I am currently doing my masters studies in financial mathematics. However, I have had a good background in number theory and I don't feel like leaving it just like that. I am thus inquiring on any applications of algebraic number theory in financial mathematics. When I finish this degree, can I be allowed to enroll for another masters in algebraic number theory? Can I marry the two and apply number theoretic concepts in finance? Please advise me.
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$\begingroup$ I think this question would be more appropriate for another site (academia?) $\endgroup$– Alex DegtyarevCommented Mar 19, 2015 at 10:50
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5$\begingroup$ @Alex Degtyarev The part that asks for a connection between two fields of mathematics is appropate for this site. The part about the master suits better in Academia as you say. Maybe, the question should be splitted. $\endgroup$– Josué Tonelli-CuetoCommented Mar 19, 2015 at 12:14
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7$\begingroup$ Having studied number theory and worked in finance, my answer to your second question is a definite no. $\endgroup$– Yaakov BaruchCommented Mar 19, 2015 at 12:22
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1$\begingroup$ With the strong caveat that I know essentially nothing about either subject, have you looked at any of the literature on algebraic statistics? $\endgroup$– RbegaCommented Mar 19, 2015 at 14:08
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I do not know any applications of algebraic number theory in financial mathematics. However, there are attempts to use string theory inspired approaches to financial markets: http://arxiv.org/abs/1109.0435 (The string prediction models as an invariants of time series in forex market, by R. Pincak and M. Repasan). On the other hand there is a deep and somewhat mysterious connection between number theory and string theory: https://www.quantamagazine.org/20150312-mathematicians-chase-moonshines-shadow/ (Mathematicians Chase Moonshine’s Shadow, by Erica Klarreich). Therefore I would not exclude that you can finally find some applications of number theory in financial mathematics.
Other possible directions to search alleged applications of number theoretic methods perhaps is path integral and gauge theory approaches to financial modeling. For applications of the path integral see, for example, http://deepblue.lib.umich.edu/handle/2027.42/44345 (The Path Integral Approach to Financial Modeling and Options Pricing, by Vadim Linetsky), http://arxiv.org/abs/1410.1611 (Path Integral and Asset Pricing, by Zurab Kakushadze) and the classic book of Hagen Kleinert: http://www.worldscientific.com/worldscibooks/10.1142/7305 (Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets). As for the applications of gauge theory, see http://www.ingentaconnect.com/content/iop/jphysa/2000/00000033/00000001/art00102 (Gauge geometry of financial markets, by K. Ilinski) and http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1458886 (Gauge Invariance, Geometry and Arbitrage, by Samuel Vazquez and Simone Farinelli).
Although this is not directly related to your question, I'd like to mention that some real world applications of number theory is considered in the book http://www.springer.com/physics/theoretical,+mathematical+%26+computational+physics/book/978-3-540-85297-1 (Number Theory in Science and Communication: With Applications in Cryptography, Physics, Digital Information, Computing, and Self-Similarity, by Manfred Schroeder).
answered Mar 19, 2015 at 13:58
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3$\begingroup$ I'm pretty skeptical of these purported "connections" between financial markets and particle physics. $\endgroup$ Commented Mar 19, 2015 at 15:59
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$\begingroup$ I'm too rather skeptical that there are direct connections. See, however, arxiv.org/abs/1307.0190 and arxiv.org/abs/1307.6727 Maybe such attempts will produce some interesting insights. At least, with regard to them, we can follow an advice "keep an open mind, but not so open your brains fall out" (the origin of this popular maxim is not quite clear: skeptic.com/insight/… ). $\endgroup$ Commented Mar 20, 2015 at 5:19
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1$\begingroup$ I feel confident that no interesting insights will be found (though perhaps some fools and their money will be separated). Every time some scientific/mathematical theory gets some popular press, bs artists write papers incorporating the relevant buzzwords. In the 70's it was catastrophe theory, then we got chaos theory and fractals, and now I suppose we get string theory. $\endgroup$ Commented Mar 20, 2015 at 5:40