Q-Html Article 

Quantum geometry is a generalization and extension of classical geometry. It incorporates various ideas and concepts of quantum physics, into the world of geometry. Quantum geometry deals with quantum spaces. In classical geometry spaces are always understandable as collections of points equipped with the appropriate additional structure. Quantum spaces are not viewable in this way. In general, quantum spaces have no points at all! They exhibit non-trivial quantum fluctuations of geometry at all scales. At the formal level, quantum spaces are described by certain non-commutative complex *-algebras. The elements of these algebras are interpretable as functions over the associated quantum spaces. Classical geometry is the commutative sector of quantum geometry. It is believed that quantum geometry could provide a consistent description of space-time at the level of ultra-small distances where classical concepts of the space-time continuum are not applicable. In principle, this could give the appropriate mathematical framework to formulate a coherent quantum theory of fundamental interactions.

Philosophical Article 

Subquantum theories pretend to go deeper than quantum mechanics. The idea is to provide a picture of physical reality which is based on individual physical systems, completely causal, and statistically compatible with quantum mechanics. Such a subquantum theory is logically possible. A natural mathematical framework for developing the subquantum theory is given by special extensions of C*-algebras of quantum observables. These structures naturally incorporate complementarity feature of quantum mechanics and allow us to introduce consistently (thanks to a special contextuality property of the subquantum world) the concept of a subquantum space. This space consists of all possible subquantum states of a given physical system. If a subquantum state of the system is known, then the values of all quantum observables for this system are completely determined. As a special topic, it is explained how to incorporate the principle of locality into the subquantum theory, using the appropriate non-classical probability theory on the subquantum space and overcoming the obstacles given by Bell's inequalities.

Fibered Article 

This section is devoted to my theory of quantum principal bundles. These objects constitute a very important class of quantum spaces. The basic conceptual picture is taken from classical geometry--the idea of a fibered space equipped with a free right action of a Lie group, so that the fibers are the orbits of the action. In the framework of quantum principal bundles, all the fibered space, the base manifold and the structure group are considered as quantum objects. The formalism of quantum principal bundles provides powerful tools for the study of the internal structure of quantum spaces, and gives a coherent mathematical framework for a formulation of the theory of fields and particles over a quantum space-time. Classical spaces are viewable in the new light, too.

Diagrammatic Article 

In this lecture, a diagrammatic formulation of generalized Hopf algebras is outlined. The formalism covers various examples of quantum spaces equipped with a group-like structure, going far beyond the framework of standard braided categories. In particular, the theory includes completely "pointless" objects, overcoming an inherent geometrical inhomogeneity of standard structures. All morphisms are built from the product and the coproduct maps. Furthermore, instead of just one braiding, an infinite system of mutually related braid operators emerges, expressing twisting properties of the product and coproduct. All braid type identities are consequences of simple axioms. Another interesting feature of the theory is its invariance under tensoring with matrix algebras.

This section includes the information about my weekly Seminar on Quantum Geometry at the Institute of Mathematics of UNAM.

It also features a basic sinopsis of my periodic one-semestral course entitled "Numbers, Space, Time and Light" presented at Mexican National Arts Center (CENART) and devoted to exploring connections between Arts, Mathematics & Physics.