Calendar

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: 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.

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.

Please register here to attend any of the lectures.

Title: Quantum Groups

Abstract: The theory of quantum groups developed in mid 1980s from attempts to construct and understand solutions of the quantum Yang-Baxter equation, an important equation arising in quantum field theory and statistical mechanics. Since then, it has grown into a vast subject with profound connections to many areas of mathematics, such as representation theory, the Langlands program, low-dimensional topology, category theory, enumerative geometry, quantum computation, algebraic combinatorics, conformal field theory, integrable systems, integrable probability, and others. I will review some of the main ideas and examples of quantum groups and try to briefly describe some of the applications.

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.

“Generalized” ’t Hooft anomalies impose new constraints on nonperturbative gauge dynamics. In confining theories with domain walls, they imply that quarks become liberated on the walls. The pertinent anomaly-inflow arguments have a formal flavor and our goal here is to shed light on dynamical aspects of domain-wall deconfinement. We use semiclassical means in a theoretically controlled setting.  While these tools do not require supersymmetry, for brevity (and elegance) we focus this talk on 4d N=1 super Yang-Mills theory. We review the set-up and study the domain walls’ properties, along the way deriving the  “N choose k” multiplicity of k-walls (connecting vacua k “steps” apart). We use the results to explain how quarks of all N-alities become deconfined on all k-walls. A similar picture applies to deconfinement on domain walls in QCD at theta=pi, adjoint QCD, and axion domain walls. We end with discussing a “wish list” of not well-understood aspects. (The bulk of this talk is based on 1909.10979, with Cox and Wong. However, it relies heavily on 1501.06773, with Anber and Sulejmanpasic, as well as 2001.03631, with Anber.)

via Zoom Video Conferencing: https://harvard.zoom.us/j/977347126

Please register here to attend any of the lectures.

Title: My life and times with the sporadic simple groups

Abstract: Five sporadic simple groups were proposed in 19th century and 21 additional ones arose during the period 1965-1975. There were many discussions about the nature of finite simple groups and how sporadic groups are placed in mathematics. While in mathematics grad school at University of Chicago,  I became fascinated with the unfolding story of sporadic simple groups. It involved theory, detective work and experiments. During this lecture, I will describe some of the people, important ideas and evolution of thinking about sporadic simple groups. Most should be accessible to a general mathematical audience.

Article

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.

will speak on:

FIELD THEORY AS A LIMIT OF INTERACTING QUANTUM BOSE GASES

We prove that the grand canonical Gibbs state of an interacting quantum Bose gas converges to the Gibbs measure of a nonlinear Schrödinger equation in the mean-field limit, where the density of the gas becomes large and the interaction strength is proportional to the inverse density. Our results hold in dimensions d = 1,2,3. For d > 1 the Gibbs measure is supported on distributions of negative regularity and we have to renormalize the interaction. The proof is based on a functional integral representation of the grand canonical Gibbs state, in which convergence to the mean-field limit follows formally from an infinite-dimensional stationary phase argument for ill-defined non-Gaussian measures. We make this argument rigorous by introducing a white-noise-type auxiliary field, through which the functional integral is expressed in terms of propagators of heat equations driven by time-dependent periodic random potentials. Joint work with Jürg Fröhlich, Benjamin Schlein, and Vedran Sohinger.

via Zoom: https://harvard.zoom.us/j/147308224

I will discuss coarse geometric properties of algebraic subvarieties of the moduli space of Riemann surfaces.  In joint work with Jenya Sapir, we prove that such a subvariety is coarsely dense, with respect to either the Teichmuller or Thurston metric, iff it has full dimension in the moduli space.  This work was motivated by an attempt to understand the geometry of the image of the projection map from a stratum of abelian or quadratic differentials to the moduli space of Riemann surfaces.  As a corollary of our theorem, we characterize when this image is coarsely dense.  A key part of the proof of the theorem involves comparing analytic plumbing coordinates at the Deligne-Mumford boundary to hyperbolic/extremal lengths of curves on nearby smooth surfaces.

via Zoom: https://harvard.zoom.us/j/972495373

I will talk about line operators of four-dimensional gauge theories on non-spin manifolds. Line operators correspond to worldlines of heavy classical particles. Specifying the spectrum of such particles/lines, leads to distinct physical theories with different discrete theta parameters. We propose a formula for the spin of line operators (boson or fermion), and classify gauge theories with simple Lie algebras on non-spin manifolds. We also discuss the one-form symmetries of these theories and their ‘t Hooft anomalies. This talk is based on https://arxiv.org/abs/1911.00589, jointly with J.P. Ang and Konstantinos Roumpedakis.

via Zoom Video Conferencing: https://harvard.zoom.us/s/977347126

We introduce a moment map picture for holomorphic string algebroids, a special class of holomorphic Courant algebroids introduced in arXiv:1807.10329. An interesting feature of our construction is that the Hamiltonian gauge action is described by means of Morita equivalences, as suggested by higher gauge theory. The zero locus of the moment map is given by the solutions of the Calabi system, a coupled system of equations which provides a unifying framework for the classical Calabi problem and the Hull-Strominger system. Our main results are concerned with the geometry of the moduli space of solutions, and assume a technical condition which is fulfilled in examples. We prove that the moduli space carries a pseudo-Kähler metric with Kähler potential given by the dilaton functional, a topological formula for the metric, and an infinitesimal Donaldson-Uhlenbeck-Yau type theorem. Finally, we relate our topological formula to a physical prediction for the gravitino mass in order to obtain a new conjectural obstruction for the Hull-Strominger system. This is joint work with Roberto Rubio and Carl Tipler.

*If you would like to attend, please email spicard@math.harvard.edu

We will be concerned with asymptotically hyperbolic ‘hyperboloidal’ initial data for the Einstein equations. Such initial data is modeled on the upper unit hyperboloid in Minkowski spacetime and consists of a Riemannian manifold (M, g) whose geometry at infinity approaches that of hyperbolic space, and a symmetric 2-tensor K representing the second fundamental form of the embedding into spacetime, such that K -> g at infinity. There is a notion of mass in this setting and a positive mass conjecture can be proven by spinor techniques. Other important results concern the case K = g, where the conjecture states that an asymptotically hyperbolic manifold whose scalar curvature is greater than or equal to that of hyperbolic space must have positive mass unless it is a hyperbolic space. In this talk, we will discuss how the method of Jang equation reduction, originally devised by Schoen and Yau to prove the positive mass conjecture for asymptotically Euclidean initial data sets, can be adapted to the asymptotically hyperbolic setting yielding a non-spinor proof of the respective positive mass conjecture. We will primarily focus on the case dim M = 3.

via Zoom: https://harvard.zoom.us/j/579137378