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Following up on the #shutdownSTEM discussions, the Department of Mathematics has launched a community web page, with evolving content to be created through community effort.
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< 2020 >
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  • CMSA EVENT: CMSA Geometry Seminar: Equivariant Floer theory and SYZ mirror symmetry
    9:30 AM-10:30 AM
    June 2, 2020

    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

  • MATHEMATICAL PICTURE LANGUAGE SEMINAR

    MATHEMATICAL PICTURE LANGUAGE SEMINAR
    Kitaev’s finite group model as an error correcting code

    10:00 AM-11:00 AM
    June 2, 2020

    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

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  • CMSA EVENT: CMSA Quantum Matter/Quantum Field Theory Seminar: Magnetic Black Holes
    10:30 AM-12:00 PM
    June 3, 2020

    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

     

  • INFORMAL GEOMETRY AND DYNAMICS SEMINAR

    INFORMAL GEOMETRY AND DYNAMICS SEMINAR
    Slices of Thurston’s Master Teapot

    4:00 PM-5:30 PM
    June 3, 2020

    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

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  • CMSA EVENT: CMSA Condensed Matter/Math Seminar: Coupled layers, p-string condensate, and exactly solvable fracton models
    10:30 AM-12:00 PM
    June 4, 2020
    In this talk, I will introduce a class of gapped fracton models, dubbed “cage-net fracton models”.  I will first review the coupled layer construction for exactly solvable fracton models. This construction leads to a general mechanism to obtain cage-net fracton models through a “p-string condensation”, where the extended one-dimensional particle strings built out of pointlike excitations are condensed. This p-string condensation generalizes the concept of anyon condensation in a conventional topological order and allows us to establish the properties of the fracton phase, such as its ground-state wave function, the spectrum and the mobility of excitations. To illustrate the main idea, I will focus on a simple example: doubled-Ising cage-net model. I will show that there are intrinsic non-Abelian excitations with restricted-mobility in this model and they cannot be understood as bound states among two-dimensional non-Abelian anyons and Abelian particles. If time permits, I will discuss another class of exactly solvable fracton models based on a generalization of the twisted gauge theory.
    via Zoom Video Conferencing: https://harvard.zoom.us/s/977347126
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  • MATHEMATICAL PICTURE LANGUAGE SEMINAR
    10:00 AM-11:00 AM
    June 9, 2020

    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

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  • CMSA EVENT: CMSA Condensed Matter/Math Seminar: Symmetry enriched U(1) quantum spin liquids and beyond
    10:30 AM-12:00 PM
    June 11, 2020
    I will present our characterization and classification of 3+1 dimensional U(1) quantum spin liquids (QSLs) enriched by symmetries. These QSLs are spin system described by a deconfined U(1) gauge theory at low energies, and we assume that the only gapless degree of freedom is the photon. I will mostly focus on the example where the symmetry includes SO(3) spin rotation and time reversal, from which I will summarize our general scheme for the characterization and classification. The characterization and classification are based on the properties of the matters coupled to the emergent U(1) gauge field, although they are gapped. I will also discuss some applications of the ideas developed here to topological phases protected by crystalline symmetries. In order to avoid potential misunderstanding due to the difference in the ideology and language between physicists with different background, I will give an overview of these differences at the beginning of the talk.
    via Zoom Video Conferencing: https://harvard.zoom.us/s/977347126
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  • CMSA EVENT: CMSA Geometry and Physics seminar: Loops in hypertoric varieties and symplectic duality
    9:30 AM-10:30 AM
    June 16, 2020
    Hypertoric varieties are algebraic symplectic varieties associated to graphs, or more generally certain hyperplane arrangements. They make many appearances in modern geometric representation theory. I will discuss certain infinite dimensional or infinite type generalizations of hypertoric varieties which occur in the study of enumerative invariants, focusing on some elementary examples. Joint work with Artan Sheshmani and Shing-Tung Yau.

     

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

  • MATHEMATICAL PICTURE LANGUAGE SEMINAR
    10:00 AM-11:00 AM
    June 16, 2020

    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

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  • MATHEMATICAL PICTURE LANGUAGE SEMINAR

    MATHEMATICAL PICTURE LANGUAGE SEMINAR
    Quantum Fourier analysis

    10:00 AM-11:00 AM
    June 23, 2020

    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

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