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The infamous sign problem leads to an exponential complexity in Monte Carlo simulations of generic many-body quantum systems. Nevertheless, many phases of matter are known to admit a sign-problem-free representative, allowing efficient simulations on classical computers. Motivated by long standing open problems in many-body physics, as well as fundamental questions in quantum complexity, the possibility of intrinsic sign problems, where a phase of matter admits no sign-problem-free representative, was recently raised but remains largely unexplored. I will describe results establishing the existence of intrinsic sign problems in a broad class of topologically ordered phases in 2+1 dimensions. Within this class, these results exclude the possibility of ‘stoquastic’ Hamiltonians for bosons, and of sign-problem-free determinantal Monte Carlo algorithms for fermions. The talk is based on arxiv: 2005.05566 and 2005.05343.

Zoom: https://harvard.zoom.us/j/977347126

TITLE: Birational geometry

ABSTRACT: About main achievements in birational geometry during the last fifty years.

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

For more information, please visit the event page.

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In the 90s’, Witten gave a physical derivation of an isomorphism between the Verlinde algebra of $GL(n)$ of level $l$ and the quantum cohomology ring of the Grassmannian $\text{Gr}(n,n+l)$. In the joint work arXiv:1811.01377 with Yongbin Ruan, we proposed a K-theoretic generalization of Witten’s work by relating the $\text{GL}_{n}$ Verlinde numbers to the level $l$ quantum K-invariants of the Grassmannian $\text{Gr}(n,n+l)$, and refer to it as the Verlinde/Grassmannian correspondence.

The correspondence was formulated precisely in the aforementioned paper, and we proved the rank 2 case (n=2) there. In this talk, I will discuss the proof for arbitrary rank. A new technical ingredient is the virtual nonabelian localization formula developed by Daniel Halpern-Leistner.  At the end of the talk, I will describe some applications of this correspondence.

Zoom: https://harvard.zoom.us/j/93926041523

TITLE: Kunihiko Kodaira and complex manifolds.

ABSTRACT: Kodaira’s motivation was to generalize the theory of Riemann surfaces in Weyl’s book to higher dimensions.  After quickly recalling the chronology of Kodaira, I will review some of Kodaira’s works in three sections on topics of harmonic analysis, deformation theory and compact complex surfaces.  Each topic corresponds to a volume of Kodaira’s collected works in three volumes, of which I will cover only tiny parts.

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

For more information, please visit the event page.

Register here to attend.

Inspired by fractional quantum Hall physics and Tannaka-Krein duality, it is conjectured that every modular tensor category (MTC) or (2+1)-topological quantum field theory (TQFT) can be realized as the representation category of a vertex operator algebra (VOA) or chiral conformal field theory (CFT).  It is obviously true for quantum group/WZW MTCs, but it is not known for MTCs appeared in subfactors such as the famous double Haagerup.  After some general discussion, I will focus on pointed MTCs or so-called abelian anyon models.  While all abelian anyon models can be realized by lattice VOAs, it is not clear whether or not they can be realized by non-lattice VOAs.  The trivial MTC is realized by the Monster moonshine module, which is a non-lattice realization.  I will provide evidence that this might be true for all abelian anyon models.  The talk is partially based on a joint work with Liang Wang: https://arxiv.org/abs/2004.12048 

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