a Voyage

Starting a field of mathematics is like taking an expedition across a vast ocean. The founders of the expedition decide where their boat will be going and they embark towards their destination. They have some idea of what they will find (if and) when they get there and they hope to find things they did not expect there as well.

 However, the ocean may be large enough that the initial crew is not there to arrive at their destination. The crew of the ship then must convince the second generation that the boat is heading in an interesting and fruitful direction and assure them of the bounties that lay ahead so that they may maintain the course of the ship. If the ocean is vast enough, the second generation may have to convince the third , and so the third the fourth, and so on.

 It may happen that, by the tenth generation, the crew does not know where the boat is going. They even do not know where the boat originally departed. They are happy maintaining the ship and aspire to become the best sailors possible.

 The ship begins to sail around in circles amongst the endless sea that encompasses their vision. They never arrive and the ship is lost en route.

 Each Mathematician, whom is not just a sailor but a true adventurer, must know both from where their boat took sail and to where their boat is due to arrive. Their duty is to keep the ship on track and to have a crew left with them to enjoy the plunder.

Unifying

For most of time the universe consisted of two distinct realms of situation: the Earth and the Sky.

The situations present on Earth are inexplicable and random. Chaotic and unprecedented. Storms, disease, droughts, floods, and scarcity plagued civilization with no certain forewarning. Things just happen and are continually happening, here, on the Earth..

The situations presented in the Sky are ordered and cyclic. Determined and precise. Constellations, planets, moon cycles, and eclipses occur like a heartbeat: steady and continuous for which they are studied so dutifully.

The Earth and Sky are separate and are governed by different rules, laws, and principles.

The only way in to understand the situations of Earth is with careful attention paid to the situations of the Sky. The appearance of constellations predict changing seasons. Regularities such as sunrise and sunset, eclipses, and colored moons give order for referencing passage of time. Knowledge and accurate measurement of the situations of the Sky are necessary, applicable, and essential to the perpetuation of life on Earth.  Accurate maps of the stars (and understanding of the properties of triangles) guided sailors across the seemingly endless seas and oceans we dared to cross.

After Newton we find that a single equation, F=M*A, can describe, with unprecedented accuracy, the situations of both realms bringing us order, understanding, and determination of the ever changing, unpredictable situations of Earth. The mathematical laws of Classical Mechanics unified our perception of our universe. We then put the first people on the moon using Calculus.

a Sense

Who discovered the sun? Thats a bad question because no one person did. We discover the sun every morning with our eyes. The answer is same for the Moon. With just our eyes alone we discovered Mercury, Venus, Mars, Jupiter, Saturn and Uranus. The naked eye alone is even sufficient enough to observe an entire separate galaxy: Andromeda. However: there are much more objects in the universe besides these and most cannot be observed with the naked eye.

How did we discover Neptune? With another of our senses: Mathematics.

Careful calculations and measurements of the orbit of Uranus displayed a violation of the Mathematical models used to predict its motion. With firm convictions in the accuracy of the model we are forced to postulate a reason for this violation.

There must exist an undiscovered massive object located in a definite position orbiting the Sun, floating alongside the other celestial beings making their way across the sky. We predicted its mass and location in space and pointed our telescopes toward that region of space. Low and behold: a new planet. Neptune.

Einstein used the language of Mathematics from the field Differential Geometry, guided by intuitions and elementary hypotheses, to run thought experiments. His theory was able to make predictions for which it took decades to have machines accurate enough to verify (i.e. Gravitational waves). But since his mental experiments, Mathematics, have lined up with experimental evidence we are afforded a new language, rich with analogies and thinking tools, with which to discuss and understand reality itself.

We use our eyes and ears to sense the world around us and we use Mathematics to do the same. Eyesight allows the concepts of light and dark, red or blue, to become part of our sense and become part of our every day language: for literal functionality and also for exploitation by analogy. Mathematics brings quantity, order, shape, symmetry, randomness, and the infinite into our sense (and is our sense of it) and achieves the same outcome (for those who speak it).

About the Author

I think of Mathematics as a non-linear narrative passed down throughout generations of humankind; mathematicians are the keepers of these stories and their associated problems that have been passed down by their ancestors. From this angle, I am interested in mathematics as a whole. Ive never met a theorem which I disliked but I have seen some theorems which I would call propositions.

My research is Algebraic Number Theory (as initiated by the great Johann Carl Friedrich Gauss). This means I am interested in (1) the solutions of polynomial equations in one or more variables over finite fields and (2) studying the arithmetic invariants associated to algebraic/arithmetic schemes over the algebraic numbers/integers . Most papers in Algebraic Number theory, however, are far removed from this setting. There is a unified way to understand both of these questions: Zeta functions (L-Functions). Reciprocity laws are (these days) represented by proving analytic continuation of the Artin L-functions, associated to nontrivial Galois representations, to the entire complex plane (Langlands Program). Studying arithmetic invariants is understood through the special values, poles, zeros, and residues of the associated L-functions. One amazing example being the Hasse-Weil Zeta function of an elliptic curve defined over the rational numbers (Modularity of Elliptic Curves and the Birch-Swinnerton Dyer Conjecture).

My current research is on mod-p congruences of systems of Hecke-Eigenvalues of genus 2 ordinary Siegel Modular forms with emphasis on proving Level Lowering theorems and computing lower bounds on the degrees of Hida Families via p-adic L-Invariants.

Purpose

How did we figure out that the Earth is a sphere? The Sun. It shines the light which imprints the Earths shadow on the face of the Moon. No matter the angle with which the light hits the Earth a shadow of a circle appears. The only three dimensional object with this property is a sphere. We cannot understand the Earth on its own because we live in/on it. We need both the sun as source of illumination and the moon as a surface to reflect the shadow of Earth onto.

The purpose of this blog is to shine a light on Mathematics, from different angles, in order to cast various shadows of its true architecture on the surface of our understanding. Each shadow is a local presentation of the global phenomenon that is Mathematics and will allow us to sense different facets, aspects, utilities, and phenomenologies associated with Mathematics, Doing Mathematics and Experiencing Mathematics while always knowing that each of the shadows are but a mere analogy and cannot, on their own, reveal the true shape of Mathematics itself.