Calendar

Quantum maps are a popular model in physics: symplectic relations on tori are quantized to produce families of $N\times N$ matrices and the high energy limit corresponds to the large $N$ limit. They share a lot of features with more complicated quantum systems but are easier to study numerically. We consider open quantum baker’s maps, whose underlying classical systems have a hole allowing energy escape. The eigenvalues of the resulting matrices lie inside the unit disk and are a model for scattering resonances of more general chaotic quantum systems.

We establish a spectral gap (that is, the spectral radius of the matrix is separated from 1 as $N\to\infty$) for all the systems considered. The proof relies on the notion of fractal uncertainty principle and uses the fine structure of the trapped sets, which in our case are given by Cantor sets, together with simple tools from harmonic analysis, algebra, combinatorics, and number theory. We also obtain a fractal Weyl upper bound for the number of eigenvalues in annuli. These results are illustrated by numerical experiments which also suggest some conjectures.

https://harvard.zoom.us/j/779283357?pwd=MitXVm1pYUlJVzZqT3lwV2pCT1ZUQT09

One can view a partial flag variety in C^n as an adjoint orbit inside the Lie algebra of n x n skew-Hermitian matrices. We use the orbit context to study the totally nonnegative part of a partial flag variety from an algebraic, geometric, and dynamical perspective. We classify gradient flows on adjoint orbits in various metrics which are compatible with total positivity. As applications, we show how the classical Toda flow fits into this framework, and prove that a new family of amplituhedra are homeomorphic to closed balls. This is joint work with Anthony Bloch.

https://harvard.zoom.us/j/94191911494?pwd=RnN3ZnIwcFYwd0QyT0MwZWVISmR5Zz09

Password: 1251442

Localization by cosection, first introduced by Kiem-Li in 2010, is one of the fundamental techniques used to study invariants in complex enumerative geometry. Donaldson-Thomas (DT) invariants counting sheaves on Calabi-Yau fourfolds were first defined by Borisov-Joyce in 2015 by combining derived algebraic and differential geometry.

In this talk, we develop the theory of cosection localization for derived manifolds in the context of derived differential geometry of Joyce. As a consequence, we also obtain cosection localization results for (-2)-shifted symplectic derived schemes. This provides a cosection localization formalism for the Borisov-Joyce DT invariant. As an immediate application, the stable pair invariants of hyperkähler fourfolds, constructed by Maulik-Cao-Toda, vanish, as expected.

https://harvard.zoom.us/j/98781914555?pwd=bmVzZGdlRThyUDREMExab20ybmg1Zz09

will speak on:

Higher rank DT theory from rank 1

Fix a Calabi-Yau 3-fold X. Its DT invariants count stable bundles and sheaves on X. The generalised DT invariants of Joyce-Song count semistable bundles and sheaves on X. I will describe work with Soheyla Feyzbakhsh showing these generalised DT invariants in any rank r can be written in terms of rank 1 invariants. By the MNOP conjecture the latter are determined by the GW invariants of X. Along the way we also show they are determined by rank 0 invariants counting sheaves supported on surfaces in X. These invariants are predicted by S-duality to be governed by (vector-valued, mock) modular forms.

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For details visit:

http://www.ims.cuhk.edu.hk/cgi-bin/SeminarAdmin/bin/Web

http://www.ims.cuhk.edu.hk/activities/seminar/joint-dg-seminar/

**Please note time is HONG KONG TIME** (3 am EST)

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A phenomenon underlying many remarkable results in number theory is the natural Galois action on the cohomology of symplectic groups of integers. In joint work with Soren Galatius and Akshay Venkatesh, we define a symplectic variant of algebraic K-theory, which carries a natural Galois action for similar reasons. We compute this Galois action and characterize it in terms of a universality property, in the spirit of the Langlands philosophy.

We use topological methods to solve special cases of a fundamental problem in group theory, the Kervaire conjecture. The conjecture asserts that, for any nontrivial group G and any element w in the free product G*Z, the quotient (G*Z)/<<w>> is still nontrivial. We interpret this as a problem of estimating the minimal complexity (in terms of Euler characteristic) of surfaces in HNN extensions. This gives a conceptually simple proof of Klyachko’s theorem that confirms the Kervaire conjecture for any G torsion-free. I will also explain new results obtained using this approach.

Zoom ID: 950 2372 5230 (Password: cmsa)

Active crystals are highly ordered structures that emerge from the nonequilibrium self-organization of motile objects, and have been widely studied in synthetic and bacterial active matter. In this talk, I will describe how swimming sea star embryos spontaneously assemble into chiral crystals that span thousands of spinning organisms and persist for tens of hours. Combining experiment, hydrodynamic theory, and simulations, we demonstrate that the formation, dynamics, and dissolution of these living crystals are controlled by the natural development of the embryos. Remarkably, due to nonreciprocal force and torque exchange between the embryos, the living chiral crystals exhibit self-sustained oscillations with dynamic signatures recently predicted to emerge in materials with odd elasticity.

https://harvard.zoom.us/j/96657833341

Password: cmsa

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Subscribe to Harvard CMSA seminar videos (more to be uploaded):
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—–
Subscribe to Harvard CMSA seminar videos (more to be uploaded):
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https://www.youtube.com/channel/UCM06KiUOw1vRrmvD8U274Ww