Lectures by Faculty
We describe some results concerning the number of connected components of nodal lines of high frequency Maass forms on the modular surface. Based on heuristics connecting these to a critical percolation model, Bogomolny and Schmit have conjectured, and numerics confirm, that this number follows an asymptotic law. While proving this appears to be very difficult, some approximations to it can be proved by developing number theoretic and analytic methods. The work report on is joint with A. Ghosh and A. Reznikov.
A classical theorem in Euclidean geometry asserts that if a set of points has the property that every line through two of them contains a third point, then they must all be on the same line. We prove several approximate versions of this theorem (and related ones), which are motivated from questions about locally correctable codes and matrix rigidity. The proofs use an interesting combination of combinatorial, algebraic and analytic tools.
Joint work with Boaz Barak, Zeev Dvir and Amir Yehudayoff
Prior to the Terror (1793–94), the French Revolution was generally viewed very positively by progressive constitutional thinkers and law reformers. On November 18, 1792, more than a hundred distinguished Anglo-American democrats, including several founders of modern feminism, gathered at the British Club in Paris to celebrate liberty, human rights, and the spread of democracy across the world—what they viewed as the assured democratic future of mankind. In this lecture, Jonathan Israel, Professor in the School of Historical Studies, explores the vast significance of the toasts drunk at this banquet and of the public address that was afterward presented to the French National Assembly. They illuminate the relationship between the French Revolution and modernity, the history of our own time, and the many ironies of the values and propositions that the “British Club” in Paris proclaimed to the world.
One of the oldest subjects in mathematics is the study of Diophantine equations, i.e., the study of whole number (or fractional) solutions to polynomial equations. It remains one of the most active areas of mathematics today. Perhaps the most basic tool is the simple idea of “congruences,” particularly congruences modulo a prime number. In this talk, Richard Taylor, Professor in the School of Mathematics, introduces prime numbers and congruences and illustrates their connection to Diophantine equations. He also describes recent progress in this area, an application, and reciprocity laws, which lie at the heart of much recent progress on Diophantine equations, including Wiles’s proof of Fermat’s last theorem.
The modern theory of dynamical systems, as well as symplectic geometry, have their origin with Poincare as one field with integrated Ideas. Since then these fields developed quite independently. Given the progress in these fields one can make a good argument why the time is ripe to bring them closer together around the core area of Hamiltonian dynamics