Related papers: A few problems on monodromy and discriminants
Affine Kac-Moody algebras give rise to interesting systems of differential equations, so-called Knizhnik-Zamolodchikov equations. The monodromy properties of their solutions can be encoded in the structure of a modular tensor category on (a…
This paper is concerned with the Dirichlet problem for an equation involving the 1--Laplacian operator $\Delta_1 u$ and having a singular term of the type $\frac{f(x)}{u^\gamma}$. Here $f\in L^N(\Omega)$ is nonnegative, $0<\gamma\le1$ and…
This manuscript represents the author's PhD dissertation thesis.The first part studies decision problems in Thompson's groups F,T,V and some generalizations. The simultaneous conjugacy problem is determined to be solvable for Thompson's…
Arithmetical invariants---such as sets of lengths, catenary and tame degrees---describe the non-uniqueness of factorizations in atomic monoids. We study these arithmetical invariants by the monoid of relations and by presentations of the…
Over the past few decades, non-monotonic reasoning has developed to be one of the most important topics in computational logic and artificial intelligence. Different ways to introduce non-monotonic aspects to classical logic have been…
We construct a version of kneading theory for families of monotonous functions on the real line. The generality of the setup covers two classical results from Milnor-Thurston's kneading theory: the first one is to dynamically characterise…
Theories of classification distinguish classes with some good structure theorem from those for which none is possible. Some classes (dense linear orders, for instance) are non-classifiable in general, but are classifiable when we consider…
A real morsification of a real plane curve singularity is a real deformation given by a family of real analytic functions having only real Morse critical points with all saddles on the zero level. We prove the existence of real…
These lecture notes provide a unified overview of most known canonical desingularization methods in characteristic zero. It starts with discussing the classical method, and then proceeds with the recently discovered ones: logarithmic…
We develop a Galois theory for systems of linear difference equations with an action of an endomorphism {\sigma}. This provides a technique to test whether solutions of such systems satisfy {\sigma}-polynomial equations and, if yes, then…
Conventional ways to solve optimization problems on low-rank matrix sets which appear in great number of applications ignore its underlying structure of an algebraic variety and existence of singular points. This leads to appearance of…
The monodromy conjecture is a mysterious open problem in singularity theory. Its original version relates arithmetic and topological/geometric properties of a multivariate polynomial $f$ over the integers, more precisely, poles of the…
A reconstruction problem is formulated for Sperner systems, and infinite families of nonreconstructible Sperner systems are presented. This has an application to a reconstruction problem for functions of several arguments and identification…
We generalize the Moishezon Teicher algorithm that was suggested for the computation of the braid monodromy of an almost real curve. The new algorithm suits a larger family of curves, and enables the computation of braid monodromy not only…
We introduce an upper semi-continuous function that stratifies the highest multiplicity locus of a hypersurface in arbitrary characteristic (over a perfect field). The blow-up along the maximum stratum defined by this function leads to a…
The main aim of this article is to establish an $L_p$-theory for elliptic operators on manifolds with singularities. The particular class of differential operators discussed herein may exhibit degenerate or singular behavior near the…
Polynomial factorization in conventional sense is an ill-posed problem due to its discontinuity with respect to coefficient perturbations, making it a challenge for numerical computation using empirical data. As a regularization, this paper…
This is a short review of some recent results obtained by the author. These results are related the problem of obtaining polynomial identities (computational formulas) for some matrix functions by means of the known polarization theorem,…
This article, written for undergraduate mathematics students, provides an accessible introduction to a few key problems in tiling theory: Heesch's problem, the isohedral number problem, and the existence of an aperiodic monotile. I…
We simplify the proof of some widely used theoretical theorems, extending their applicability, while correcting some erroneous results. We also generalize key results and present new results that contribute to the development of the theory.…