upcoming events

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  • MATHEMATICAL PICTURE LANGUAGE SEMINAR

    MATHEMATICAL PICTURE LANGUAGE SEMINAR
    The information in a wave

    10:00 AM-11:00 AM
    May 5, 2020

    Suppose that some information is transmitted by an undulatory signal.

    In Classical Field Theory, the stress-energy tensor provides the energy-momentum

    density of the wave packet at any time. But, how to measure the information, or

    entropy, carried by the wavepacket in a certain region at given time?

    Surprisingly, one can answer the above (entirely classical) question by means of

    Operator Algebras and Quantum Field Theory. In fact, in second quantisation a

    wave packet gives rise to a sector of the Klein-Gordon Quantum Field Theory on

    the Rindler spacetimeW. The associated vacuum noncommutative entropy of the

    global von Neumann algebras of W is the entropy of the wave packet in the

    wedge region W of the Minkowski spacetime. One can then read this result in first

    quantisation via a notion of entropy of a vectorof a Hilbert space with respect to a

    real linear subspace.

    I give a path to the above results by an overview of some of basic results in

    Operator Algebras and Quantum Field Theory and of the relation with the

    Quantum Null Energy Inequality.

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

  • CMSA EVENT: Math Science Literature Lecture Series
    11:00 AM-12:30 PM
    May 5, 2020
    Please register here to attend any of the lectures.

     

    Title: Black Hole Formation

    Abstract: Can black holes form through the focusing of gravitational waves?
    This was an outstanding question since the early days of general relativity. In his breakthrough result of 2008, Demetrios Chrstodoulou answered this question with “Yes!”
    In order to investigate this result, we will delve deeper into the dynamical mathematical structures of the Einstein equations. Black holes are related to the presence of trapped surfaces in the spacetime manifold.
    Christodoulou proved that in the regime of pure general relativity and for arbitrarily dispersed initial data, trapped surfaces form through the focusing of gravitational waves provided the incoming energy is large enough in a precisely defined way. The proof combines new ideas from geometric analysis and nonlinear partial differential equations as well as it introduces new methods to solve large data problems. These methods have many applications beyond general relativity. D. Christodoulou’s result was generalized in various directions by many authors. It launched mathematical activities going into multiple fields in mathematics and physics. In this talk, we will discuss the mathematical framework of the above question. Then we will outline the main ideas of Christodoulou’s result and its generalizations, show relations to other questions and give an overview of implications in other fields.

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

    The schedule will be updated as talks are confirmed.

  • CMSA EVENT: Math Science Literature Lecture Series
    3:00 PM-4:30 PM
    May 5, 2020
    Please register here to attend any of the lectures.

     

    Title: Quantum Groups

    Abstract: The theory of quantum groups developed in mid 1980s from attempts to construct and understand solutions of the quantum Yang-Baxter equation, an important equation arising in quantum field theory and statistical mechanics. Since then, it has grown into a vast subject with profound connections to many areas of mathematics, such as representation theory, the Langlands program, low-dimensional topology, category theory, enumerative geometry, quantum computation, algebraic combinatorics, conformal field theory, integrable systems, integrable probability, and others. I will review some of the main ideas and examples of quantum groups and try to briefly describe some of the applications.

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

    The schedule will be updated as talks are confirmed.

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  • CMSA EVENT: CMSA Quantum Matter/Quantum Field Theory Seminar: Domain walls, anomalies, and deconfinement
    10:30 AM-12:00 PM
    May 6, 2020

    “Generalized” ’t Hooft anomalies impose new constraints on nonperturbative gauge dynamics. In confining theories with domain walls, they imply that quarks become liberated on the walls. The pertinent anomaly-inflow arguments have a formal flavor and our goal here is to shed light on dynamical aspects of domain-wall deconfinement. We use semiclassical means in a theoretically controlled setting.  While these tools do not require supersymmetry, for brevity (and elegance) we focus this talk on 4d N=1 super Yang-Mills theory. We review the set-up and study the domain walls’ properties, along the way deriving the  “N choose k” multiplicity of k-walls (connecting vacua k “steps” apart). We use the results to explain how quarks of all N-alities become deconfined on all k-walls. A similar picture applies to deconfinement on domain walls in QCD at theta=pi, adjoint QCD, and axion domain walls. We end with discussing a “wish list” of not well-understood aspects. (The bulk of this talk is based on 1909.10979, with Cox and Wong. However, it relies heavily on 1501.06773, with Anber and Sulejmanpasic, as well as 2001.03631, with Anber.)

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

     

  • CMSA EVENT: Math Science Literature Lecture Series
    1:00 PM-2:30 PM
    May 6, 2020
    Please register here to attend any of the lectures.

     

    TitleMy life and times with the sporadic simple groups

    Abstract: Five sporadic simple groups were proposed in 19th century and 21 additional ones arose during the period 1965-1975. There were many discussions about the nature of finite simple groups and how sporadic groups are placed in mathematics. While in mathematics grad school at University of Chicago,  I became fascinated with the unfolding story of sporadic simple groups. It involved theory, detective work and experiments. During this lecture, I will describe some of the people, important ideas and evolution of thinking about sporadic simple groups. Most should be accessible to a general mathematical audience.

    Article

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

    The schedule will be updated as talks are confirmed.

  • RANDOM MATRIX SEMINAR
    2:00 PM-3:00 PM
    May 6, 2020

    will speak on:

    FIELD THEORY AS A LIMIT OF INTERACTING QUANTUM BOSE GASES

    We prove that the grand canonical Gibbs state of an interacting quantum Bose gas converges to the Gibbs measure of a nonlinear Schrödinger equation in the mean-field limit, where the density of the gas becomes large and the interaction strength is proportional to the inverse density. Our results hold in dimensions d = 1,2,3. For d > 1 the Gibbs measure is supported on distributions of negative regularity and we have to renormalize the interaction. The proof is based on a functional integral representation of the grand canonical Gibbs state, in which convergence to the mean-field limit follows formally from an infinite-dimensional stationary phase argument for ill-defined non-Gaussian measures. We make this argument rigorous by introducing a white-noise-type auxiliary field, through which the functional integral is expressed in terms of propagators of heat equations driven by time-dependent periodic random potentials. Joint work with Jürg Fröhlich, Benjamin Schlein, and Vedran Sohinger.

    via Zoom: https://harvard.zoom.us/j/147308224

  • NUMBER THEORY SEMINAR
    3:00 PM-4:00 PM
    May 6, 2020
     Langlands functoriality predicts the transfer of automorphic representations along maps of L-groups. In particular, the symmetric power representation Symm^{n-1} of GL(2) should give rise to a lifting from automorphic representations of GL(2) to automorphic representations of GL(n). I will discuss joint work with Jack Thorne, in which we prove the existence of all symmetric power lifts for many cuspidal Hecke eigenforms (for example, those of square-free level).
  • INFORMAL GEOMETRY AND DYNAMICS SEMINAR

    INFORMAL GEOMETRY AND DYNAMICS SEMINAR
    Coarse density of subsets of moduli space

    4:00 PM-5:30 PM
    May 6, 2020

    I will discuss coarse geometric properties of algebraic subvarieties of the moduli space of Riemann surfaces.  In joint work with Jenya Sapir, we prove that such a subvariety is coarsely dense, with respect to either the Teichmuller or Thurston metric, iff it has full dimension in the moduli space.  This work was motivated by an attempt to understand the geometry of the image of the projection map from a stratum of abelian or quadratic differentials to the moduli space of Riemann surfaces.  As a corollary of our theorem, we characterize when this image is coarsely dense.  A key part of the proof of the theorem involves comparing analytic plumbing coordinates at the Deligne-Mumford boundary to hyperbolic/extremal lengths of curves on nearby smooth surfaces.

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

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  • CMSA EVENT: CMSA Condensed Matter/Math Seminar: Line Operators of Gauge Theories on Non-Spin Manifolds
    10:30 AM-12:00 PM
    May 7, 2020

    I will talk about line operators of four-dimensional gauge theories on non-spin manifolds. Line operators correspond to worldlines of heavy classical particles. Specifying the spectrum of such particles/lines, leads to distinct physical theories with different discrete theta parameters. We propose a formula for the spin of line operators (boson or fermion), and classify gauge theories with simple Lie algebras on non-spin manifolds. We also discuss the one-form symmetries of these theories and their ‘t Hooft anomalies. This talk is based on https://arxiv.org/abs/1911.00589, jointly with J.P. Ang and Konstantinos Roumpedakis.

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

  • DIFFERENTIAL GEOMETRY SEMINAR

    DIFFERENTIAL GEOMETRY SEMINAR
    Gauge theory for string algebroids

    4:00 PM-5:00 PM
    May 7, 2020

    We introduce a moment map picture for holomorphic string algebroids, a special class of holomorphic Courant algebroids introduced in arXiv:1807.10329. An interesting feature of our construction is that the Hamiltonian gauge action is described by means of Morita equivalences, as suggested by higher gauge theory. The zero locus of the moment map is given by the solutions of the Calabi system, a coupled system of equations which provides a unifying framework for the classical Calabi problem and the Hull-Strominger system. Our main results are concerned with the geometry of the moduli space of solutions, and assume a technical condition which is fulfilled in examples. We prove that the moduli space carries a pseudo-Kähler metric with Kähler potential given by the dilaton functional, a topological formula for the metric, and an infinitesimal Donaldson-Uhlenbeck-Yau type theorem. Finally, we relate our topological formula to a physical prediction for the gravitino mass in order to obtain a new conjectural obstruction for the Hull-Strominger system. This is joint work with Roberto Rubio and Carl Tipler.

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

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  • CMSA GENERAL RELATIVITY SEMINAR CMSA EVENT
    11:00 AM-12:00 PM
    May 8, 2020

    We will be concerned with asymptotically hyperbolic ‘hyperboloidal’ initial data for the Einstein equations. Such initial data is modeled on the upper unit hyperboloid in Minkowski spacetime and consists of a Riemannian manifold (M, g) whose geometry at infinity approaches that of hyperbolic space, and a symmetric 2-tensor K representing the second fundamental form of the embedding into spacetime, such that K -> g at infinity. There is a notion of mass in this setting and a positive mass conjecture can be proven by spinor techniques. Other important results concern the case K = g, where the conjecture states that an asymptotically hyperbolic manifold whose scalar curvature is greater than or equal to that of hyperbolic space must have positive mass unless it is a hyperbolic space. In this talk, we will discuss how the method of Jang equation reduction, originally devised by Schoen and Yau to prove the positive mass conjecture for asymptotically Euclidean initial data sets, can be adapted to the asymptotically hyperbolic setting yielding a non-spinor proof of the respective positive mass conjecture. We will primarily focus on the case dim M = 3.

    via Zoom: https://harvard.zoom.us/j/579137378

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  • MATHEMATICAL PICTURE LANGUAGE SEMINAR
    10:00 AM-11:00 AM
    May 12, 2020

    The partition functions of euclidean quantum field theory can be described as functions on the moduli space of compact manifolds with Riemanninan metric that have few generalized derivatives. The conventional derivative with respect to the metric yields the energy-momentum tensor. All fields can be described in an analogous fashion, but one has to introduce derivatives that can change the topology, The idea is tested for the (2,5) minimal model in two-dimensional conformal field theory, where the partition function yields a natural generalization of the Rogers-Ramanujan functions to arbitrary genus.

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

  • DIFFERENTIAL GEOMETRY SEMINAR

    DIFFERENTIAL GEOMETRY SEMINAR
    Angular momentum in general relativity

    3:00 PM-4:00 PM
    May 12, 2020

    In the theory of general relativity, defining a valid notion of angular momentum is proven to be an even more challenging task than the definition of energy/mass. In this talk, I shall discuss this fundamental notion from the quasilocal level to null infinity.

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

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  • CMSA EVENT: CMSA Quantum Matter/Quantum Field Theory Seminar: Continuum Quantum Field Theory for Fractons, Part I
    10:30 AM-12:00 PM
    May 13, 2020

    Starting with a lattice system with local interactions at short distances, its long-distance behavior is captured by a continuum Quantum Field Theory (QFT).  This description is universal, i.e. it is independent of most of the details of the microscopic system. Surprisingly, certain recently discovered lattice systems, and in particular models of fractons, seem to violate this general dogma.  Motivated by this apparent contradiction, we will present exotic continuum QFTs that describe these systems.

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

  • RANDOM MATRIX SEMINAR
    2:00 PM-3:00 PM
    May 13, 2020

    will speak on:

    Quantized quantum transport and Abelian anyons

    I’ll discuss recent developments in the study of quantized quantum transport, focussing on the quantum Hall effect. Beyond presenting an index taking rational values, and which is the Hall conductance in the adapted setting, I will explain how the index is intimately paired with the existence of quasi-particle excitations having non-trivial braiding properties.

    via Zoom: https://harvard.zoom.us/j/147308224

  • RANDOM MATRIX SEMINAR
    2:00 PM-3:00 PM
    May 13, 2020

    will speak on:

    Quantized quantum transport and Abelian anyons

    I’ll discuss recent developments in the study of quantized quantum transport, focussing on the quantum Hall effect. Beyond presenting an index taking rational values, and which is the Hall conductance in the adapted setting, I will explain how the index is intimately paired with the existence of quasi-particle excitations having non-trivial braiding properties.

    via Zoom: https://harvard.zoom.us/j/147308224

  • NUMBER THEORY SEMINAR

    NUMBER THEORY SEMINAR
    Prismatic Dieudonné theory

    3:00 PM-4:00 PM
    May 13, 2020
    I would like to explain a classification result for p-divisible groups, which unifies many of the existing results in the literature. The main tool is the theory of prisms and prismatic cohomology recently developed by Bhatt and Scholze. This is joint work with Johannes Anschütz.
  • INFORMAL GEOMETRY AND DYNAMICS SEMINAR
    4:00 PM-5:30 PM
    May 13, 2020

    It is a beautiful fact that any holomorphic one-form on a genus g Riemann surface can be presented as a collection of polygons in the plane with sides identified by translation. Since GL(2, R) acts on the plane (and polygons in it), it follows that there is an action of GL(2, R) on the collection of holomorphic one-forms on Riemann surfaces. This GL(2, R) action can also be described as the group action generated by scalar multiplication and Teichmuller geodesic flow. By work of McMullen in genus two, and Eskin, Mirzakhani, and Mohammadi in general, given any holomorphic one-form, the closure of its GL(2, R) orbit is an algebraic variety. While McMullen classified these orbit closures in genus two, little is known in higher genus.

    In the first part of the talk, I will describe the Mirzakhani-Wright boundary of an invariant subvariety (using mostly pictures) and a new result about reconstructing an orbit closure from its boundary. In the second part of the talk, I will define the rank of an invariant subvariety – a measure of size related to dimension – and explain why invariant subvarieties of rank greater than g/2 are loci of branched covers of lower genus Riemann surfaces. This will address a question of Mirzakhani.

    No background on Teichmuller theory or dynamics will be assumed. This material is work in progress with Alex Wright.

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

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  • CMSA GENERAL RELATIVITY SEMINAR CMSA EVENT

    CMSA GENERAL RELATIVITY SEMINAR CMSA EVENT
    Polynomial tails and conservation laws of waves on black holes

    11:00 AM-12:00 PM
    May 15, 2020

    In 1972, Price suggested that inverse polynomial tails should be present in the late-time behaviour of scalar fields on Schwarzschild black holes. In the decades since, many features of these tails have been explored both numerically and heuristically in more general settings. The presence of polynomial tails in the context of the Einstein equations has important implications for the nature of singularities inside dynamical black holes and the late-time behaviour of gravitational waves observed at infinity. In this talk I will discuss recent work in collaboration with Y. Angelopoulos and S. Aretakis that establishes rigorously the existence of Price’s polynomial late-time tails in the context of scalar fields on black holes. I will moreover describe how late-time tails are connected to the existence of conservation laws for scalar fields in asymptotically flat spacetimes.

    via Zoom: https://harvard.zoom.us/j/579137378

  • CMSA EVENT: CMSA Mathematical Physics Seminar – Eigenstate thermalization and disorder averaging in gravity
    1:00 PM-2:00 PM
    May 15, 2020

    It has long been believed that progress in understanding the black hole information paradox would require coming to terms with microscopic details of quantum gravity, beyond the reach of semiclassical effective field theory. In that light, one of the most surprising discoveries of the last year has been that signature features of the unitary evaporation of black holes can already be seen within effective field theory, albeit with the inclusion of ‘euclidean wormholes’. However, these novel contributions are best understood when the gravitational theory is not a single microscopic theory, but an average over many different theories. To save unitarity must we then simultaneously throw it away? I will explain how the same story can be recovered within a single microscopic theory by thinking carefully about the right effective theory for finite-lifetime observers.

    Zoom: https://harvard.zoom.us/j/837429475

     

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  • MATHEMATICAL PICTURE LANGUAGE SEMINAR
    10:00 AM-11:00 AM
    May 19, 2020

    We present explicit constructions of infinite families of CW- complexes of arbitrary dimension with buildings as the universal covers. These complexes give rise to new families of C*-algebras, classifiable by their K-theory. The underlying building structure allows explicit computation of the K-theory. We will also present new higher dimensional generalizations of the Thompson groups, which are usually difficult to distinguish, but the K-theory of C*-algebras gives new invariants to recognize non-isomorphic groups. We will also discuss new directions of generalizations to higher dimensions of the work of Vaughan Jones and his collaborators on connections of the Thomson’s group and Conformal Field Theory.

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

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  • CMSA EVENT: CMSA Condensed Matter/Math Seminar: Gauge-invariant path-integral measure for the overlap Weyl fermions in 16 of SO(10)
    10:30 AM-12:00 PM
    May 21, 2020
    We consider a lattice formulation of the SO(10) chiral gauge theory with left-handed Weyl fermions in the sixteen dimensional spinor representation 16  in the framework of the overlap fermion/the Ginsparg-Wilson relation.  We propose a manifestly gauge-invariant path-integral measure for the left-handed Weyl field on the lattice using all the components of the Dirac field, but the right-handed part of which is just saturated completely by inserting a suitable product of the SO(10)-invariant ‘t Hooft vertices in terms of the right-handed field. The definition of the measure applies to all possible topological sectors. The measure possesses all required transformation properties under lattice symmetries and the induced effective action is CP invariant. The global U(1) symmetry of the left-handed field is anomalous due to the non-trivial transformation of the measure, while that of the right-handed field is explicitly broken by the ’t Hooft vertices. There remains the issue of smoothness/locality in the gauge-field dependence of the Weyl fermion measure.
    We also discuss the relations of our formulation to other approaches/proposals to decouple the species-doubling/mirror degrees of freedom. Those include Eichten-Preskill model, Ginsparg-Wilson Mirror-fermion model, Domain wall fermion model with the boundary Eichten-Preskill term, and 4D Topological Insulator/Superconductor with gapped boundary phase. We clarify the similarity and the difference in technical detail and show that our proposal is a well-defined and unified testing ground for that basic question.

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

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  • MATHEMATICAL PICTURE LANGUAGE SEMINAR

    MATHEMATICAL PICTURE LANGUAGE SEMINAR
    From Archimedes to Quantum Supremacy

    10:00 AM-11:00 AM
    May 26, 2020

    I’ll tell a mathematical story that runs from Archimedes’ hat-box theorem of ~200BC (which he used to calculate the surface area of the sphere), to the properties of “Porter-Thomas” probability distributions, all the way to my and others’ work establishing the theoretical foundations of Google’s quantum supremacy experiment from this past fall.  While this admittedly has little to do with mathematical picture languages, there will be pictures of spheres.

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

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  • CMSA EVENT: CMSA Condensed Matter/Math Seminar: Supermetal from a high-order Van Hove singularity
    10:30 AM-12:00 PM
    May 28, 2020

    A Van Hove singularity (VHS) of the density of states (DOS) is universal in a periodic system.  In two dimensions, a saddle point of energy dispersion yields a logarithmic divergence in the DOS.  Here, we introduce a new kind of VHS, motivated by the recent development of moiré materials.  We define a high-order VHS, which gives a power-law DOS divergence [1].  It requires only a single tuning parameter, such as a twist angle of a moiré material, pressure, and strain.  We further perform a renormalization group analysis near a high-order VHS to study the effect of electron interactions [2].  We reveal a nontrivial metallic state, where various divergent susceptibilities coexist, but no long-range order appears.  We call such a metallic state as a supermetal.  Our controlled analysis shows that a supermetal at the interacting fixed point is a non-Fermi liquid.

    [1] N. F. Q. Yuan, H. Isobe, and L. Fu, Nat. Commun. 10, 5769 (2019).
    [2] H. Isobe and L. Fu, Phys. Rev. Research 1, 033206 (2019).

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

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