Calendar

In this talk, we will first review a symplectic realization of the SYZ program and some of its applications. Then I will explain some recent works on equivariant Lagrangian Floer theory and disc potentials of immersed SYZ fibers. They are joint works with Hansol Hong, Yoosik Kim and Xiao Zheng.

via Zoom: https://harvard.zoom.us/j/94717938264

Kitaev’s quantum double models in 2D provide some of the most commonly studied examples of topological quantum order.  In particular, the ground space is thought to yield a quantum error-correcting code.  We offer an explicit proof that this is the case for arbitrary finite groups. Actually, a stronger claim is shown: any two states with zero energy density in some contractible region must have the same reduced state in that region.  Alternatively, the local properties of a gauge-invariant state are fully determined by specifying that its holonomies in the region are trivial. This implies that Kitaev’s model satisfies both TQO1 and TQO2 conditions of Bravyi-Hastings-Michalakis, and so it is a topological order in the sense of B-H-M. We note that the methods developed by P. Naaijkens (PhD thesis, 2012) under a different context can be adapted to provide another proof of this result. We also note that more recently Q. Yang and Z. Wang proved the same result for the more general class of Levin-Wen models, but their method of proof is very different.

via Zoom: https://harvard.zoom.us/j/779283357

We discuss properties of magnetically charged black holes in the Standard Model. We will discuss how the electroweak symmetry is restored around the black hole. In addition, the Hawking evaporation rate is greatly enhanced by a factor of the charge of the black hole.
These provide interesting candidates for primordial black holes which can have a relatively low mass.

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

Thurston’s Master Teapot is the closure of the set of all points $(z,\lambda) \in \mathbb{C} \times \mathbb{R}$ such that $\lambda$ is the growth rate of a critically periodic unimodal self-map of an interval and $z$ is a Galois conjugate of $\lambda$. I will present a new characterization of which points are in this set. This characterization gives a way to think of each horizontal slice of the Master Teapot as an analogy of the Mandelbrot set for a “restricted iterated function system.”  An application of this characterization is that the Master Teapot is not invariant under the map $(z,\lambda) \mapsto (-z,\lambda)$. This presentation is based on joint work with Chenxi Wu.

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

Legendrian graphs naturally appear in the study of Weinstein manifolds with a singular Lagrangian skeleton, and a tangle decomposition of Legendrian submanifolds. I will introduce various invariant of Legendrian graphs including DGA type, polynomial type, sheaf theoretic one, and their relationship. This is joint work with Byunghee An, and partially with Tamas Kalman and Tao Su.

via Zoom: https://harvard.zoom.us/j/94717938264

The partition function of a quantum system without a sign problem can be represented by a path integral in which every amplitude is efficiently computable and nonnegative, which is a substantial simplification from the interference of complex amplitudes in the general quantum case.  In quantum computing the presence of a sign problem has been recast as a virtue, because it helps to increase the complexity of the quantum system beyond the range of classical simulation.  This is particularly important for quantum adiabatic algorithms based on ground states, where the run time depends on the scaling of the spectral gap above the ground state.  This motivates us to study the relation of the sign problem to the spectral gap, using methods such as random matrix theory and spectral graph theory.  The latter relates the discrete geometry of ground states (in a world where vertices are basis elements and edges are Hamiltonian matrix elements) to the level spacings in the low energy spectrum using the higher-order signed Cheeger inequalities.  This talk will include analytical results from 1703.10133 and 2004.07681.

via Zoom: https://harvard.zoom.us/j/779283357

In my talk I discuss traversable wormholes in four and two dimensions.

Based on arXiv: 1807.04726 and 1912.03276

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

Zoom: https://harvard.zoom.us/j/94717938264

Interference is an essential part of quantum mechanics. However, an important class of Hamiltonians considered are those with “no sign problem”, where all off-diagonal matrix elements of the Hamiltonian are non-negative. This means that the ground state wave function can be chosen to have all amplitudes real and positive. In a sense, no destructive interference is possible for these Hamiltonians so that they are “almost classical”, and there are several simulation algorithms which work well in practice on classical computers today. In this talk, I’ll discuss what happens when one considers adiabatic evolution of such Hamiltonians, and show that they still have some power that cannot be efficiently simulated on a classical computer; to be precise and formal, I’ll show this “relative to an oracle”, which I will explain. I’ll discuss implications for simulation of these problems and open questions.

Zoom: https://harvard.zoom.us/j/779283357

More than 10 years ago, I showed that non-supersymmetric QCD with adjoint fermions admits a (non-thermal) compactification on $R^3 \times S^1$ where non-perturbative gauge dynamics becomes calculable. The mass gap, linear confinement and discrete chiral symmetry breaking are sourced by magnetic bions, which are correlated monopole-instanton anti-instanton pairs which have non-vanishing magnetic charge but have zero topological charge. This construction led to the idea of the double-trace deformation of pure Yang-Mills theory on $R^3 x S^1$, which is continuously connected to pure YM on $R^4$ in the sense of continuity of all gauge invariant order parameters. It turns out that all qualitative non-perturbative properties of deformed YM theory are in agreement with all of our non-perturbative expectations concerning pure YM theory. Over the last year, numerical simulations have shown that the topological susceptibility of deformed YM on small $S^1 \times R^3$ are in precise agreement with the numerical results on large $S^1 \times R^3$. Therefore, it is very likely that there is more truth in deformed YM construction than that meets the eye. I will give a lecture style talk on these topics.

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

I will describe two new descriptions of gapped fracton topological order. These descriptions are generic and make minimal modifications to ordinary TQFT in order to obtain fracton physics. The first description constructs a fracton order by embedding a network of topological defects (aka interfaces) within an ordinary topological order, which results in the restricted fracton mobility. The second description takes the continuum limit by viewing the defect layers as infinitesimally separated. This is done by coupling a TQFT, such as BF theory, to a new kind of foliated gauge field. The first description is based on arXiv:2002.05166, while the second is based on a forthcoming work.

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In this talk, we construct a dgBV algebra PV*(X) associated to a possibly degenerate Calabi–Yau variety X equipped with local thickening data. This gives a version of the Kodaira–Spencer dgLa which is applicable to degenerated spaces including both log smooth or maximally degenerated Calabi–Yau. We use this to prove an unobstructedness result about the smoothing of degenerated Log Calabi–Yau varieties X satisfying Hodge–deRham degeneracy property for cohomology of X, in the spirit of Kontsevich–Katzarkov–Pantev. This is a joint work with Kwokwai Chan and Naichung Conan Leung.

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Quantum Fourier analysis is a subject that combines an algebraic Fourier transform (pictorial in the case of subfactor theory and topological quantum field theory) with analytic estimates. We give an overview of quantum Fourier analysis in this talk. We highlight an application in recent work joint with Sebastien Palcoux and Jinsong Wu: we find new, surprisingly efficient, analytic obstructions of unitary categorification of fusion rings by applying quantum Fourier analysis to the Drinfeld center of unitary fusion categories.

Zoom: https://harvard.zoom.us/j/779283357

Yang-Mills theory. In my talk I will tell how to derive a holomorphic
anomaly equation for its partition function on CP^2 by two different,
but somewhat analogous, methods. First is the derivation from the path
integral of the effective theory on the Coulomb branch. The second is
the derivation from the path-integral in the effective 2d theory
obtained by compactification of the corresponding 6d (2,0) theory on the
ternsor branch.

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

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“Type-II” fracton phases are exotic, gapped phases of matter in which all point-like excitations are deconfined but completely immobile. We present two approaches to constructing and analyzing models for type-II fractons using tools from conventional topological order. First, we use networks of defects in topological quantum field theories to describe Haah’s B code, which is a type-II fracton model. We use this fact as evidence for a conjecture that all fracton models can be described by a suitable defect network. Second, we show how gauging global symmetries of Abelian type-II fracton models produces new, exactly solvable models with non-Abelian type-II fractons, and we analyze the unusual excitations.

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We show that every quantum computation can be described by Bayesian update of a probability distribution on a finite state space. Negativity in a quasiprobability function is not required, neither in states nor the operations. Our result is consistent with Gleason’s Theorem and the PBR theorem. This is joint work with Michael Zurel and Cihan Okay; see arXiv:2004.01992.

Zoom: https://harvard.zoom.us/j/779283357

\hat{Z} is an invariant of 3-manifolds valued in q-series (i.e. power series in q with integer coefficients), which has interesting modular properties. While originally from physics, this invariant has been mathematically constructed for a big class of 3-manifolds, and conjecturally it can be extended to all 3-manifolds. In this talk, I will give a gentle introduction to \hat{Z} and what is known about it, as well as highlighting some recent developments, including the use of R-matrix, generalization to higher rank, large N-limit and interpretation as open topological string partition functions.

Zoom: https://harvard.zoom.us/j/94717938264