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We describe general methods for enumerating subsemigroups of finite semigroups and techniques to improve the algorithmic efficiency of the calculations. As a particular application we use our algorithms to enumerate all transformation…
The toric Hilbert scheme parametrizes all algebras isomorphic to a given semigroup algebra as a multigraded vectorspace. All components of the scheme are toric varieties, and among them, there is a fairly well understood coherent component.…
A short survey on applications of algebraic geometry in topological data analysis.
Invariants withstand transformations and, therefore, represent the essence of objects or phenomena. In mathematics, transformations often constitute a group action. Since the 19th century, studying the structure of various types of…
We discuss the principle tools and results and state a few open problems concerning the classification and topology of plane sextics and trigonal curves in ruled surfaces.
Persistent homology is a common technique in topological data analysis providing geometrical and topological information about the sample space. All this information, known as topological features, is summarized in persistence diagrams, and…
We propose an algorithm to estimate the topology of an embedded metric graph from a well-sampled finite subset of the underlying graph.
In this paper, we revisit the problem of classifying real algebraic and semialgebraic sets by their topological types, focusing on establishing the effectiveness of bounds rather than deriving new quantitative estimates. Building on Hardt's…
A graph drawing in the plane is called an almost embedding if the images of any two non-adjacent simplices (i.e. vertices or edges) are disjoint. Almost embeddings (more precisely, their higher-dimensional analogues) naturally appear in…
The empirical results suggest that the learnability of a neural network is directly related to its size. To mathematically prove this, we borrow a tool in topological algebra: Betti numbers to measure the topological geometric complexity of…
Loop invariants are properties of a program loop that hold before and after each iteration of the loop. They are often employed to verify programs and ensure that algorithms consistently produce correct results during execution.…
This research addresses a new tool for data analysis known as Topological Data Analysis TDA It underlies an area of Mathematics known as Combinatorial Algebra or more recently Algebraic Topology which through making strong use of…
We review the method of symplectic invariants recently introduced to solve matrix models loop equations, and further extended beyond the context of matrix models. For any given spectral curve, one defined a sequence of differential forms,…
Geometry problem solving, a crucial aspect of mathematical reasoning, is vital across various domains, including education, the assessment of AI's mathematical abilities, and multimodal capability evaluation. The recent surge in deep…
This is a survey article on the currently very active research area of free (=non-commutative) real algebra and geometry. We first review some of the important results from the commutative theory, and then explain similarities and…
There are many numerical methods for solving partial different equations (PDEs) on manifolds such as classical implicit, finite difference, finite element, and isogeometric analysis methods which aim at improving the interoperability…
We consider the problem of constructing semisimple subalgebras of real (semi-) simple Lie algebras. We develop computational methods that help to deal with this problem. Our methods boil down to solving a set of polynomial equations. In…
In this paper and upcoming ones, we initiate a systematic study of Bethe ansatz equations for integrable models by modern computational algebraic geometry. We show that algebraic geometry provides a natural mathematical language and…
Consider the set of solutions to a system of polynomial equations in many variables. An algebraic manifold is an open submanifold of such a set. We introduce a new method for computing integrals and sampling from distributions on algebraic…
We survey results in algebraic complexity theory, focusing on matrix multiplication. Our goals are (i.) to show how open questions in algebraic complexity theory are naturally posed as questions in geometry and representation theory, (ii.)…