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I've discovered Algebraic General Topology (AGT), a new field of math which generalizes old General Topology. Mathematical Synthesis is how I call Algebraic General Topology applied to study of Mathematical Analysis.
PDF Slides prepared by me in order to give a talk at a research conference. Use these slides to quickly familiarize yourself with basics of Algebraic General Topology.
Algebraic General Topology. Volume 1 (PDF, preprint) My entire theory in one PDF file. You are strongly recommended to read this book, not draft articles below.
Let's consider generalizations of proximity spaces and uniform spaces.
Also in this article continuity is defined in algebraic manner which hides old epsilon-delta notion under a smart algebra. It generalizes continuity, uniform continuity, and proximity-continuity in one formula.
Orderings of filters which extend Rudin-Keisler preorder of ultrafilters are defined in terms of reloids. Also there is defined isomorphism of filters which extends Rudin-Keisler equivalence of ultrafilters.
It's defined the notion of convergence and limit for funcoids.
It's defined (generalized) limit of arbitrary (not necessarily continuous) functions under certain conditions.
It's defined the notion of connectedness for special binary relations called connectors. This generalizes topological connectedness, path connectedness, connectedness of digraphs, proximal connectedness, and some other kinds of connectedness.
Consider a generalization of funcoids.
This document lists all yet unsolved problems and conjectures in the field of AGT. You can discuss these open problems in Open Problem Garden and in Algebraic General Topology Research forum. If you solved any of these problems, read here.
They are defined multifuncoids. Their basis properties are researched.
I define product of two morphisms for certain categories. (These products are pointfree funcoids.) As a special case of this product of two funcoids and product of two reloids is defined.
I haven't yet shown that these are direct products in categorical sense.
As a generalization of product of two funcoids I define product of an arbitrary family of funcoids.
I consider some categories related with pointfree funcoids.
An additional open problem, said concisely: upgrading a multifuncoid is a multifuncoid. (The concepts of upgrading and multifuncoid are defined in this article.)
In this article I prove the conjecture for n=0,1,2. n=3 and above is unknown.
Operations with values of singularities. Conjectures about application of this to General Relativity and black holes.
Below are supplementary articles (Definitions and facts from these articles are used in the articles above.):
It's about filters on arbitrary lattices and posets as well as certain generalizations thereof.
Filters on Posets and Generalizations.
See also: Research Plans and Ideas in Algebraic General Topology (an informal document).
Quasi-cartesian functions - a blind valley of research, a theory which I am not going to publish.
Algebraic General Topology is about how to act with abstract topological objects expressing infinities with algebraic operations.
AGT is to math/functional analysis like algebra to old
prose
arithmetic.
Achievements and advantages:
We now we can get rid of math analysis as now it becomes synthesis, I would say. So I call AGT applied to study of such things as continuity, limits, and differentials Mathematical Synthesis.
AGT isn't a continuation of former functional analysis research, it is re-research anew beginning almost from scratch. This makes the knowledge level of the First course of a math faculty enough to understand it.
This new research field is both just generalizing former analysis and new theorems/concepts not having analogs in old theories. Several different theorems of analysis often collapse into one AGT equation of which they are obvious consequences.
AGT is very abstract, indeed even the current level of AGT knowledge often allowed me to find simple solutions of practical tasks (such as calculations of infinite sums). I have not yet reached the level of integrals in the synthesis research.
AGT is a kind of thinking with formulas. No real numbers analysis expressiveness with visual images preserved. That is not needed anyway as the equations of AGT are even more clear than graphics of old analysis. AGT is simple, natural, and beautiful.
Note that Algebraic General Topology being a generalization of General Topology has nothing in common (except of the name) with Algebraic Topology. Math synthesis is a generalization of functional analysis.
Some of my further research directions - collaborate
Funcoids and Reloids project at the projects site
Algebraic General Topology at Facebook.
Algebraic General Topology at WikInfo.
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