems in applied algebraic geometry and algebraic statistics.
Complexity and decidability of logics is a major research area involving a huge range of different logical systems. This calls for a unified and systematic approach for the field. We introduce a research program based on an algebraic…
Adaptive stochastic gradient algorithms in the Euclidean space have attracted much attention lately. Such explorations on Riemannian manifolds, on the other hand, are relatively new, limited, and challenging. This is because of the…
Graphical models and factor analysis are well-established tools in multivariate statistics. While these models can be both linked to structures exhibited by covariance and precision matrices, they are generally not jointly leveraged within…
Motivated by the concept of Euclidean Distance Degree, which measures the complexity of finding the nearest point to an algebraic set in Euclidean space, we introduce the notion of Grassmann Distance Complexity (GDC). This concept…
A Schubert class is called rigid if it can only be represented by Schubert varieties. The rigid Schubert classes have been classified in Grassmannians and orthogonal Grassmannians. In this paper, we study the rigidity problem in partial…
This paper is a documentation of author's reseach, focusing on the topic Grassmann Algebra spanning over July, August 2025 under mentorship provided by DRP Turkiye 2025. Grassmann algebra is a fundamental structure in mathematics with…
This article is concerned with an example of complex planar geometry arising from flat origami challenges. The complexity of solution algorithms is illustrated, depending on the depth of the initial analysis of the problem, starting from…
A strong link between information geometry and algebraic statistics is made by investigating statistical manifolds which are algebraic varieties. In particular it it shown how first and second order efficient estimators can be constructed,…
Logic-based argumentation is a well-established formalism modelling nonmonotonic reasoning. It has been playing a major role in AI for decades, now. Informally, a set of formulas is the support for a given claim if it is consistent,…
In this paper we investigate the exactness of the Grassmannian BGG complexes introduced in a previous work (arXiv:1211.2486), and obtain some inequalities between some Hodge numbers of some irregular varieties. In particular, we obtain…
Finding, counting and/or listing triangles (three vertices with three edges) in large graphs are natural fundamental problems, which received recently much attention because of their importance in complex network analysis. We provide here a…
We compute the automorphism groups of finite and cofinite ind-grassmannians, as well as of the ind-variety of maximal flags indexed by Z_{>0}. We pay special attention to differences with the case of ordinary flag varieties.
Recent work in theoretical computer science and scientific computing has focused on nearly-linear-time algorithms for solving systems of linear equations. While introducing several novel theoretical perspectives, this work has yet to lead…