This document outlines a research agenda to develop mathematical models of financial systems using concepts from geometry and topology. It proposes using "convex preference maps" to model individual choices, allowing parameters to vary to produce "saddle preference maps" with both convex and concave regions. The goal is to develop models where the divergence of preference maps equals zero, achieving a "steady state." Later work would introduce "solitons" representing dynamic equilibria, allow metrics to vary, and use the "Ricci flow" equation to model how metrics evolve over time in response to endogenous or exogenous changes. The overall agenda is to solve problems in mathematical finance using techniques from geometry and topology, with results targeted for publication between 2014-2020

1. Preliminary Draft:
For submission to
Economics Letters
upon receipt of comments
Convexity, Solitons and some economics
leading to Zero Divergence and Harmonicity
June, 2010
Richard A. Baum
Adjunct Faculty
Department of Mathematics
Santa Barbara City College
721 Cliff Drive
Santa Barbara, California
Office Phone (805) 965-0581 Ext. 3845 (message only)
baum@sbcc.edu

2. Distribution
Marilynn Spaventa, M.A.T.
Dean, Mathematics and English Programs, SBCC
Full Time Math Faculty at SBCC
Peter Georgakis, M.A., J.D.
James Kruidenier, M.A.
Ronald Wopat, M.A.
Robert Elmore, M.A.
Peter Rojas, M.A.
Lindsey Bramlett-Smith, Ph.D.
Ignacio Alarcon, M.A.S., President of Academic Senate
Pamela Guenther, M.A.
Sharareh Masooman, Ph.D.
Bronwen Moore, M.A., Math Department Chair
Jason Miner, M.S
David Gilbert, M.A.
Anna Parmely, M.S
Noureddine Laanaoui, M.S.
Elizabeth Cunningham, M.S.
James Campbell, M.S.
Jared Hersh, M.A.
Lee Chang, M.A.
Full Time Economics Faculty at SBCC
Collette Barr, M.A., UCLA, Economics
Full Time Political Science Faculty at SBCC
Manoutchehr M. Eskandari-Qajar, Ph.D.
Joseph Martorana, M.A.
Economics Faculty at CSUN
Shirley V. Svorny, Ph.D. UCLA
Math Faculty at UCSB
Dr. John E. Doner
Dr. John Douglas Moore
Dr. Thomas C. Sideris
Dr. Xianzhe Dai
Economics Faculty at UCLA
Dr. Michael D. Intriligator
Dr. John G. Riley
Dr. Bryan C. Ellickson
Mathematics Faculty at UCLA
Dr. Robert E. Greene

3. Convexity, Solitons and some economics
leading to Zero Divergence and Harmonicity
June 14, 2010
An atlas of convex preference maps will naturally display negative divergence of the
unit gradient evaluated with respect to goods comprising each map within the atlas.
What must happen to get our desired result of zero divergence of the unit gradient of
a chart of maps within the atlas? First, we must extend the domain of evaluation of
divergence beyond goods to include all relevant objects of choice. We do not allow choice
of parameters, though these may vary continuously within the atlas. But given an
individual’s set of sets of parameters, we do allow choice with respect to each individual’s
particular set of parameters. By this device, certain financial instruments become objects of
choice with respect to parameters. Further, not all such instruments may be goods.
A simple example of an object of choice that may be regarded as a bad is coinsurance
where the insured bears part of the risk or indemnified value. Everything else held constant,
less coinsurance is preferred to more, so it’s a bad. But not everything else is constant in the
case of coinsurance if an increase in coinsurance leads to an insurance premium rebate, or
what is the same thing, a lower net insurance premium. The rebate is a good. Varying
parameters within a chart of preference maps may lead to objects of choice such as
coinsurance yielding concavity as well as convexity of those cross sections of preference
maps taken with respect to coinsurance as well as an underlying good. In 3
R resulting

4. partially convex, partially concave preference maps are called saddle preference maps. In
4
R a level set of such preference maps may look like a tower of saddles.
Hence the stage is set where divergence of the unit gradient of some chart of saddle
preference maps taken with respect to all relevant objects of choice may become identically
zero given a smooth variation of parameters. This is the promised land of harmonicity,
harmonic maps or generically minimally immersed isometric manifolds sought since 1988.
Once the promised land is found, one’s greed for generality as Misha Gromov has
remarked (Gromov’s Preface: Harmony and Harmonicity in Harmonic Maps Between
Riemannian Polyhedra: James Eells and Bent Fuglede, (Cambridge Tracts in Mathematics:
142; CUP: 2001)) may not be satisfied, and we may extend our search from the steady state
to solitons which are a continua of equilibria allowing for dynamics within the steady state.
Once solitons are introduced, we may vary metrics in addition to variation of parameters.
Metrics are the means of measuring distance, area, volume and angles within our atlas or
chart. Variation in metric leads us to consider singularities which may form with an
evolution of metric such as short time blow up or collapse. Finally, bringing us to 2003,
evolutions of metric according to the Ricci flow leads us to consider simultaneous
concentrations and dissipations of curvature arising from a particular time evolution of
metric. The Ricci flow is a device for evolving the metric. Surgery with regard to the flow is
allowed and may be a means for avoiding short time blow up or collapse of the flow or in our
case of a soliton representing a steady state equilibria path or flow. The Ricci flow is type of
reaction-diffusion evolution equation which is useful for modeling dynamics, in our case, the
dynamics of a variety of mathematical finance given what economists have regarded as an

5. exogenous shock to the system but within our framework of the Ricci flow may be an
endogenous evolution of metric.
Returning to school at UC Santa Barbara in Fall, 2011, it is believed with a ten year
curricula of research and study the above agenda will yield a solution to a problem of
endogenous variation of metric in mathematical finance by 2014 and a more general set of
solutions by 2019, yielding a published article by 2020 and a book surveying results within
twenty-some years of the 2003 solution of the Poincaré Conjecture by Grisha Perelman
following twenty years of work by Richard S. Hamilton’s Program. An exposition in co-
ordinates of the work of the Hamilton-Perelman Theory of Ricci Flow is given in text in the
June, 2006 issue (Volume 10 Number 2) of the Asian Journal of Mathematics published
quarterly by International Press, P.O. Box 43502, Somerville, MA 02143, USA.