Calendar

Recent years have seen tremendous advances in our understanding of perturbative quantum field theory—fueled largely by discoveries (and eventual explanations and exploitation) of shocking simplicity in the mathematical form of the predictions made for experiment. Among the most important frontiers in this progress is the understanding of loop amplitudes—their mathematical form, underlying geometric structure, and how best to manifest the physical properties of finite observables in general quantum field theories. This work is motivated in part by the desire to simplify the difficult work of doing Feynman integrals. I review some of the examples of this progress, and describe some ongoing efforts to recast perturbation theory in terms that expose as much simplicity (and as much physics) as possible.

https://harvard.zoom.us/j/91799784675?pwd=MS9LV25DWk9RcmJoRVM0K3RGWkFRdz09

Password: 1251442

In this talk, I will survey the P=W conjecture, which describes certain structures of the cohomology of the moduli space of Higgs bundles on a curve in terms of the character variety of the curve.  I will then explain how certain symmetries of this cohomology, which are predictions of this conjecture, can be constructed using techniques from non-abelian Hodge theory in positive characteristic.  Based on joint work with Mark de Cataldo, Junliang Shen, and Siqing Zhang.

https://cuhk.zoom.us/j/99708889880

(Meeting ID: 997 0888 9880; Passcode: 20220126)

In 1975, Stephen Hawking showed that black holes radiate away in a manner that violates quantum theory. Starting in 1997, it was observed that black holes in string theory did not have the form expected from general relativity: in place of “empty space will all the mass at the center”, one finds a “fuzzball” where the mass is distributed throughout the interior of the horizon. This resolves the paradox, but opposition to this resolution came from groups who sought to extrapolate some ideas in holography. In 2009 it was shown, using some theorems from quantum information theory, that these extrapolations were incorrect, and the fuzzball structure was essential for resolving the puzzle. Opposition continued along different lines, with a postulate that information would leak out through wormholes. Recently, it was shown that this wormhole idea had some basic flaws, leaving the fuzzball paradigm as the natural resolution of Hawking’s puzzle.

https://harvard.zoom.us/j/95767170359

(Password: cmsa)

Can machine learning be a useful tool for research mathematicians? There are many examples of mathematicians pioneering new technologies to aid our understanding of the mathematical world: using very early computers to help formulate the Birch and Swinnerton-Dyer conjecture and using computer aid to prove the four colour theorem are among the most notable. Up until now there hasn’t been significant use of machine learning in the field and it hasn’t been clear where it might be useful for the questions that mathematicians care about. In this talk we will discuss the results of our recent Nature paper, where we worked together with top mathematicians to use machine learning to achieve two new results – proving a new connection between the hyperbolic and geometric structure of knots, and conjecturing a resolution to a 50-year problem in representation theory, the combinatorial invariance conjecture. Through these examples we demonstrate a way that machine learning can be used by mathematicians to help guide the development of surprising and beautiful new conjectures.

---

Fluids pervade complex systems, ranging from fish schools, to bacterial colonies and nanoparticles in drug delivery. Despite its importance, little is known about the role of fluid mechanics in such applications. Is schooling the result of vortex dynamics synthesized by individual fish wakes or the result of behavioral traits? Is fish schooling energetically favorable?  I will present multifidelity computational studies of collective swimming in 2D and 3D flows. Our studies demonstrate that classical models of collective swimming (like the Reynolds model) fail to maintain coherence in the presence of long range hydrodynamic interactions. We demonstrate in turn that collective swimming can be achieved through reinforcement learning. We extend these studies to 2D and 3D viscous flows governed by the Navier Stokes equations. We examine various hydrodynamic benefits with a progressive increase of the school size and demonstrate the importance of controlling the vorticity field generated by up to 300 synchronized swimmers.

The core entropy of a postcritically finite quadratic polynomial is the topological entropy of its restriction to its Hubbard tree.  Core entropy is also the logarithm of the largest eigenvalue of the matrix associated to the Markov partition of the Hubbard tree obtained by cutting the tree at the postcritical set.  Tiozzo proved that core entropy is a continuous function of external angle for the Mandelbrot set.  How do the other (non-largest) eigenvalues of the Markov transition matrix vary with external angle?  In a recent preprint, G. Tiozzo, C. Wu and I answered this question. We defined “Master Teapots” associated to principal veins in the Mandelbrot set and proved that the eigenvalues outside the unit circle move continuously while roots inside the unit circle “persist.”  This talk will discuss this circle of ideas and results, and is based on joint work with G. Tiozzo and C. Wu.

The core entropy of a postcritically finite quadratic polynomial is the topological entropy of its restriction to its Hubbard tree.  Core entropy is also the logarithm of the largest eigenvalue of the matrix associated to the Markov partition of the Hubbard tree obtained by cutting the tree at the postcritical set.  Tiozzo proved that core entropy is a continuous function of external angle for the Mandelbrot set.  How do the other (non-largest) eigenvalues of the Markov transition matrix vary with external angle?  In a recent preprint, G. Tiozzo, C. Wu and I answered this question. We defined “Master Teapots” associated to principal veins in the Mandelbrot set and proved that the eigenvalues outside the unit circle move continuously while roots inside the unit circle “persist.”  This talk will discuss this circle of ideas and results, and is based on joint work with G. Tiozzo and C. Wu.

In this talk, I will describe many-body quantum chaos in a recently proposed large-N theory for critical Fermi surfaces in two spatial dimensions, by computing out-of-time-order correlation functions. I will use the ladder identity proposed by Gu and Kitaev, and show that the chaos Lyapunov exponent in this system takes on the maximum possible value of 2πkBT/ℏ, where T is the absolute temperature. In addition, by varying the dynamic critical exponent, I will show that the maximal chaos persists only in the regime where quasiparticles are absent. When quasiparticles are present, the Lyapunov exponent scales with the temperature as ~ T^a, where a < 1, which is parametrically smaller than the maximal rate.

https://harvard.zoom.us/j/977347126
Password: cmsa