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Cyclic codes over finite fields are widely implemented in data storage systems, communication systems, and consumer electronics, as they have very efficient encoding and decoding algorithms. They are also important in theory, as they are…
We develop the fundamentals of a new theory of convex geometry -- which we call "broken line convex geometry". This is a theory of convexity where the ambient space is the rational tropicalization of a cluster variety, as opposed to an…
This overview paper is intended as a quick introduction to Lie algebras of vector fields. Originally introduced in the late 19th century by Sophus Lie to capture symmetries of ordinary differential equations, these algebras, or…
In this note we present a survey on some classical and modern approaches on Pythagorean triples. Some questions are also posed in direction of some materials under review. In particular some non commutative and operator theoretical…
Most of the literature of computational geometry concerns geometric properties of sets of static points. M.J. Atallah introduced dynamic computational geometry, concerned with both momentary and long-term geometric properties of sets of…
A mostly expository account of old questions about the relationship between polyhedra and topological manifolds. Topics are old topological results, new gauge theory results (with speculations about next directions), and history of the…
In recent years, attempts to generalize lattice gauge theories to model topological order have been carried out through the so called $2$-gauge theories. These have opened the door to interesting new models and new topological phases which…
An idea to present a classical Lie group of positive dimension by generators and relations sounds dubious, but happens to be fruitful. The isometry groups of classical geometries admit elegant and useful presentations by generators and…
Quantum geometry, i.e., the quantum theory of intrinsic and extrinsic spatial geometry, is a cornerstone of loop quantum gravity. Recently, there have been many new ideas in this field, and I will review some of them. In particular, after a…
This is the introductive paper to the volume "Symmetries in Physics: Philosophical Reflections", Cambridge University Press, 2003. We begin with a brief description of the historical roots and emergence of the concept of symmetry that is at…
One of the themes in algebraic geometry is the study of the relation between the ``topology'' of a smooth projective variety and a (``general'') hyperplane section. Recent results of Nori produce cohomological evidence for a conjecture that…
In this paper we construct several arrangements of lines and/or conics that are derived from the geometry of the Klein arrangement of $21$ lines in the complex projective plane.
An overview of new 4d supersymmetric gauge theories with 2-form gauge potentials constructed by various authors during the past five years is given. The key role of three particular types of interaction vertices is emphasized. These…
Supersymmetric gauge theories in four dimensions have taught us many important physics lessons. These can both inform and be informed by future work on the lattice. I focus on three issues: the properties of supersymmetric Yang-Mills theory…
These draft notes are from a graduate course given by the author in Berkeley during the spring semester of 2005. They cover the basic ideas of a new, geometric approach to geometric measure theory. They begin with a new theory of exterior…
In this thesis we study classical aspects of superconformal field theory via symmetry principles. Specifically, by employing the powerful setup of conformal superspace, we obtain a plethora of new results in the fields of geometric and…
In this paper, we survey constructions of and nonexistence results on combinatorial/geometric structures which arise from unions of cyclotomic classes of finite fields. In particular, we survey both classical and recent results on…
Cyclic contractions generalize the usual contractivities in metric spaces and $b$-MSs. In this paper, we enhance several fixed point theorems related to cyclic (i) Banach self-maps, (ii) Chatterjea contractivities, (iii) Kannan…
Mathematical aspects of contemporary classical and quantum gauge theory are sketched.
The classical Lorenz system is considered. For many years, this system has been the subject of study by numerous authors. However, until now the structure of the Lorenz attractor is not clear completely yet, and the most important question…