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  • MATHEMATICAL PICTURE LANGUAGE SEMINAR

    MATHEMATICAL PICTURE LANGUAGE SEMINAR
    Topological order, tensor networks and subfactors

    10:00 AM-11:00 AM
    December 1, 2020

    We present recent progress on studies of 2-dimensional topological order in terms of tensor networks and its connections to subfactor theory. We explain how Drinfel’d centers and higher relative commutants naturally appear in this context and use of picture language in this study.

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

  • CMSA EVENT: CMSA Computer Science for Mathematicians: Some extensions on argumentation frameworks via hypergraphs
    11:30 AM-12:30 PM
    December 1, 2020
    The Dung Abstract Argumentation Framework (AAF) is an effective formalism for modelling disputes between two or more agents. Generally, the Dung AF is extended to include some unique interactions between agents. This has further been explained with the Bipolar Argumentation Framework (BAF). In the academic space, the use of AAF is highly signified. We can use the AF as a means to resolve disagreements that allows for the determination of a winning argument. In general, there can be imperfect ontologies that affect how reasoning is defined. Typical logic-based AFs apply to the incoherent/uncertain ontologies. However, Dung demonstrated a stable extension of AF to support an “acceptable standard of behavior”. This talk will align with present endeavors on extending the Dung AAF to consider the notion of conflict-freeness in relation to persistence over a hypergraph. With a generic type of argumentation, there are some methods that can exploit certain complex decision procedures. Argument and attack relations within the Dung AAF, thus are further defined to obtain a graphical formula of Kripke groundedness. The incorporating of multiple levels of knowledge aligns with a computational linguistics aspect for the defining of a classification criteria for AAF. In the construction, I will provide some treatment of ‘good’ model-theoretic properties that bridge AAF with Zarankiewicz’s problem to introduce how arguments are consistent with bipartite hypergraphs. The Zarankiewicz problem appears with the communication complexity on AF graphs.

    Zoom: https://harvard.zoom.us/j/98231541450

  • HARVARD-MIT ALGEBRAIC GEOMETRY SEMINAR

    HARVARD-MIT ALGEBRAIC GEOMETRY SEMINAR
    Positroid varieties and q,t-Catalan numbers

    3:00 PM-4:00 PM
    December 1, 2020

    Positroid varieties are subvarieties of the Grassmannian obtained by intersecting cyclic rotations of Schubert varieties.  We show that the “top open positroid variety” has mixed Hodge polynomial given by the q,t-rational Catalan numbers (up to a simple factor).  Unlike the Grassmannian, the cohomology of open positroid varieties is not pure.

    The q,t-rational Catalan numbers satisfy remarkable symmetry and unimodality properties, and these arise from the Koszul duality phenomenon in the derived category of the flag variety, and from the curious Lefschetz phenomenon for cluster varieties.  Our work is also related to knot homology and to the cohomology of compactified Jacobians.

    This talk is based on joint work with Pavel Galashin.

    Zoom: https://harvard.zoom.us/j/91794282895?pwd=VFZxRWdDQ0VNT0hsVTllR0JCQytoZz09

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  • CMSA EVENT: CMSA Math Science Literature Lecture Series
    8:00 AM-9:30 AM
    December 2, 2020

    TITLE: Is relativity compatible with quantum theory?

    ABSTRACT: We review the background, mathematical progress, and open questions in the effort to determine whether one can combine quantum mechanics, special relativity, and interaction together into one mathematical theory. This field of mathematics is known as “constructive quantum field theory.” Physicists believe that such a theory describes experimental measurements made over a 70 year period and now refined to 13-decimal-point precision—the most accurate experiments ever performed.

    Talk chair: Zhengwei Liu

    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.
  • CMSA EVENT: CMSA Strongly Correlated Quantum Materials and High-Temperature Superconductors Series: Interplay between superconductivity and non-Fermi liquid at a quantum critical point in a metal
    12:00 PM-1:30 PM
    December 2, 2020

    I discuss the interplay between non-Fermi liquid behaviour and pairing near a quantum-critical point (QCP) in a metal. These tendencies are intertwined in the sense that both originate from the same interaction mediated by gapless fluctuations of a critical order parameter. The two tendencies compete because fermionic incoherence destroys the Cooper logarithm, while the pairing eliminates scattering at low energies and restores fermionic coherence. I discuss this physics for a class of models with an effective dynamical interaction V (Ω) ~1/|Ω|^γ (the γ-model). This model describes, in particular, the pairing at a 2D Ising-nematic critical point in (γ=1/3), a 2D antiferromagnetic critical point (γ=1/2) and the pairing by an Einstein phonon with vanishing dressed Debye frequency (γ=2). I argue the pairing wins, unless the pairing component of the interaction is artificially reduced, but because of fermionic incoherence in the normal state, the system develops a pseudogap, preformed pairs behaviour in the temperature range between the onset of the pairing at Tp and the onset of phase coherence at the actual superconducting Tc. The ratio Tc/Tp decreases with γ and vanishes at γ =2. I present two complementary arguments of why this happens. One is the softening of longitudinal gap fluctuations, which become gapless at γ =2. Another is the emergence of a 1D array of dynamical vortices, whose number diverges at γ =2. I argue that once the number of vortices becomes infinite, quasiparticle energies effectively get quantized and do not get re-arranged in the presence of a small phase variation. I show that a new non-superconducting ground state emerges at γ >2.

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

  • RANDOM MATRIX SEMINAR
    2:00 PM-3:00 PM
    December 2, 2020

    The talk discusses a toy model for phase transitions in mixtures of incompressible droplets. The model consists of non-overlapping hypercubes of side-lengths 2^j, j\in \N_0. Cubes belong to an admissible set such that if two cubes overlap, then one cube is contained in the other, a picture reminiscent of Mandelbrot’s fractal percolation model. I will present exact formulas for the entropy and pressure, discuss phase transitions from a fluid phase with small cubes towards a condensed phase with a macroscopic cube, and briefly sketch some broader questions on renormalization and cluster expansions that motivate the model. Based on arXiv:1909.09546 (J. Stat. Phys. 179 (2020), 309-340).

    Zoom: https://harvard.zoom.us/j/98520388668

  • NUMBER THEORY SEMINAR
    3:00 PM-4:00 PM
    December 2, 2020

    The Cohen–Lenstra–Martinet conjectures have been verified in
    only two cases. Davenport–Heilbronn compute the average size of the
    3-torsion subgroups in the class group of quadratic fields and Bhargava
    computes the average size of the 2-torsion subgroups in the class groups of cubic fields. The values computed in the above two results are remarkably stable. In particular, work of Bhargava–Varma shows that they do not change if one instead averages over the family of quadratic or cubic fields satisfying any finite set of splitting conditions.

    However for certain “thin” families of cubic fields, namely, families of
    monogenic and n-monogenic cubic fields, the story is very different. In
    this talk, we will determine the average size of the 2-torsion subgroups of
    the class groups of fields in these thin families. Surprisingly, these
    values differ from the Cohen–Lenstra–Martinet predictions! We will also
    provide an explanation for this difference in terms of the Tamagawa numbers of naturally arising reductive groups. This is joint work with Manjul Bhargava and Jon Hanke.

    Zoom: https://harvard.zoom.us/j/96767001802

    Password: The order of the permutation group on 9 elements.

  • MATH TABLE
    4:30 PM-5:30 PM
    December 2, 2020

    p-Adic numbers have always been primarily associated with pure Mathematics, and have become especially relevant in algebra and modern number theory. But why did Computer Scientists become interested in them? In this talk we will introduce p-adic numbers and survey their main properties. We will then introduce Dixon’s algorithm, which is the first algorithm that used p-adic numbers to compute the exact rational solution to an integer linear system of equations. We will also explore the latest runtime improvements in p-adic linear algebra algorithms, and discuss whether we can solve linear equation systems faster than matrix multiplication.

    Zoom: https://harvard.zoom.us/j/96759150216?pwd=Tk1kZlZ3ZGJOVWdTU3JjN2g4MjdrZz09

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  • CMSA EVENT: CMSA Math Science Literature Lecture Series
    8:00 AM-9:30 AM
    December 4, 2020

    TITLE: Michael Atiyah: Geometry and Physics

    ABSTRACT: In mid career, as an internationally renowned mathematician, Michael Atiyah discovered that some problems in physics responded to current work in algebraic geometry and this set him on a path to develop an active interface between mathematics and physics which was formative in the links which are so active today. The talk will focus, in a fairly basic fashion, on some examples of this interaction, which involved both applying physical ideas to solve mathematical problems and introducing mathematical ideas to physicists.

    Talk chair: Peter Kronheimer

    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.
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  • CMSA EVENT: CMSA Strongly Correlated Quantum Materials and High-Temperature Superconductors Series: Signatures of anomalous symmetry breaking in the cuprates
    10:30 AM-12:00 PM
    December 9, 2020

    The temperature versus doping phase diagram of the cuprate high-Tsuperconductors features an enigmatic pseudogap region whose microscopic origin remains a subject of intensive study. Experimentally resolving its symmetry properties is imperative for narrowing down the list of possible explanations. In this talk I will give an overview of how optical second harmonic generation (SHG) can be used as a sensitive probe of symmetry breaking, and recap the ways it has been used to solve outstanding problems in condensed matter physics. I will then describe how we have been applying SHG polarimetry and spectroscopy to interrogate the cuprate pseudogap. In particular, I will discuss our data on YBa2Cu3Oy [1], which show an order parameter-like increase in SHG intensity below the pseudogap temperature T* across a broad range of doping levels. I will then focus on our more recent results on a model parent cuprate Sr2CuO2Cl2 [2], where evidence of anomalous broken symmetries surprisingly also exists. Possible connections between these observations will be speculated upon.

    [1] L. Zhao, C. A. Belvin, R. Liang, D. A. Bonn, W. N. Hardy, N. P. Armitage and D. Hsieh, “A global inversion-symmetry-broken phase inside the pseudogap region of YBa2Cu3Oy,” Nature Phys. 13, 250 (2017).

    [2] A. de la Torre, K. L. Seyler, L. Zhao, S. Di Matteo, M. S. Scheurer, Y. Li, B. Yu, M. Greven, S. Sachdev, M. R. Norman and D. Hsieh. “Anomalous mirror symmetry breaking in a model insulating cuprate Sr2CuO2Cl2,” Preprint at https://arxiv.org/abs/2008.06516.

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

  • NUMBER THEORY SEMINAR
    3:00 PM-4:00 PM
    December 9, 2020

    There are several ways to specify a number field. One can provide the minimal polynomial of a primitive element, the multiplication table of a $\bf Q$-basis, the traces of a large enough family of elements, etc. From any way of specifying a number field one can hope to deduce a bound on the number $N_n(H)$ of number fields of given degree $n$ and discriminant bounded by $H$. Experimental data suggest that the number of isomorphism classes of number fields of degree $n$ and discriminant bounded by $H$ is equivalent to $c(n)H$ when $n\geqslant 2$ is fixed and $H$ tends to infinity. Such an estimate has been proved for $n=3$ by Davenport and Heilbronn and for $n=4$, $5$ by Bhargava. For an arbitrary $n$ Schmidt proved a bound of the form $c(n)H^{(n+2)/4}$ using Minkowski’s theorem. Ellenberg et Venkatesh have proved that the exponent of $H$ in $N_n(H)$ is less than sub-exponential in $\log (n)$. I will explain how Hermite interpolation (a theorem of Alexander and Hirschowitz) and geometry of numbers combine to produce short models for number fields and sharper bounds for $N_n(H)$.

    Zoom: https://harvard.zoom.us/j/96767001802

    Password: The order of the permutation group on 9 elements.

  • CMSA EVENT: CMSA New Technologies in Mathematics: Machine learning and SU(3) structures on six manifolds
    3:00 PM-4:00 PM
    December 9, 2020

    In this talk we will discuss the application of Machine Learning techniques to obtain numerical approximations to various metrics of SU(3) structure on six manifolds. More precisely, we will be interested in SU(3) structures whose torsion classes make them suitable backgrounds for various string compactifications. A variety of aspects of this topic will be covered. These will include learning moduli dependent Ricci-Flat metrics on Calabi-Yau threefolds and obtaining numerical approximations to torsional SU(3) structures.

    Zoom: https://harvard.zoom.us/j/96047767096?pwd=M2djQW5wck9pY25TYmZ1T1RSVk5MZz09

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January
January