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WHY GEOMETRY AND TOPOLOGY?

I find Geometric solutions to practical Financial problems particularly powerful (here is an example). The connection between Quantitative Finance and Geometry may not be evident, however Geometry can be found at the root of fields as distant as Finance and Biology (see a connection here). Geometry allows you to formulate a problem visually and search for a solution in non-symbolic terms. How else were the Ancient Greeks able to solve complex Mathematical problems without ever using equations? Archimedes solved integral calculus questions using Geometric arguments, more than 1900 years before calculus was rediscovered by Leibniz and Newton. Gauss' first breakthrough was as a Geometer, and I suspect that the reason why Gauss' personal diaries are devoid of motivation is partly because he first approached problems Geometrically (like a Greek mathematician), and knew exactly where the arguments were leading him.

 Pythagoras' Theorem has received at least 370 different proofs over the last 25 centuries, many of them Geometrical. This theorem, and its generalization into the Law of Cosines, are at the center of most Risk models

Similarly, Topology helps unveil the complex structure governing a system. The "Seven Bridges of Königsberg" problem inaugurated that field in Mathematics, when Euler recognized that understanding a system required modeling the logical and hierarchical interrelations between its components. While at Cornell University, Richard Feynman revolutionized nearly every aspect of theoretical physics thanks to his Topological approach to handling the morass of calculations involved in quantum electrodynamics (QED) problems.

The goal of my research is to forecast financial systems and make optimal decisions under uncertainty. In order to achieve that, I may apply probability theory, inferential statistics, vector spaces, stochastic calculus, etc. However, when possible, I prefer to state a problem Geometrically or Topologically, because it gives me the leverage to comprehend it before committing equations to a paper. For example, here is a Geometry work from Ronald Fisher dealing with Karl Pearson's (a Geometry professor at Gresham College) most famous invention: Correlation. Geometry is the right way to think about correlations. And when it comes to understanding complex dynamic systems with logical and hierarchical relationships, Topology is the way to go!

 Financial markets are prime examples of complex dynamic networks: We can only understand the behavior of one price after modeling the dynamics of all prices. The above figure shows the Stochastic Flow Diagram of the global financial system, following an Energy shock

James Simons, one of the greatest living Geometers, also happens to be the founder of Renaissance Technologies. Since 1989, their flagship fund (Medallion) has averaged 35% annual returns, after fees of 5% (management) and 44% (performance). Renaissance's secret weapon may not so much consist in secret models, but a unique way of thinking about financial markets and coming up with novel trading ideas, inspired by Geometry and Topology. For instance, the dynamic co-dependence relationships across U.S. Equities can be modeled as a massive truss structure. Accordingly, an investment opportunity appears when that truss becomes an impossible object, or a faulty structure likely to collapse if overloaded in a particular direction. This approach is richer than standard statistical models.

Academic genealogy studies how schools of thought are transmitted over generations, from advisors to doctoral students. Studying one's academic ancestry is partly an amusement, a tribute or a homage, however it can also explain how some of us came (consciously or unconsciously) to think about these problems Geometrically and Topologically. Mathematicians with shared academic ancestors are more likely to read each others' publications and eventually work together. The tables below show my line of doctoral advisors, which quickly converges from Mathematical Finance to Geometry and Topology (here is a chart according to the Math Genealogy Project).

Source: American Mathematical Society, Complutense University's Dissertations Catalogue, Spain's Ministry of Science and Royal Academy of Economic Science.

 Differential Geometry plays a critical role in the Theory of Relativity. Miguel Vegas y Puebla-Collado wrote his 1888 doctoral dissertation on the Geometry of Curved Spaces, and became a leading researcher in that field. In this photo we can see Vegas (first from the left, seated) and Einstein (seated at the center) together, meeting at a Conference in 1923. Complutense's Blas Cabrera is seated at the right end

FOLLOWING THE BRANCH OF PROF. MIGUEL VEGAS

 STUDENT ADVISOR DISSERTATION FIELD YEAR UNIVERSITY COMMENTS Eduardo Torroja y Caballe 1888 Complutense University Prof. Vegas y Puebla-Collado's Analytic Geometry was an internationally acclaimed tractatus of Geometry, which followed the influences of Prof. Torroja, a disciple of Prof. Staudt. Member of the Royal Academy of Sciences since 1905. Karl Georg Christian von Staudt Geometry 1873 Complutense University He obtained his Math degree from Complutense in 1864. Very early in his studies he became a disciple of Karl Georg Christian von Staudt, whose ideas of Geometry he embraced and promoted among his fellow mathematicians for the rest of his life. Member of the Royal Academy of Sciences since 1891. Carl Friedrich Gauss Geometry 1822 The book Geometrie der Lage (1847) was a landmark in projective geometry. Staudt went beyond real projective geometry and into complex projective space in his three volumes of Beiträge zur Geometrie der Lage published from 1856 to 1860. The Staudt-Clausen theorem is partially named after him. Johann Friedrich Pfaff 1799 Sometimes referred to as Princeps Mathematicorum. In his 1799 doctorate in absentia, A new proof of the theorem that every integral rational algebraic function of one variable can be resolved into real factors of the first or second degree, Gauss proved the Fundamental Theorem of Algebra. Mathematicians including d'Alembert had produced false proofs before him, and Gauss's dissertation contains a critique of d'Alembert's work. Ironically, by today's standard, Gauss's own attempt is not acceptable, owing to implicit use of the Jordan curve theorem. However, he subsequently produced three other proofs, the last one in 1849 being generally rigorous.

Source: American Mathematical Society, Complutense University's Dissertations Catalogue and Spain's Royal Academy of Sciences.

 Seated, from left to right, Complutense mathematicians Julio Rey Pastor, Octavio de Toledo, Jose Maria Plans, Miguel Vegas and

FOLLOWING THE BRANCH OF PROF. JULIO REY PASTOR