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Let $X=G/H$ be a spherical variety over an algebraically closed field of characteristic $p\ge0$. We compute the $p'$-parts of $\pi_0(H)$ and $\pi_1(X)$ from the spherical system of $X$.

We study connections between the topology of generic character varieties of fundamental groups of punctured Riemann surfaces, Macdonald polynomials, quiver representations, Hilbert schemes on surfaces, modular forms and multiplicities in…

On the basis of the covariant description of the canonical formalism for quantization, we present the basic elements of the symplectic geometry for a restricted class of topological defects propagating on a curved background spacetime. We…

We give a concrete description for the boundary of the central quadratic hyperbolic component. The connectedness of the Julia sets of the boundary maps are also considered.

We establish a characterization of adequate knots in terms of the degree of their colored Jones polynomial. We show that, assuming the Strong Slope conjecture, our characterization can be reformulated in terms of "Jones slopes" of knots and…

From a finite set in a lattice, we can define a toric variety embedded in a projective space. In this paper, we give a combinatorial description of the dual defect of the toric variety using the structure of the finite set as a Cayley sum…

Hypertoric varieties are quaternionic analogues of toric varieties, important for their interaction with the combinatorics of matroids as well as for their prominent place in the rapidly expanding field of algebraic symplectic and…

We present two diagnostic methods based on ideas of Principal Component Analysis and demonstrate their efficiency for sophisticated processing of multicolour photometric observations of variable objects.

We give examples of complex hyperplane arrangements for which the top characteristic variety contains positive-dimensional irreducible components that do not pass through the origin of the character torus. These examples answer several…

The goal of these lectures is to explain speaker's results on uniqueness properties of spherical varieties. By a uniqueness property we mean the following. Consider some special class of spherical varieties. Define some combinatorial…

Invariants underlying shape inference are elusive: a variety of shapes can give rise to the same image, and a variety of images can be rendered from the same shape. The occluding contour is a rare exception: it has both image salience, in…

This article presents a simple characterization for entangled vectors in a finite dimensional Hilbert space $H$. The characterization is in terms of the coefficients of an expansion of the vector relative to an orthonormal basis for $H$.…

We define an analog in characteristic $p>0$ of the proalgebraic completion of the topological fundamental group of a complex manifold.

We announce results on a compactification of general character varieties that has good topological properties and give various interpretations of its ideal points. We relate this to the Weyl chamber length compactification and apply our…

In this paper, based on the value of central character on the transposition, we find structure and large genus asymptotics of certain Hurwitz numbers.

We provide a natural interpretation of the secondary Euler characteristic and introduce higher Euler characteristics. For a compact oriented manifold of odd dimension, the secondary Euler characteristic recovers the Kervaire…

We observe that the characteristic cycle of a D-module gives bounds for decomposition numbers of intersection cohomology complexes.

In this note, we discuss a number of parametricity features and what their requirements are in terms of complexity of the type system and its model.

In this note, finite type epimorphisms of rings are characterized.

How does the first order language of fields encode birational invariants of varieties?... This question is related to rational points on varieties and effectiveness in algebraic/arithmetic geometry.