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A celebrated theorem of Nash completed by Kuiper implies that every smooth Riemannian surface has a C¹ isometric embedding in the Euclidean 3-space E³. An analogous result, due to Burago and Zalgaller, states that every polyhedral surface, obtained by gluing Euclidean triangles, has an isometric PL embedding in E³. In particular, this provides PL isometric embeddings for every flat torus (a quotient of E² by a rank 2 lattice). However, the proof of Burago and Zalgaller is partially constructive, relying on the Nash-Kuiper theorem. In practice, it produces PL embeddings with a huge number of vertices, moreover distinct for every flat torus. Based on a construction of Zalgaller and on recent works by Arnoux et al. we exhibit a universal triangulation with less than 10.000 vertices, admitting for any flat torus an isometric embedding that is linear on each triangle. Based on joint work with Florent Tallerie.

Zoom ID: 950 2372 5230 (Password: cmsa)

Metals that do not fit Landau’s famous Fermi liquid paradigm of quasiparticles are plentiful in experiments, but constructing their theoretical description is a major challenge in modern quantum many-body physics. I will describe new models that can systematically describe such non-Fermi liquid metals at quantum critical points, and that allow for the accurate computation of a whole host of experimentally measurable static and dynamic quantities despite the presence of both strong correlations and disorder. I will further demonstrate that disorder coupling to interaction operators can lead to the experimentally observed linear-in-temperature (T-linear) resistivity seen at metallic quantum critical points, and can also generate the observed universal “Planckian” transport scattering rate of kBT/ℏ. Finally, I will show that “perfect” T-linear resistivity is associated with an energy invariant quantity defined in the many-body microcanonical ensemble, which motivates the existence of a deep connection between the T-linear resistivity seen at high temperatures and low temperatures with the same slope in many quantum critical materials.

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

Markov decision processes are a model for several artificial intelligence problems, such as games (chess, Go…) or robotics. At each timestep, an agent has to choose an action, then receives a reward, and then the agent’s environment changes (deterministically or stochastically) in response to the agent’s action. The agent’s goal is to adjust its actions to maximize its total reward. In principle, the optimal behavior can be obtained by dynamic programming or optimal control techniques, although practice is another story.

Here we consider a more complex problem: learn all optimal behaviors for all possible reward functions in a given environment. Ideally, such a “controllable agent” could be given a description of a task (reward function, such as “you get +10 for reaching here but -1 for going through there”) and immediately perform the optimal behavior for that task. This requires a good understanding of the mapping from a reward function to the associated optimal behavior.

We prove that there exists a particular “map” of a Markov decision process, on which near-optimal behaviors for all reward functions can be read directly by an algebraic formula. Moreover, this “map” is learnable by standard deep learning techniques from random interactions with the environment. We will present our recent theoretical and empirical results in this direction.

Zoom ID: 950 2372 5230 (Password: cmsa)

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)

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