WORKSHOP ON TOPOLOGY: IDENTIFYING ORDER IN COMPLEX SYSTEMS

http://www.math.ias.edu/files/WorkshopIntro.pdf

In physics, the twentieth century started with the twin revolutions of relativity and quantum mechanics. Much of the second half of the century was devoted to the construction of a theoretical structure unifying these radical ideas, confirmed experimentally to exquisite precision over the past three decades. Yet questions remain. The union of quantum mechanics and gravity strongly suggests that space-time is doomed—but what replaces it?

http://math.ias.edu/files/seminars/GriffithsTwoTalks.pdf

These two talks will be about automorphic cohomology in the non-classical

case.

http://math.ias.edu/files/seminars/GriffithsTwoTalks.pdf
 

The correspondence between homotopy types and higher categorical analogs of groupoids which was first conjectured by Alexander Grothendieck naturally leads to a view of mathematics where sets are used to parametrize collections of objects without "internal structure" while collections of objects with "internal structure" are parametrized by more general homotopy types. Univalent Foundations are based on the combination of this view with the discovery that it is possible to directly formalize reasoning about homotopy types using Martin-Lof type theories.

In this lecture, Enrico Bombieri, IBM von Neumann Professor in the School of Mathematics, attempts to give an idea of the numerous different notions of truth in mathematics. Using accessible examples, he explains the difference between truth, proof, and verification. Bombieri, one of the world’s leading authorities on number theory and analysis, was awarded the Fields Medal in 1974 for his work on the large sieve and its application to the distribution of prime numbers. Some of his work has potential practical applications to cryptography and security of data transmission and identification.

In this talk, Professor Matias Zaldarriaga discusses the development of the modern study of cosmology, beginning with the discovery of the expansion of the Universe by Edwin Hubble, through current efforts to map the cosmic microwave background, test ideas about the initial conditions of the Universe, and explain the accelerated expansion of the Universe.