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The colloquium typically meets Mondays at 4:00 PM in Room 617 on the sixth floor of Wachman Hall.
The colloquium is preceded by tea starting at 3:30 in the Faculty Lounge, adjacent to Room 617. Click on title for abstract.
Christian Schafmeister, Department of Chemistry, Temple University
TBA
Ilya Kapovich, CUNY
The problem of counting closed geodesics of bounded length, originally in the setting of negatively curved manifolds, goes back to the classic work of Margulis in 1960s about the dynamics of the geodesic flow. Since then Margulis' results have been generalized to many other contexts where some whiff of hyperbolicity is present. Thus a 2011 result of Eskin and Mirzakhani shows that for a closed hyperbolic surface S of genus $g\ge 2$, the number $N(L)$ of closed Teichmuller geodesics of length $\le L$ in the moduli space of $S$ grows as $e^{hL}/(hL)$ where $h=6g-6$. The number $N(L)$ is also equal to the number of conjugacy classes of pseudo-Anosov elements $\phi$ in the mapping class group $MCG(S)$ with $\log\lambda(\phi)\le L$, where $\lambda(\phi)>1$ is the "dilation" or "stretch factor" of $\phi$.
We consider an analogous problem in the $Out(F_r)$ setting, for the action of $Out(F_r)$ on a "cousin" of Teichmuller space, called the Culler-Vogtmann outer space $X_r$. In this context being a "fully irreducible" element of $Out(F_r)$ serves as a natural counterpart of being pseudo-Anosov. Every fully irreducible $\phi\in Out(F_r)$ acts on $X_r$ as a loxodromic isometry with translation length $\log\lambda(\phi)$, where again $\lambda(\phi)$ is the stretch factor of $\phi$. We estimate the number $N_r(L)$ of fully irreducible elements $\phi\in Out(F_r)$ with $\log\lambda(\phi)\le L$. We prove, for $r\ge 3$, that $N_r(L)$ grows doubly exponentially in $L$ as $L\to\infty$, in terms of both lower and upper bounds. These bounds reveal new behavior not present in classic hyperbolic dynamical systems. The talk is based on a joint paper with Catherine Pfaff.
Ioannis Karatzas, Columbia University
We introduce models for financial markets and, in their context, the notions of portfolio rules and of arbitrage. The normative assumption of absence of arbitrage is central in the modern theories of mathematical economics and finance. We relate it to probabilistic concepts such as "fair game", "martingale", "coherence" in the sense of deFinetti, and "equivalent martingale measure".
We also survey recent work in the context of the Stochastic Portfolio Theory pioneered by E.R. Fernholz. This theory provides descriptive conditions under which opportunities for arbitrage, or outperformance, do exist; then constructs simple portfolios that implement them. We also explain how, even in the presence of such arbitrage, most of the standard mathematical theory of finance still functions, though in somewhat modified form.
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