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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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  • MATHEMATICAL PICTURE LANGUAGE SEMINAR
    10:00 AM-11:00 AM
    April 7, 2020

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

    Circularly polarized light (i.e. helicity) is a concept defined in terms of
    plane wave expansions of solutions to Maxwell’s equations.  We wish to find  an analogous concept for classical and quantized Yang-Mills fields. Since the classical (hyperbolic) Yang-Mills equation is a non-linear equation, a gauge invariant  plane wave expansion does not exist.  We will first
    show, in electromagnetism,  an equivalence between the usual plane wave characterization  of helicity and a characterization in terms of (anti-)self  duality of a gauge potential on a half space of Euclidean R^4. The transition from Minkowski space to Euclidean space is implemented by the
    Maxwell-Poisson equation. We will then replace the Maxwell- Poisson equation by the Yang-Mills-Poisson equation to find a decomposition of the Yang-Mills configuration space into submanifolds arguably corresponding to positive and negative helicity. This is a report on the paper [1].
    References
    [1] https://doi.org/10.1016/j.nuclphysb.2019.114685

  • DIFFERENTIAL GEOMETRY SEMINAR

    DIFFERENTIAL GEOMETRY SEMINAR
    Collapsing Calabi-Yau Manifolds

    4:15 PM-5:15 PM
    April 7, 2020

    via Zoom Video Conferencing:  link TBA

    I will report on some recent progress on the problem of understanding the collapsing behavior of Ricci-flat Kahler metrics on Calabi-Yau manifolds that admit a fibration structure, when the volume of the fibers shrinks to zero. Based on joint works with Gross-Zhang and with Hein.

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  • CMSA EVENT: CMSA Quantum Matter/Quantum Field Theory Seminar: Anomaly of the Electromagnetic Duality of Maxwell Theory
    10:30 AM-12:00 PM
    April 8, 2020

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

    Every physicist knows that the classical electromagnetism is described by Maxwell’s equations and that it is invariant under the electromagnetic duality S: (E, B) → (B, −E). However, the properties of the electromagnetic duality in the quantum theory might not be as well known to physicists in general, and in fact are not very well understood in the literature. This is particularly true when going around a nontrivial path in the spacetime results in a duality transformation. In our recent work, we uncovered a feature of the Maxwell theory and its duality symmetry in such a situation, namely that it has a quantum anomaly. We found that the anomaly of this system in a particular formulation is 56 times that of a Weyl fermion. Our result reproduces, as a special case, the known anomaly of the all-fermion electrodynamics—a version of the Maxwell theory where particles of odd (electric or magnetic) charge are fermions—discovered in the last few years.

  • NUMBER THEORY SEMINAR
    3:00 PM-4:00 PM
    April 8, 2020

    via Zoom Video Conferencing: https://harvard.zoom.us/j/136830668

    (joint w/ Arul Shankar) We discuss a new method to bound 5-torsion in class groups of quadratic fields using the refined BSD conjecture for elliptic curves. The most natural “trivial” bound on the n-torsion is to bound it by the size of the entire class group, for which one has a global class number formula. We explain how to make sense of the n-torsion of a class group intrinsically as a selmer group of a Galois module. We may then similarly bound its size by the Tate-Shafarevich group of an appropriate elliptic curve, which we can bound using the BSD conjecture. This fits into a general paradigm where one bounds selmer groups of finite Galois modules by embedding into global objects, and using class number formulas. If time permits, we explain how the function field picture yields unconditional results and suggests further generalizations.
  • INFORMAL GEOMETRY AND DYNAMICS SEMINAR

    INFORMAL GEOMETRY AND DYNAMICS SEMINAR
    Effective density for values of generic quadratic forms

    4:00 PM-5:30 PM
    April 8, 2020

    via Zoom Video Conferencing: https://harvard.zoom.us/j/972495373

    The Oppenheim Conjecture, proved by Margulis, states that any irrational quadratic form, has values (at integer coordinates) that are dense on the real line. However, obtaining effective estimates for any given form is a very difficult problem. In this talk I will discuss recent results, where such effective estimates are obtained for generic forms using a combination of methods from dynamics and analytic number theory. I will also discuss some results on analogous problems for inhomogenous forms and more general higher degree polynomials.

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  • CMSA EVENT: CMSA Mathematical Physics Seminar: Comments on the lattice-continuum correspondence
    12:00 PM-1:00 PM
    April 13, 2020

    will speak on:

    The goal of this talk is to precisely describe how certain operator properties of continuum QFT (e.g. operator product expansions, current algebras, vertex operator algebras) emerge from an underlying lattice theory.  The main lesson will be that a “continuum limit” must always involve two or more cutoffs being taken to zero in a specific order.  In other words, the naive statement that continuum theories are obtained from lattice ones by letting a “lattice spacing” go to zero is never sufficient to describe the lattice-continuum correspondence.  Using these insights, I will show in detail how the Kac-Moody algebra arises from a nonperturbatively well defined, fully regularized model of free fermions, and I will comment on generalizations and applications to bosonization.  Time permitting, I will describe more intricate examples involving scalar fields, and I will discuss several open questions.

    via Zoom Video Conferencing:  https://harvard.zoom.us/j/837429475

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  • MATHEMATICAL PICTURE LANGUAGE SEMINAR

    MATHEMATICAL PICTURE LANGUAGE SEMINAR
    Evolution of NLS with Bounded Data

    10:00 AM-11:00 AM
    April 14, 2020

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

    We study the nonlinear Schroedinger equation (NLS) with bounded initial data which
    does not vanish at infinity. Examples include periodic, quasi-periodic and random initial data.
    On the lattice we prove that solutions are polynomially bounded in time for any bounded data.
    In the continuum, local existence is proved for real analytic data by a Newton iteration scheme.
    Global existence for NLS with a regularized nonlinearity follows by analyzing a local energy norm.

    This is joint work with B. Dodson and A. Soffer.

  • DIFFERENTIAL GEOMETRY SEMINAR
    4:15 PM-5:15 PM
    April 14, 2020

    via Zoom Video Conferencing: link TBA

    The anisotropic Calderon inverse problem consists in recovering the metric of a compact connected Riemannian manifold with boundary from the knowledge of the Dirichlet-to-Neumann map at fixed energy. A fundamental result due to Lee and Uhlmann states that there is uniqueness in the analytic case. We shall present counterexamples to uniqueness in cases when: 1) The metric smooth in the interior of the manifold, but only Holder continuous on one connected component of the boundary, with the Dirichlet and Neumann data being measured on the same proper subset of the boundary. 2) The metric is smooth everywhere and Dirichlet and Neumann data are measured on disjoint subsets of the boundary. This is joint work with Thierry Daude (Cergy-Pontoise) and Francois Nicoleau (Nantes).

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  • MATHEMATICAL PICTURE LANGUAGE SEMINAR

    MATHEMATICAL PICTURE LANGUAGE SEMINAR
    Graphical proofs for fault-tolerant computation

    10:00 AM-11:00 AM
    April 28, 2020

    Experimentalists are getting better and better at building qubits, but no matter how hard they try, their qubits will never be perfect. In order to build a large quantum computer, we will almost certainly need to encode the qubits using quantum error-correcting codes and encode the quantum circuits using fault-tolerant protocols. The central result of the theory of fault tolerance is the threshold theorem, which states that arbitrarily long and reliable quantum computations are possible if the error rate per gate or time step is below some constant threshold value. Fault tolerance can be nicely defined using graphical techniques, allowing for a relatively straightforward proof of the threshold theorem.

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

  • DIFFERENTIAL GEOMETRY SEMINAR
    3:00 PM-4:00 PM
    April 28, 2020

    The SYZ conjecture predicts that for polarised Calabi-Yau manifolds undergoing the large complex structure limit, there should be a special Lagrangian torus fibration. A weak version asks if this
    fibration can be found in the generic region. I will discuss my recent work proving this weak SYZ conjecture for the degenerating hypersurfaces in the Fermat family. Although these examples are quite
    special, this is the first construction of generic SYZ fibrations that works uniformly in all complex dimensions.

    If you would like to attend, please email spicard@math.harvard.edu

     

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