Calendar

Title: Rationality questions in algebraic geometry

Abstract: Over the course of the history of algebraic geometry, rationality questions — motivated by both geometric and arithmetic problems — have often driven the subject forward. The rationality or irrationality of cubic hypersurfaces in particular have led to the development of abelian integrals (dimension one), birational geometry (dimension two) and Hodge theory (dimension 3). But there is still much we don’t understand about the condition of rationality — we don’t know the answer for cubic fourfolds, for example; and it’s not known whether rationality is an open condition or a closed condition in families. In this talk I’ll try to give an overview of the history of rationality and the current state of our knowledge.

Written articles will accompany each lecture in this series and be available as part of the publication “History and Literature of Mathematical Science.”

The schedule will be updated as talks are confirmed.

Please register here to attend any of the lectures.

We will revisit the computations of Stokes matrices for tt*-structures done by Cecotti and Vafa in the 90’s in the context of Frobenius manifolds and the so-called monodromy identity.  We will argue that those cases provide examples of non-commutative Hodge structures of exponential type in the sense of Katzarkov, Kontsevich and Pantev.

via Zoom Video Conferencing:  https://harvard.zoom.us/j/837429475

Title: The ADHM construction of Yang-Mills instantons

Abstract: In 1978 (Physics Letters 65A) Atiyah, Hitchin, Drinfeld and Manin (ADHM) described a construction of the general solution of the Yang-Mills instanton equations over the 4-sphere using linear algebra. This was a major landmark in the modern interaction between geometry and physics,  and the construction has been the scene for much  research activity up to the present day. In this lecture we will review the background and the original ADHM proof,  using Penrose’s twistor theory and results on algebraic vector bundles over projective 3-space. As time permits, we will also discuss some further developments, for example the work of Nahm on monopoles and connections to Mukai duality for bundles over complex tori.

Written articles will accompany each lecture in this series and be available as part of the publication “History and Literature of Mathematical Science.”

The schedule will be updated as talks are confirmed.

Please register here to attend any of the lectures.

Suppose that some information is transmitted by an undulatory signal.

In Classical Field Theory, the stress-energy tensor provides the energy-momentum

density of the wave packet at any time. But, how to measure the information, or

entropy, carried by the wavepacket in a certain region at given time?

Surprisingly, one can answer the above (entirely classical) question by means of

Operator Algebras and Quantum Field Theory. In fact, in second quantisation a

wave packet gives rise to a sector of the Klein-Gordon Quantum Field Theory on

the Rindler spacetimeW. The associated vacuum noncommutative entropy of the

global von Neumann algebras of W is the entropy of the wave packet in the

wedge region W of the Minkowski spacetime. One can then read this result in first

quantisation via a notion of entropy of a vectorof a Hilbert space with respect to a

real linear subspace.

I give a path to the above results by an overview of some of basic results in

Operator Algebras and Quantum Field Theory and of the relation with the

Quantum Null Energy Inequality.

via Zoom: https://harvard.zoom.us/j/779283357

Please register here to attend any of the lectures.

Title: Black Hole Formation

Abstract: Can black holes form through the focusing of gravitational waves?
This was an outstanding question since the early days of general relativity. In his breakthrough result of 2008, Demetrios Chrstodoulou answered this question with “Yes!”
In order to investigate this result, we will delve deeper into the dynamical mathematical structures of the Einstein equations. Black holes are related to the presence of trapped surfaces in the spacetime manifold.
Christodoulou proved that in the regime of pure general relativity and for arbitrarily dispersed initial data, trapped surfaces form through the focusing of gravitational waves provided the incoming energy is large enough in a precisely defined way. The proof combines new ideas from geometric analysis and nonlinear partial differential equations as well as it introduces new methods to solve large data problems. These methods have many applications beyond general relativity. D. Christodoulou’s result was generalized in various directions by many authors. It launched mathematical activities going into multiple fields in mathematics and physics. In this talk, we will discuss the mathematical framework of the above question. Then we will outline the main ideas of Christodoulou’s result and its generalizations, show relations to other questions and give an overview of implications in other fields.

Written articles will accompany each lecture in this series and be available as part of the publication “History and Literature of Mathematical Science.”

The schedule will be updated as talks are confirmed.