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Minerva Foundation Lectures
Tuesdays 4:10–5:25 PM
Room 622 Mathematics
Professor Rama Cont
Oxford University
Rough Calculus
The Ito calculus may be viewed as an extension of the Newton-Leibniz calculus to smooth functions of paths with non-zero quadratic variation. This analytical viewpoint is exploited to develop a calculus for smooth function(al)s of irregular paths with non-zero p-th variation for arbitrary p>1. Although this “rough calculus” is strictly pathwise in nature and does not involve any probabilistic ingredient, it is applicable to stochastic processes with irregular paths.
We illustrate the concepts and results of this theory in the setting of the Ito-Föllmer calculus for smooth function(al)s of paths with finite quadratic variation. We will then show how these results may be extended to the more general setting of smooth functionals of paths with non-zero p-th variation for arbitrary p>1, leading to a higher order Ito-type calculus. Finally, we will sketch some examples of applications to transport equations, optimal control and rough dynamics on manifolds.
I. Ito calculus without probability
II. Ito-Föllmer calculus for functionals of paths with finite quadratic variation.
Pathwise isometry and rough-smooth decompositions.
III. Rough calculus for function(al)s of path with finite p-th variation.
IV. The case of paths with fractional regularity (*)
V. Transport of measures along rough trajectories.
VI. Pathwise optimal control of dynamical systems driven by rough signals (*)
VII. Rough dynamics on manifolds
* : if time permits
First lecture: Tuesday, January 16th, 2025
Meeting on Tuesdays at 4:10 p.m.
Room 622, Mathematics Hall
2990 Broadway (117th Street)
Lecture Series References List
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Joseph Fels Ritt Lectures,
Spring 2025
Professor Gunther Uhlmann
University of Washington
Lecture 1: Wednesday, April 16, 4:30-5:30 PM,
Math Hall 520
Title: 45 Years of Calderon’s Problem
Abstract: Calderon’s inverse problem asks whether one can determine the conductivity of a medium by making voltage and current measurements at the boundary. In mathematical terms it consists in recovering coefficients of a PDE by making boundary measurements of solutions. This question arises in several areas of applications including medical imaging and geophysics. It has also led to a proposal for making objects invisible. I will report on some of the progress that has been made on this problem since Calderon proposed it in 1980, including recent developments on similar problems for nonlinear equations and nonlocal operators.
Lecture 2: Thursday, April 17, 2:45-3:45 PM,
Math Hall 520
Title: Journey to the Center of the Earth
Abstract: We will consider the inverse problem of determining the sound speed or index of refraction of a medium by measuring the travel times of waves going through the medium. This problem arises in global seismology in an attempt to determine the inner structure of the Earth by measuring travel times of earthquakes. It also has several applications in optics and medical imaging among others. The problem can be recast as a geometric problem: Can one determine the Riemannian metric of a Riemannian manifold with boundary by measuring the distance function between boundary points? This is the boundary rigidity problem. We will survey some of the known results about this problem.
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