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We propose and discuss how basic notions (quadratic modules, positive elements, semialgebraic sets, Archimedean orderings) and results (Positivstellensaetze) from real algebraic geometry can be generalized to noncommutative $*$-algebras. A…
We consider several classes of complete intersection numerical semigroups, aris- ing from many different contexts like algebraic geometry, commutative algebra, coding theory and factorization theory. In particular, we determine all the…
Real algebraic geometry is the study of semi-algebraic sets, subsets of $\R^k$ defined by Boolean combinations of polynomial equalities and inequalities. The focus of this thesis is to study quantitative results in real algebraic geometry,…
We introduce and investigate a category-theoretic abstraction of the standard "system-solution" adjunction in affine algebraic geometry. We then look further into these geometric adjunctions at different levels of generality, from syntactic…
We describe a method of computing equivariant and ordinary intersection cohomology of certain varieties with actions of algebraic tori, in terms of structure of the zero- and one-dimensional orbits. The class of varieties to which our…
We prove an equivalence of categories from formal complex structures with formal holomorphic maps to homotopy algebras over a simple operad with its associated homotopy morphisms. We extend this equivalence to complex manifolds. A complex…
We study the geometry of flag manifolds under different embeddings into a product of Grassmannians. We show that differential geometric objects and operations -- tangent vector, metric, normal vector, exponential map, geodesic, parallel…
Many statistical models are algebraic in that they are defined by polynomial constraints or by parameterizations that are polynomial or rational maps. This opens the door for tools from computational algebraic geometry. These tools can be…
We construct the differential geometry of smooth manifolds equipped with an algebraic curvature map acting as an area measure. Area metric geometry provides a spacetime structure suitable for the discussion of gauge theories and strings,…
In this text, we outline a theory of schemes associated with a site, which generalizes a variety of geometries, such as manifolds, schemes, analytic spaces, simplicial complexes, and more. We present an abstract process of gluing model…
Enumerative Geometry is concerned with the number of solutions to a structured system of polynomial equations, when the structure comes from geometry. Enumerative real algebraic geometry studies real solutions to such systems, particularly…
For any finite dimensional basic associative algebra, we study the presentation spaces and their relation with the representation spaces. We prove two propositions about a general presentation, one on its subrepresentations and the other on…
The Grassmann manifold of linear subspaces is important for the mathematical modelling of a multitude of applications, ranging from problems in machine learning, computer vision and image processing to low-rank matrix optimization problems,…
We give an accessible introduction into the theory of lower central series of associative algebras, exhibiting the interplay between algebra, geometry and representation theory that is characteristic for this subject, and to discuss some…
We provide a generalization of the reduction and Robertson positive maps in matrix algebras. They give rise to a new class of optimal entanglement witnesses. Their structural physical approximation is analyzed. As a byproduct we provide a…
Hausdorff relation, topologically identifying points in a given space, belongs to elementary tools of modern mathematics. We show that if subtle enough mathematical methods are used to analyze this relation, the conclusions may be…
The paper is essentially a continuation of B.Plotkin, G.Zhitomirski, "Some logical invariants of algebras and logical relations between algebras", St.Peterburg Math. J., {19:5}, (2008) 859 -- 879, whose main notion is that of…
Perceptual manifolds arise when a neural population responds to an ensemble of sensory signals associated with different physical features (e.g., orientation, pose, scale, location, and intensity) of the same perceptual object. Object…
By a numerical continuation method called a diagonal homotopy we can compute the intersection of two positive dimensional solution sets of polynomial systems. This paper proposes to use this diagonal homotopy as the key step in a procedure…
Amoebas are projections of complex algebraic varieties in the algebraic torus under a Log-absolute value map, which have connections to various mathematical subjects. While amoebas of hypersurfaces have been intensively studied in recent…