TBA

The ideas of quantum physics have had a huge impact on the development of mathematics, all its fields have been influenced. Many notions have emerged, such as quantum groups and algebras, quantum calculus, many special functions. Numbers, the most elementary and ancient concept at the heart of mathematics since the Babylonians, should also have their place in the quantum landscape. This talk is an elementary and accessible overview of the emerging theory of quantum numbers, including motivations, first results, and the connection to other parts of mathematics.

Recall that a square matrix P is called a projection matrix if P^2 = P.  It makes sense to talk about projection matrices with coefficients in any commutative ring; the image of a projection matrix is called a projective module.  This seemingly innocuous notion intercedes in geometric questions in the same spirit as the famous Hodge conjecture because of Serre's dictionary: projective modules are ``vector bundles''. If X is a smooth complex affine variety, we can consider the rings of algebraic or holomorphic functions on X.  Which of the holomorphic vector bundles on X admit an algebraic structure?  I will discuss recent progress on these questions, using motivic homotopy theory, and based on joint work with Tom Bachmann and Mike Hopkins.

The search for an ultimate axiomatization of mathematics is inevitably incomplete. However, this does not preclude the possibility of strong and natural extensions to the standard axioms of set theory (ZFC) which shed light on many unresolved mathematical questions. Gödel suggested searching for strong axioms of infinity (known as large cardinals) as potential candidates for such extensions -- a pursuit that has continued for over half a century. This talk will survey several central developments in this search, including results concerning finite graphs, questions about Lebesgue measurability of projective sets of reals, the inner model program, and current efforts toward identifying an "ultimate" model of set theory via canonical inner models.

A Riemannian metric is said to be Einstein if it has constant Ricci curvature. Certain peculiar features of 4-dimensional geometry make dimension four into a “Goldilocks zone” for Einstein metrics, with just the right amount of local flexibility managing to coexist with strong global rigidity results. This talk will first describe some aspects of the interplay between Einstein metrics and smooth topology on compact symplectic 4-manifolds without boundary. We will see how ideas from Kähler and conformal geometry allow us to construct Einstein metrics on many such manifolds, while a complimentary tool-box shows that these existence results are optimal in certain specific contexts. The talk will then conclude with a brief discussion of analogous results concerning complete Ricci-flat 4-manifolds.