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We study quotients of quadratic forms and associated polar lines in the projective plane. Our results, applied pointwise to quadratic differential forms, shed some light on classical binary differential equations (BDEs) associated to…
This note provides an affirmative answer to a question of Viterbo concerning the existence of nondiffeomorphic contact forms that share the same Reeb vector field. Starting from an observation by Croke-Kleiner and Abbondandolo that such…
We provide a unified geometric realization of the classical deformation complexes. We construct GL-equivariant bilinear incidence varieties whose diagonal slices recover the varieties of associative, commutative, Leibniz, and Lie algebra…
In the field of enumeration of weighted walks confined to the quarter plane, it is known that the generating functions behave very differently depending on the chosen step set; in practice, the techniques used in the literature depend on…
We classify elliptic fibrations birational to a nonsingular, minimal cubic surface over a field of characteristic zero. Our proof is adapted to provide computational techniques for the analysis of such fibrations, and we describe an…
This dissertation presents a multifaceted look into the structural decomposition of permutation classes. The theory of permutation patterns is a rich and varied field, and is a prime example of how an accessible and intuitive definition…
Many statistical models are algebraic in that they are defined in terms of polynomial constraints, or in terms of polynomial or rational parametrizations. The parameter spaces of such models are typically semi-algebraic subsets of the…
This note provides an insight to the diophantine properties of abelian surfaces with quaternionic multiplication over number fields. We study the fields of definition of the endomorphisms on these abelian varieties and the images of the…
We enumerate complex algebraic hypersurfaces in $P^n$, of a given (high) degree with one singular point of a given singularity type. Our approach is to compute the (co)homology classes of the corresponding equi-singular strata in the…
Constructive-deductive method for plane Euclidean geometry is proposed and formalized within Coq Proof Assistant. This method includes both postulates that describe elementary constructions by idealized geometric tools (pencil, straightedge…
We customize the existing models for the bounded derived category of gentle algebras to obtain simple graph theoretic tools to analyze indecomposable objects, Auslander-Reiten triangles, and their interaction with the associated homological…
We give explicit computations of the $\Gamma$-Euler characteristic of several families of orbit space definable translation groupoids. These include the translation groupoids associated to finite-dimensional linear representations of the…
We study quantum deformed $gl(n)$ and $igl(n)$ algebras on a quantum space discussing multi-parametric extension. We realize elements of deformed $gl(n)$ and $igl(n)$ algebras by a quantum fermionic space. We investigate a map between…
Cylindrical algebraic decomposition is a classical construction in real algebraic geometry. Although there are many algorithms to compute a cylindrical algebraic decomposition, their practical performance is still very limited. In this…
Many classical geometric inequalities on functionals of convex bodies depend on the dimension of the ambient space. We show that this dimension dependence may often be replaced (totally or partially) by different symmetry measures of the…
Multisymplectic geometry is an adequate formalism to geometrically describe first order classical field theories. The De Donder-Weyl equations are treated in the framework of multisymplectic geometry, solutions are identified as integral…
This is a noncommutative version of the previous work entitled "Deformation Expression for Elements of Algebras (I)." In general in a noncommutative algebra, there is no canonical way to express elements in univalent way, which is often…
The study of solutions to polynomial equations over finite fields has a long history in mathematics and is an interesting area of contemporary research. In recent years the subject has found important applications in the modelling of…
We formulate quantum group Riemannian geometry as a gauge theory of quantum differential forms. We first develop (and slightly generalise) classical Riemannian geometry in a self-dual manner as a principal bundle frame resolution and a dual…
The elliptic gamma function is a generalization of the Euler gamma function. Its trigonometric and rational degenerations are the Jackson q-gamma function and the Euler gamma function. We prove multiplication formulas for the elliptic gamma…