I made several scientific discoveries (mainly in abstract mathematics but also in XML files processing). I claim to be a genius based on this fact. But what I mean saying “I am a genius”? I mean just that all the rest experts showed themselves… idiots. Most of the formulas are too simple. Nobody guessed, nobody followed the rules taught to sophomores if not freshmen.
The first thing I discovered is the theory of filters. Mathematicians defined filters on sets and more generally on lattices and on posets and proved some theorems about them. Many tens of years nobody wrote an article with a systematic study of filters. I did this. I did this before I knew how filters are related with topologies (Stone duality).
Later I discovered what I call “filtrators”. A filtrator is just a pair of a poset and its subset (yes, that’s the definition of a filtrator, can it be easier?) If we take the poset the set of filters ordered reversely to set theoretic inclusion and the subset the set of principal filters, then essentially the set of filters becomes a filtrator. So I generalized many of my theorems about filters for filtrators.
My main math discovery is “funcoids”. I did it at the first year of a university. I decided: I will discover an algebraic way of general topology. I tried to understand the “mystery” of topological spaces. A topological space maps points to filters. I generalized it to mapping filters to filters and wrote the following formulas (note that I changed the notation below to be easier to explain instead of my first year self-contradictory notation):
- (I equated sets such as with principal filters.)
- (or was it initially ?)
Here , , are lattice and poset operators on the poset of filters. and denote arbitary filters.
Can it be simpler? A sophomore could write these formulas.
The third formula was superfluous and I later removed it.
The next step was to consider two such “Deltas”: and and realize they bijectively correspond to each other by the formula: .
It was almost the definition of a funcoid. To make it (almost) the modern definition of funcoid, it is enough to take and arbitrary functions on the set of filters on some fixed set. Then the formulas and follow automatically.
I can be proud that I did this before I became a sophomore.
Then I proved that a funcoid can be alternatively (up to an isomorphism, if I knew the word isomorphism that time, I am not sure) defined as a binary relation between sets conforming to the axioms:
The above formulas are worth the title of a genius. Many tens of years nobody guessed to write them down. Every third year student would be able to.
Then I developed the theory of funcoids in details. It was an extremely interesting journey with many personal discoveries. Some conjectures are yet unsolved.
Another “big” thing I guessed was to consider the set of filters on a cartesian square of a set. I proudly called this “reloids” (as a generalization of binary relations). It seems nobody before me researched this obvious in idea topic. Reloids are comparably interesting to funcoids (however, I’d say funcoids are my main discovery). I tried to research reloids like funcoids in as much details as I can.
Then I discovered than funcoids and reloids are related and founds several mappings between the class of funcoids and the class of reloids. First I mapped reloids into funcoids with a certain maps that I denoted . It was a stunning for me discovery that there are two distinct “easy and natural” ways to map funcoids into reloids. I also researched all these three mappings and their relations with each other.
There were many interesting details and specific “discoveries” on the way, but the above describes the essence of this most starting stage of my research. I will also note that I soon discovered that both funcoids and reloids are partially ordered (and are even lattices).
Pointfree funcoids were a simple but not quite trivial for me to guess generalization of funcoids. For them I defined a partial order in a specific case by a trivial generalization of the order on the set of funcoids. Later I generalized this order for arbitrary pointfree funcoids with a simple but taking time to guess formula.
One more big idea was “staroids”. Staroids are are generalization of pointfree funcoids. Definition is also simple but hard to guess and took some time to write down in the modern concise form. Later I also guessed prestaroids and completary staroids. Another man proved that completary staroids are not the same as staroids. Another because I am bad at solving problems and particularly at finding counter-examples, what he did.
Before staroids I also discovered related concept, “multifuncoids”.
It took some time to research some properties of staroids in many details. But in great degree staroids are a mystery yet.
Yet later I discovered the remarkable fact that the set of funcoids is isomorphic to the set of filters on a certain (boolean) lattice of sets. Am I the first person who described this simple lattice in details?
I also discovered continuity and connectedness for funcoids and for reloids. Continuity is remarkable by being described by simple algebraic (without quantifiers and epsilon-delta) formulas. It generalizes continuity, uniform continuity, and proximity continuity in the same formulas. I was greatly excited by this discovery. I found that these three formulas coincide for monovalued functions (and monovalued funcoids/reloids, which I later also described in simple algebraic way) but continuity of non-monovalued functions/funcoids/reloids is defined by me in three different and non-equivalent algebraic formulas. Yes, I defined continuity for non-monovalued functions.
Connectedness definitions also provided several yet unsolved open problems.
I easily defined limit of a funcoid, generalizing limit of a function on a filter.
The next stunning thing was generalized limit. I defined limit for an arbitrary (discontinuous) function and even for an arbitrary funcoid (as such it includes also any multivalued function). Here is the extermely simple (but not easy to guess) formula:
Here is a topological structure (funcoid) on the space where our function or more generally funcoid acts to and is a group such as the group of translations of our “source” vector topological space or like this.
Not too hard to prove that for continuous functions the generalized limit and customary limit are bijectively related. The generalized limit has algebraic operations (like addition, multiplication and like) which easily extend the operations on the points of our space.
Now we have every function differentiable (just like as we have square root of -1). But I didn’t succeed (yet) to put this generalized derivative into a differential equation, because the set of values of the limit is not the same as the set of values of the differentiated function. I tried this but it appeared hard: a function with singularity when differentiated produces an “infinitely big” derivative. If we differentiate this derivative, it becomes an “infinity of infinities”, etc. We have a countable sequence of bigger and bigger kinds of infinities or singularities (name it whatever you want). In my opinion, it is probably the most important problem of modern mathematics how to put them into both left and right part of a differential equation and define what is equality for them. I have only guesses what are non-smooth and discontinuous differential equations. Start working on this problem!
I wrote a book about all this. I wrote also some draft notes for the future and a partial draft of the second volume of the book. The second volume contains category theory things related with my theory. In the way I despite not being an expert in category theory discovered some new fundamental category theory concepts.
Also I introduces some simple but novel order theory concepts and theorems.
I wrote all above into an 393 pages book (also contains introductory materials for beginning students) + addons and second volume. I omitted many particular details above. It is freely available at my site.
Later I discovered a way to turn a category (with certain additional structure) into a semigroup. This way I “translate”/generalize funcoids and reloids from the language of category theory into the language of semigroup theory. In the way I understood what are “identities” in a rather wide generalization of them.
I also discovered two things unrelated to this:
- Theory of formulas. It is a very simple axiomatic system describing anything with different kinds (like the left and the right part of a formula) with constituent parts (like subformulas). I only started this research. I strongly suspect that I discovered something related with trinity of God. It has three parts and these parts seems to describe everything and are somehow related with things like logic and thus reason. Somethings like three ways to reason about anything or like this.
- A new way to process XML files. I dreamed for a long time to make processing of XML files in several stages based on elements namespaces automatically. Implementing this nearly obvious idea (I think I am not the first person thinking of it) turned out to be a work on many details with no big subideas, but many small but interesting subideas. I wrote a relatively complex data and algorithm specification. This is dissimilar to my math research where I had big ideas, but small ideas were less important. I was able to do both kinds of things: big ideas in math but details in programming. I successfully implemented the most important parts of in Python programming language. My command line utility took about a second to start what is too much. I tried several programming languages and finally chose D language to rewrite it in it to make it fast (and more reliable). This is a work in progress. It will replace HTML and other modern XML technologies. Yes, that’s the future, a specific way to implement the dreams of people who invented XML for it.
Oh, in the way once I acted like a crackpot. I assumed that I made an even greater discovery that I called “21st century math method” which I put without a proof at my site. Later I found these unproved theorems erroneous and deleted it from the site.
The conclusion: I made a series of discoveries worth at least trillions of dollars in this way. But if it would come that nobody guessed yet 50 years, it may cost more. I am probably two times more smart than Einstein. Only I guessed easy formulas, only I followed the sophomore rules. Special relativity would be guessed by somebody other in a few years. But to repeat my discoveries it takes tens of years. I closed a big hole in the humankind’s mathematics.
Now I want to win the legal battle with Russia.