Category Archives: Mathematics

University of Copenhagen Gold Medals

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Recipients of the University of Copenhagen’s Gold Medal in Mathematics and Economics for the years 1940–1980.

Year Mathematics Economics (“Statsvidenskab”)
1980 Lars Tyge Nielsen None
1979 None Lars Tyge Nielsen
1978 None None
1977 Jesper Laub None
1976 None None
1975 None None
1974 Susanne Aref None
1973 Dorte Olesen None
1972 None None
1971 None None
1970 Søren Jøndrup, Knud Lønsted None
1969 Christian Berg Birgit Grodal
1968 None None
1967 None None
1966 Søren Johansen None
1965 None None
1964 Arne Brøndsted Erling B Andersen
1963 None None
1962 None None
1961 None None
1960 None None
1959 None None
1958 None None
1957 None None
1956 None None
1955 None None
1954 None None
1953 Frans Handest None
1952 None None
1951 None None
1950 Sigurdur Helgason None
1949 None None
1948 None None
1947 None None
1946 None None
1945 Ole Rindung None
1944 None Bent Hansen
1943 None None
1942 Thøger Bang None
1941 Helge A. B. Clausen Hans Julius Brems
1940 None None

Sources include the University of Copenhagen Annual Review (Københavns Universitets Årbog), the Royal Library (Det Kongelige Bibliotek, KB), and the Faculty of Social Sciences, among others.

MathJax

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This site uses MathJax as implemented in the wordpress plugin MathJax-LaTeX.

Here is an example of how MathJax renders mathematics:

Given \((\mu,\sigma)\) and a setup \((\Omega,\mathcal{F},P,F,W)\), a solution on \((\Omega,\mathcal{F},P,F,W)\) of the SDE \((\mu,\sigma)\) is an \(N\) dimensional Itô process \(X\) on \((\Omega,\mathcal{F},P,F,W)\) with a potentially random initial value \(X(0)\), such that the process \(\mu(X,t)\) belongs to \(\mathcal{L}^{1}\), the process \(\sigma(X,t)\) belongs to \(\mathcal{L}^{2}\), and for all \(t \in [0,\infty)\),
$$X(t) = X(0) + \int_{0}^{t} \mu(X,s) \, ds + \int_{0}^{t} \sigma(X,s) \, dW$$

To use LaTex in comments on this site, you can enclose it in \(\backslash ( \ldots \backslash ) \) (for inline LaTex) or \( \$\$ \ldots \$\$ \) (for display LaTex).

Disavowing Lebesgue

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No, you cannot disavow Lebesgue.

Lebesgue integration (and measure theory) are indispensable in stochastic calculus, especially for finance applications.

Take for example the martingale representation theorem, which is central to the theory of dynamic hedging and replication. This version is essentially Rogers and Williams (1987), Theorem 36.5:

Theorem
The martingale representation theorem
Let \(W\) be a standard Brownian motion of dimension \(K\). If \(X\) is a martingale with respect to the augmented filtration generated by \(W\), then there exists a process \(b \in \mathcal{L}^{2}\) such that
$$X(t) -X(0) = \int_{0}^{t} b \, dW$$ (in the sense that the two processes are indistinguishable).

But what does it mean to say that the process \(b\) is in \(\mathcal{L}^{2}\)? It means that \(b\) is measurable and adapted and pathwise square integrable on bounded intervals. In other words,
$$\int_{0}^{t} \|b\|^{2} \, ds < \infty $$ with probability one, for all \(t \in [0,\infty)\). Or, to write it out in detail, if the underlying probability space is \((\Omega,\mathcal{F},P)\), then the requirement is that
$$\int_{0}^{t} \|b(\omega,s)\|^{2} \, ds < \infty $$ for \(P\)-almost all \(\omega \in \Omega\), for all \(t \in [0,\infty)\).

For each \(\omega\), the time integral is a Lebesgue integral. The martingale representation theorem does not in any way guarantee that the function \(\|b\|^{2}\) is pathwise Riemann integrable. How could it?

So the statement of the martingale representation theorem requires Lebesgue integration and measure theory.

Mathematical Genealogy

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My Ph.D. is in economics, not in mathematics, but I did write a master’s thesis in differential topology, some of which was published in: Transversality and the Inverse Image of a Submanifold with Corners. Mathematica Scandinavica 49 (1981), 211-221.

My thesis advisor was Vagn Lundsgaard Hansen, then at the University of Copenhagen, now at the Technical University of Denmark. Through him, I trace my mathematical ancestry back to Birkhoff, and possibly all the way back to Poisson, Lagrange, Euler, Johan and Jacob Bernoulli, and, finally, Leibnitz.

Apart from being my advisor, Lundsgaard Hansen was also my teacher in first-year undergraduate mathematical analysis. The course started with an unforgettably vivid exposition of the topology of metric spaces — pulling back open and closed sets and mapping compact sets forward and so on. Later on I took his course in algebraic topology, where the main agenda was to use category theory to translate topological questions into easier algebraic ones.

Category theory was apparently invented by Samuel Eilenberg together with Saunders Mac Lane.

Samuel Eilenberg

Samuel Eilenberg

This bust of Samuel Eilenberg stands in the corner of the Faculty Lounge in the Columbia Mathematics Department. Eilenberg was a professor in the department from 1947 and until his retirement in 1982.