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  • CMSA EVENT: CMSA Quantum Matter in Mathematics and Physics Seminar: Kardar-Parisi-Zhang dynamics in integrable quantum magnets

    Speaker: Francisco Machado – Berkeley/Harvard

    9:00 AM-10:30 AM
    November 1, 2022

    Although the equations of motion that govern quantum mechanics are well-known, understanding the emergent macroscopic behavior that arises from a particular set of microscopic interactions remains remarkably challenging. One particularly important behavior is that of hydrodynamical transport; when a quantum system has a conserved quantity (i.e. total spin), the late-time, coarse-grained dynamics of the conserved charge is expected to follow a simple, classical hydrodynamical description. However the nature and properties of this hydrodynamical description can depend on many details of the underlying interactions. For example, the presence of additional dynamical constraints can fundamentally alter the propagation of the conserved quantity and induce slower-than-diffusion propagation. At the same time, the presence of an extensive number of conserved quantities in the form of integrability, can imbue the system with stable quasi-particles that propagate ballistically through the system.

    In this talk, I will discuss another possibility that arises from the interplay of integrability and symmetry; in integrable one dimensional quantum magnets with complex symmetries, spin transport is neither ballistic nor diffusive, but rather superdiffusive. Using a novel method for the simulation of quantum dynamics (termed Density Matrix Truncation), I will present a detailed analysis of spin transport in a variety of integrable quantum magnets with various symmetries. Crucially, our analysis is not restricted to capturing the dynamical exponent of the transport dynamics and enables us to fully characterize its universality class: for all superdiffusive models, we find that transport falls under the celebrated Kardar-Parisi-Zhang (KPZ) universality class.

    Finally, I will discuss how modern atomic, molecular and optical platforms provide an important bridge to connect the microscopic interactions to the resulting hydrodynamical transport dynamics. To this end, I will present recent experimental results, where this KPZ universal behavior was observed using atoms confined to an optical lattice.

    [1] Universal Kardar-Parisi-Zhang dynamics in integrable quantum systems
    B Ye†, FM*, J Kemp*, RB Hutson, NY Yao
    (PRL in press) – arXiv:2205.02853

    [2] Quantum gas microscopy of Kardar-Parisi-Zhang superdiffusion
    D Wei, A Rubio-Abadal, B Ye, FM, J Kemp, K Srakaew, S Hollerith, J Rui, S Gopalakrishnan, NY Yao, I Bloch, J Zeiher
    Science (2022) — arXiv:2107.00038

     


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  • CMSA EVENT: CMSA Topological Quantum Matter Seminar: Optical axion electrodynamics

    Speaker: Junyeong Ahn – Harvard

    9:00 AM-10:00 AM
    November 2, 2022
    1 Oxford Street, Cambridge, MA 02138 USA

    Electromagnetic fields in a magneto-electric medium behave in close analogy to photons coupled to the hypothetical elementary particle, the axion. This emergent axion electrodynamics is expected to provide novel ways to detect and control material properties with electromagnetic fields. Despite having been studied intensively for over a decade, its theoretical understanding remains mostly confined to the static limit. Formulating axion electrodynamics at general optical frequencies requires resolving the difficulty of calculating optical magneto-electric coupling in periodic systems and demands a proper generalization of the axion field. In this talk, I will introduce a theory of optical axion electrodynamics that allows for a simple quantitative analysis. Then, I will move on to discuss the issue of the Kerr effect in axion antiferromagnets, refuting the conventional wisdom that the Kerr effect is a measure of the net magnetic moment. Finally, I will apply our theory to a topological antiferromagnet MnBi2Te4.

  • CMSA EVENT: CMSA Colloquium: Doping and inverting Mott insulators on semiconductor moire superlattices

    Speaker: Liang Fu – MIT

    12:45 PM-1:45 PM
    November 2, 2022
    20 Garden Street, Cambridge, MA 02138

    Semiconductor bilayer heterostructures provide a remarkable platform for simulating Hubbard models on an emergent lattice defined by moire potential minima. As a hallmark of Hubbard model physics, the Mott insulator state with local magnetic moments has been observed at half filling of moire band. In this talk, I will describe new phases of matter that grow out of the canonical 120-degree antiferromagnetic Mott insulator on the triangular lattice. First, in an intermediate range of magnetic fields, doping this Mott insulator gives rise to a dilute gas of spin polarons, which form a pseudogap metal. Second, the application of an electric field between the two layers can invert the many-body gap of a charge-transfer Mott insulator, resulting in a continuous phase transition to a quantum anomalous Hall insulator with a chiral spin structure. Experimental results will be discussed and compared with theoretical predictions.


     

  • NUMBER THEORY SEMINAR: Endoscopy for symmetric varieties

    NUMBER THEORY SEMINAR
    Endoscopy for symmetric varieties

    Speaker: Spencer Leslie – Boston College

    3:00 PM-4:00 PM
    November 2, 2022
    1 Oxford Street, Cambridge, MA 02138 USA

    Relative trace formulas are central tools in the study of relative functoriality. In many cases of interest, basic stability problems have not previously been addressed. In this talk, I discuss a theory of endoscopy in the context of symmetric varieties with the global goal of stabilizing the associated relative trace formula. I outline how, using the dual group of the symmetric variety, one can give a good notion of endoscopic symmetric variety and conjecture a matching of relative orbital integrals in order to stabilize the relative trace formula, which can be proved in some cases. Time permitting, I will explain my proof of these conjectures in the case of unitary Friedberg–Jacquet periods.

  • SEMINARS: Informal Seminar: A norm for the homology of 3-manifolds

    Speaker: Rafael Saavedra – Harvard

    4:00 PM-5:00 PM
    November 2, 2022

    This seminar will be held in Science Center 530 at 4:00pm on Wednesday, November 2nd.

    Please see the seminar page for more details: https://www.math.harvard.edu/~ctm/sem

     

  • HARVARD-MIT COMBINATORICS SEMINAR: Harvard-MIT Combinatorics: $K$-rings of wonderful varieties and matroids

    Speaker: Shiyue Li –

    4:15 PM-5:15 PM
    November 2, 2022

    The wonderful variety of a realizable matroid and its Chow ring have played key roles in solving many long-standing open questions in combinatorics and algebraic geometry. Yet, its $K$-rings are underexplored until recently. I will be sharing with you some discoveries on the $K$-rings of the wonderful variety associated with a realizable matroid: an exceptional isomorphism between the $K$-ring and the Chow ring, with integral coefficients, and a Hirzebruch–Riemann–Roch-type formula. These generalize a recent construction of Berget–Eur–Spink–Tseng on the permutohedral variety. We also compute the Euler characteristic of every line bundle on wonderful varieties, and give a purely combinatorial formula. This in turn gives a new valuative invariant of an arbitrary matroid. As an application, we present the $K$-rings and compute the Euler characteristic of arbitrary line bundles of the Deligne–Mumford–Knudsen moduli spaces of rational stable curves with distinct marked points. Joint with Matt Larson, Sam Payne and Nick Proudfoot.


    For more information on the speaker, please see: http://www.shiyue.li

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  • CMSA EVENT: CMSA Active Matter Seminar: Force transmission informs the collective behavior of active cell layers

    Speaker: Siavash Monfared – Niels Bohr Institute, Copenhagen

    1:00 PM-2:00 PM
    November 3, 2022
    20 Garden Street, Cambridge, MA 02138

    Collective cell migration drives numerous physiological processes such as tissue morphogenesis, wound healing, tumor progression and cancer invasion. However, how the interplay of mechanical interactions and the modes of collective self-organization among cells informs such processes is yet to be established. In this talk, I will focus on the role of three-dimensional force transmission, from a theoretical and computational perspective, on two phenomena: (1) cell extrusion from a cellular monolayer and (2) density-independent solid-like to fluid-like transition of active cell layers. For the first topic, I will focus on how increasing cell-cell adhesion relative to cell-substrate adhesion enables cells to collectively exploit distinct mechanical pathways – leveraging defects in nematic and hexatic phases associated with cellular arrangement – to eliminate an unwanted cell. For the second topic, I will show how solid-like to fluid-like transition in active cell layers is linked to the percolation of isotropic stresses. This is achieved via two distinct and independent paths to model this transition by increasing (a) cell-cell adhesion and (b) active traction forces. Additionally, using finite-size scaling analyses, the phase transition associated with each path is mapped onto the 2D site percolation universality class. Our results highlight the importance of force transmission in informing the collective behavior of living cells and opens the door to new sets of questions for those interested in connecting the physics of cellular self-organization to the dynamics of biological systems.


    This seminar will be held in person and on Zoom. For more information on how to join, please see: https://cmsa.fas.harvard.edu/event/active-matter-seminar

     

  • SEMINARS: Algebraic Dynamics: The pentagram map

    Speaker: Max Weinreich – Havard University

    4:00 PM-6:00 PM
    November 3, 2022

    The pentagram map was introduced by Schwartz as a dynamical system on convex polygons in the real projective plane. The map sends a polygon to the shape formed by intersecting certain diagonals. This simple operation turns out to define a discrete integrable system, meaning roughly that it can be viewed as a translation map on a family of real tori. We will explain how the real, complex, and finite field dynamics of the pentagram map are all related by the following generalization: the pentagram map’s first or second iterate is birational to a translation on a family of Jacobian varieties (except possibly in characteristic 2). The second hour will get into the details of the proof, especially the definition of the Lax representation and the spectral curve.


     

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  • SEMINARS: Gauge Theory and Topology: Immersed curve invariants for knot complements

    Speaker: Jonathan Hanselman – Princeton University

    3:30 PM-4:30 PM
    November 4, 2022
    1 Oxford Street, Cambridge, MA 02138 USA

    Bordered Floer homology is an extension of Heegaard Floer homology to manifolds with parametrized boundary, and in the case of manifolds with torus boundary knot Floer homology gives another such extension. In earlier joint work with J. Rasmussen and L. Watson, it was shown that in this setting the bordered Floer invariant, which is equivalent to the UV=0 truncation of the knot Floer complex, can be encoded geometrically as a collection of immersed curves in the punctured torus and a pairing theorem recovers HF-hat (the simplest version of Heegaard Floer homology) of a glued manifold via Floer homology of immersed curves. In this talk, we will survey some applications of this result and then discuss a generalization that encodes the full knot Floer complex of a knot as a collection of decorated immersed curves in the torus. When two manifolds with torus boundary are glued, a pairing theorem computes HF^- of the resulting manifold as the Floer homology of certain immersed curves associated with each side. We remark that the curves we describe are invariants of knots, but we expect they are in fact invariants of the knot complements; if this is true, they may be viewed as defining a minus type bordered Floer invariant for manifolds with torus boundary.


     

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  • CMSA EVENT: CMSA Topological Quantum Matter Seminar: Controlling Quantum Matter with Quantum Cavity Fields

    Speaker: Vasil Rokaj – Harvard University

    9:00 AM-10:00 AM
    November 9, 2022
    20 Garden Street, Cambridge, MA 02138

    Cavity modification of material properties and phenomena is a novel research field motivated by the advances in strong light-matter interactions~[1]. For condensed matter systems it has been demonstrated experimentally that the transport properties of 2D materials can be modified via coupling to vacuum fields~[2,3]. While in polaritonic chemistry it has been shown that ground state chemical properties can be controlled with cavity fields~[4].  In the first part of my talk, I will present how the quantized cavity field can alter the conduction  properties of a condensed matter system by focusing on the paradigmatic Sommerfeld model of the free electron gas~[5]. The exact analytic solution of the Sommerfeld model in the cavity will be presented as well as its fundamental properties. Then, in the second part of the talk, I will focus on a many-particle system of cold ions in a harmonic trap coupled to the cavity field. I will show how this system couples collectively to the cavity and that hybrid states between light and matter, known as polaritons, emerge. The formation of polaritons leads to the modification of the properties of the cold ions and enhances the localization of the many-body wave function~[6]. Connections to experiments will be discussed as well.

    [1] F. Garcia-Vidal, C. Ciuti, T. W. Ebbesen, Science, 373, 178 (2021)

    [2] G. L. Paravicini-Bagliani et al., Nat. Phys. 15, 186-190 (2019)

    [3] F. Appugliese et al., Science 375 (6584), 1030-1034 (2022)

    [4] T. W. Ebbesen, Acc. Chem. Res. 49, 11, 2403–2412 (2016)

    [5] V. Rokaj, M. Ruggenthaler, F. G. Eich, A. Rubio, Phys. Rev. Research 4, 013012 (2022)

    [5] V. Rokaj, S.I. Mistakidis, H.R. Sadeghpour, arXiv:2207.03436 (2022)

  • CMSA EVENT: CMSA/Tsinghua Math-Science Literature Lecture

    Speaker: Hugh Woodin – Harvard University

    9:30 AM-11:00 AM
    November 9, 2022
    1 Oxford Street, Cambridge, MA 02138 USA

    CMSA/Tsinghua Math-Science Literature Lecture

    woodin_portait_books

    Prof. Hugh Woodin will present a lecture in the CMSA/Tsinghua Math-Science Literature Lecture Series.

    Date: Wednesday, November 9, 2022

    Time: 9:30 – 11:00 am ET

    Location: Via Zoom Webinar and Room G10, CMSA, 20 Garden Street, Cambridge MA 02138

    Registration is required.

     

    Title: Large cardinals and small sets: The AD+ Duality Program

    Abstract: The determinacy axiom, AD, was introduced by Mycielski and Steinhaus over 60 years ago as an alternative to the Axiom of Choice for the study of arbitrary sets of real numbers.  The modern view is that determinacy axioms concern generalizations of the borel sets, and deep connections with large cardinal axioms have emerged.

    The study of determinacy axioms has led to a specific technical refinement of AD, this is the axiom AD+. The further connections with large axioms have in turn implicitly led to a duality program, this is the AD+ Duality Program.

    The main open problems here are intertwined with those of the Inner Model Program, which is the central program in the study of large cardinal axioms.

    This has now all been distilled into a series of specific conjectures.

     

    Talk chair: Horng-Tzer Yau (Harvard Mathematics & CMSA)

    Moderator: Alejandro Poveda Ruzafa (Harvard CMSA)

     

    Beginning in Spring 2020, the CMSA began hosting a lecture series on literature in the mathematical sciences, with a focus on significant developments in mathematics that have influenced the discipline, and the lifetime accomplishments of significant scholars.

  • NUMBER THEORY SEMINAR: Number Theory: Cohomological degree-shifting operators on Shimura varieties

    Speaker: Gyujin Oh – Columbia

    3:00 PM-4:00 PM
    November 9, 2022
    1 Oxford Street, Cambridge, MA 02138 USA

    An automorphic form can appear in multiple degrees of the cohomology of arithmetic manifolds, and this happens mostly when the arithmetic manifolds are not algebraic. This phenomenon is a part of the “derived” structures of the Langlands program, suggested by Venkatesh. However, even over algebraic arithmetic manifolds, certain automorphic forms like weight-one elliptic modular forms possess a derived structure. In this talk, we discuss this idea over Shimura varieties. A part of the story is the construction of archimedean/p-adic “derived” operators on the cohomology of Shimura varieties, using complex/p-adic Hodge theory.

  • CMSA EVENT: CMSA Probability: Liouville quantum gravity from random matrix dynamics

    Speaker: Hugo Falconet – Courant Institute, NYU

    3:30 PM-4:30 PM
    November 9, 2022
    20 Garden Street, Cambridge, MA 02138

    The Liouville quantum gravity measure is a properly renormalized exponential of the 2d GFF. In this talk, I will explain how it appears as a limit of natural random matrix dynamics: if (U_t) is a Brownian motion on the unitary group at equilibrium, then the measures $|det(U_t – e^{i theta}|^gamma dt dtheta$ converge to the 2d LQG measure with parameter $gamma$, in the limit of large dimension. This extends results from Webb, Nikula and Saksman for fixed time. The proof relies on a new method for Fisher-Hartwig asymptotics of Toeplitz determinants with real symbols, which extends to multi-time settings. I will explain this method and how to obtain multi-time loop equations by stochastic analysis on Lie groups.

     

    Based on a joint work with Paul Bourgade.


     

  • CMSA EVENT: CMSA Probability: Liouville quantum gravity from random matrix dynamics

    Speaker: Hugo Falconet – Courant Institute, NYU

    3:30 PM-4:30 PM
    November 9, 2022
    20 Garden Street, Cambridge, MA 02138

    The Liouville quantum gravity measure is a properly renormalized exponential of the 2d GFF. In this talk, I will explain how it appears as a limit of natural random matrix dynamics: if (U_t) is a Brownian motion on the unitary group at equilibrium, then the measures $|det(U_t – e^{i theta}|^gamma dt dtheta$ converge to the 2d LQG measure with parameter $gamma$, in the limit of large dimension. This extends results from Webb, Nikula and Saksman for fixed time. The proof relies on a new method for Fisher-Hartwig asymptotics of Toeplitz determinants with real symbols, which extends to multi-time settings. I will explain this method and how to obtain multi-time loop equations by stochastic analysis on Lie groups.

     

    Based on a joint work with Paul Bourgade.


     

  • SEMINARS: Informal Seminar: From mapping classes to dynamics on character varieties

    Speaker: Max Weinreich – Harvard

    4:00 PM-5:00 PM
    November 9, 2022

    This seminar will be held in Science Center 530 at 4:00pm on Wednesday, November 9th.

    Please see the seminar page for more details: https://www.math.harvard.edu/~ctm/sem

     

  • HARVARD-MIT COMBINATORICS SEMINAR: Harvard-MIT Combinatorics: Bijections for the regions of hyperplane arrangements of Coxeter type

    Speaker: Olivier Bernardi – Brandeis

    4:15 PM-5:15 PM
    November 9, 2022

    A hyperplane arrangement of braid type is a collection of hyperplanes in R^n of the form {x_i-x_j=s}, where i,j are indices in [n] and s is an integer. Classical families include the Catalan, Shi, Semi-order and Linial arrangements. In this talk we will discuss some bijections between the regions of braid type arrangements and some labeled plane trees. This bijective framework applies to the braid type arrangements which satisfy a particular property that we call “transitivity” (the above classical families are all transitive). Time permitting we will then discuss some recent progress in extending this bijective framework in two directions: (a) extension of the bijections to lower dimensional faces, and (b) extension to arrangements of other Coxeter types (which include hyperplanes of the form {x_i+x_j=s}). Part of this work is joint with Te Cao.


     

  • OPEN NEIGHBORHOOD SEMINAR: Open Neighborhood Seminar

    OPEN NEIGHBORHOOD SEMINAR
    Open Neighborhood Seminar

    Speaker: Álvaro Lozano-Robledo – UConn

    4:30 PM-5:30 PM
    November 9, 2022
    1 Oxford Street, Cambridge, MA 02138 USA

    Title: TBA

    Abstract: TBA

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  • CMSA EVENT: CMSA General Relativity: Schwarzschild-like Topological Solitons in Gravity

    Speaker: Pierre Heidmann – Johns Hopkins

    9:30 AM-10:30 AM
    November 10, 2022

    We present large classes of non-extremal solitons in gravity that are asymptotic to four-dimensional Minkowski spacetime plus extra compact dimensions. They correspond to smooth horizonless geometries induced by topology in spacetime and supported by electromagnetic flux, which characterize coherent states of quantum gravity. We discuss a new approach to deal with Einstein-Maxwell equations in more than four dimensions, such that they decompose into a set of Ernst equations. We generate the solitons by applying different techniques associated with the Ernst formalism. We focus on solitons with zero net charge yet supported by flux, and compare them to Schwarzschild black holes. These are also ultra-compact geometries with very high redshift but differ in many aspects. At the end of the talk, we discuss the stability properties of the solitons and their gravitational signatures.


    This seminar will be held over Zoom. For more information on how to join, please see: https://cmsa.fas.harvard.edu/event/general-relativity-2021-22/

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  • SEMINARS: Gauge Theory and Topology: Surface singularities, unexpected fillings, and line arrangements

    Speaker: Olga Plamenevskaya – Stony Brook University

    3:30 PM-4:30 PM
    November 11, 2022
    1 Oxford Street, Cambridge, MA 02138 USA

    A link of an isolated complex surface singularity (X, 0) is a 3-manifold Y which is the boundary of the intersection of X with a small ball centered at 0. Smoothings of the singularity give non-singular 4-manifolds, the Milnor fibers, with the same boundary Y. The Milnor fibers carry symplectic (even Stein) structures, and thus provide fillings of the canonical contact structure on Y; another Stein filling comes from the minimal resolution of (X, 0). An important question is whether all Stein fillings of the link come from this algebraic construction: this is true in some simple cases such as lens spaces. However, even in the “next simplest” case, for many rational singularities, we are able to construct “unexpected” Stein fillings that do not arise from Milnor fibers. To this end, we encode Stein fillings via curve arrangements, motivated by  T.de Jong-D.van Straten’s description of smoothings  of certain rational surface singularities in terms of deformations of associated singular plane curves. We then use classical projective geometry  to construct unexpected line arrangements and unexpected fillings. This is a topological story, with minimal input from algebraic geometry.  Joint work with L. Starkston.


     

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  • CMSA EVENT: CMSA Topological Quantum Matter: Vacuum fluctuations in cavities: breakdown of the topological protection in the integer Quantum Hall effect

    Speaker: Jerome Faist – Institute of Quantum Electronics, ETH Zurich

    10:00 AM-11:30 AM
    November 16, 2022
    20 Garden Street, Cambridge, MA 02138

    When a collection of electronic excitations are strongly coupled to a single mode cavity, mixed light-matter excitations called polaritons are created. The situation is especially interesting when the strength of the light-matter coupling ΩR is such that the coupling energy becomes close to the one of the bare matter resonance ω0. For this value of parameters, the system enters the so-called ultra-strong coupling regime, in which a number of very interesting physical effects were predicted caused by the counter-rotating and diamagnetic terms of the Hamiltonian.

    In a microcavity, the strength of the electric field caused by the vacuum fluctuations, to which the strength of the light-matter coupling ΩRis proportional, scales inversely with the cavity volume. One very interesting feature of the circuit-based metamaterials is the fact that this volume can be scaled down to deep subwavelength values in all three dimension of space.1 Using metamaterial coupled to two-dimensional electron gases under a strong applied magnetic field, we have now explored to which extend this volume can be scaled down and reached a regime where the stability of the polariton is limited by diffraction into a continuum of plasmon modes2.

    We have also used transport to probe the ultra-strong light-matter coupling3, and show now that the latter can induce a breakdown of the integer quantum Hall effect4. The phenomenon is explained in terms of cavity-assisted hopping, an anti-resonant process where an electron can scatter from one edge of the sample to the other by “borrowing” a photon from the cavity5. We are also evaluating a proposal suggesting that the value of the quantization voltage can be renormalized by the cavity6.

     

    1. Scalari, G. et al. Ultrastrong Coupling of the Cyclotron Transition of a 2D Electron Gas to a THz Metamaterial. Science 335, 1323–1326 (2012).
    2. Rajabali, S. et al. Polaritonic Nonlocality in Light Matter Interaction. Nat Photon 15, 690–695 (2021).
    3. Paravicini-Bagliani, G. L. et al. Magneto-Transport Controlled by Landau Polariton States. Nat. Phys. 15, 186–190 (2019).
    4. Appugliese, F. et al. Breakdown of topological protection by cavity vacuum fields in the integer quantum Hall effect. Science 375, 1030–1034 (2022).
    5. Ciuti, C. Cavity-mediated electron hopping in disordered quantum Hall systems. Phys. Rev. B 104, 155307 (2021).
    6. Rokaj, V., Penz, M., Sentef, M. A., Ruggenthaler, M. & Rubio, A. Polaritonic Hofstadter butterfly and cavity control of the quantized Hall conductance. Phys. Rev. B 105, 205424 (2022).

     

    For more information on how to join, please see: https://cmsa.fas.harvard.edu/event_category/topological-quantum-matter-seminar/

  • CMSA EVENT: CMSA Colloquium: Noether’s Learning Dynamics: Role of Symmetry Breaking in Neural Networks

    Speaker: Hidenori Tanaka – Harvard University

    12:30 PM-1:30 PM
    November 16, 2022
    20 Garden Street, Cambridge, MA 02138

    In nature, symmetry governs regularities, while symmetry breaking brings texture. In artificial neural networks, symmetry has been a central design principle, but the role of symmetry breaking is not well understood. Here, we develop a Lagrangian formulation to study the geometry of learning dynamics in neural networks and reveal a key mechanism of explicit symmetry breaking behind the efficiency and stability of modern neural networks. Then, we generalize Noether’s theorem known in physics to describe a unique symmetry breaking mechanism in learning and derive the resulting motion of the Noether charge: Noether’s Learning Dynamics (NLD). Finally, we apply NLD to neural networks with normalization layers and discuss practical insights. Overall, through the lens of Lagrangian mechanics, we have established a theoretical foundation to discover geometric design principles for the learning dynamics of neural networks.

  • NUMBER THEORY SEMINAR: Number Theory: Covers of reductive groups and functoriality

    Speaker: Tasho Kaletha – University of Michigan

    3:00 PM-4:00 PM
    November 16, 2022
    1 Oxford Street, Cambridge, MA 02138 USA

    To a connected reductive group G over a local field F we define a compact topological group π_1~(G) and an extension G(F)_ of G(F) by π_1~(G). From any character x of π_1~(G) of order n we obtain an n-fold cover G(F)_x of the topological group G(F). We also define an L-group for G(F)_x, which is a usually non-split extension of the Galois group by the dual group of G, and deduce from the linear case a refined local Langlands correspondence between genuine representations of G(F)_x and L-parameters valued in this L-group.

     

    This construction is motivated by Langlands functoriality. We show that a subgroup of the L-group of G of a certain kind naturally leads to a smaller quasi-split group H and a double cover of H(F). Genuine representations of this double cover are expected to be in functorial relationship with representations of G(F). We will present two concrete applications of this, one that gives a characterization of the local Langlands correspondence for supercuspidal L-parameters when p is sufficiently large, and one to the theory of endoscopy.


     

  • CMSA EVENT: CMSA Probability Seminar: Outlier-Robust Algorithms for Clustering Non-Spherical Mixtures

    Speaker: Ainesh Bakshi – MIT

    3:30 PM-4:30 PM
    November 16, 2022
    20 Garden Street, Cambridge, MA 02138
    In this talk, we describe the first polynomial time algorithm for robustly clustering a mixture of statistically-separated, high-dimensional Gaussians. Prior to our work this question was open even in the special case of 2 components in the mixture. Our main conceptual contribution is distilling analytic properties of distributions, namely hyper-contractivity of degree-two polynomials and anti-concentration of linear projections, which are necessary and sufficient for clustering.
    Based on joint work with Pravesh Kothari.

     

  • SEMINARS: Informal Seminar: Bers, Henon, Painleve and Schrodinger

    Speaker: Max Weinreich – Harvard

    4:00 PM-5:00 PM
    November 16, 2022

    This seminar will be held in Science Center 530 at 4:00pm on Wednesday, November 16th.

    Please see the seminar page for more details: https://www.math.harvard.edu/~ctm/sem

     

  • HARVARD-MIT COMBINATORICS SEMINAR: Harvard-MIT Combinatorics: Configuration spaces on graphs, phylogenetic trees, and moduli spaces of tropical curves

    Speaker: Melody Chan – Brown University

    4:15 PM-5:15 PM
    November 16, 2022

    I will discuss joint work with Christin Bibby, Nir Gadish, and Claudia Yun.  The historical antecedents are in earlier work of Billera-Holmes-Vogtmann around 2000 and others, who study an interesting space of metric trees on n labelled taxa. This is a space that is shellable and whose top homology was calculated, as an S_n-representation, by Robinson-Whitehouse.  These spaces have a reinterpretation as moduli of tropical curves of genus 0.  Other historical antecedents of our work are in the study of configuration spaces of n points on a graph, whose topological invariants are quite interesting. For example, Ko-Park proved that failure of planarity of a graph can be detected by torsion in H_1 of its unordered configuration space.

    The work I will then describe concerns a genus g>0 analogue of the space of phylogenetic trees: the moduli space of tropical curves of genus g. Roughly speaking, this space parametrizes n-marked graphs of first Betti number g.  These spaces are no longer shellable for g>1, and their homology groups, as S_n-representations, are quite mysterious.  I will explain how making precise connections to compactified configuration spaces on graphs made it possible for us to make calculations in Sage when g=2 in a range beyond what was previously accessible.  These in turn produced new calculations and conjectures concerning the rational cohomology of M_{2,n}. No tropical geometry prerequisites are assumed in this talk.


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  • CMSA EVENT: CMSA General Relativity: Ringdown and geometry of trapping for black holes

    Speaker: Semyon Dyatlov – MIT

    9:30 AM-10:30 AM
    November 17, 2022
    20 Garden Street, Cambridge, MA 02138

    Quasi-normal modes are complex exponential frequencies appearing in long time expansions of solutions to linear wave equations on black hole backgrounds. They appear in particular during the ringdown phase of a black hole merger when the dynamics is expected to be driven by linear effects. In this talk I give an overview of various results in pure mathematics which relate asymptotic behavior of quasi-normal modes at high frequency to the geometry of the set of trapped null geodesics, such as the photon sphere in Schwarzschild(-de Sitter). These trapped geodesics have two kinds of behavior: the geodesic flow is hyperbolic in directions normal to the trapped set (a feature stable under perturbations) and it is completely integrable on the trapped set. It turns out that normal hyperbolicity gives information about the rate of decay of quasi-normal modes, while complete integrability gives rise to a quantization condition.


    This seminar will be held in person and on Zoom. For more information on how to join, please see: https://cmsa.fas.harvard.edu/event/general-relativity-2021-22/

  • CMSA EVENT: CMSA Active Matter Seminar: Dynamic and multicolor electron microscopy

    Speaker: Max Prigozhin – Harvard University

    1:00 PM-2:00 PM
    November 17, 2022
    20 Garden Street, Cambridge, MA 02138

    My lab is developing biophysical methods to achieve multicolor and dynamic biological imaging at the molecular scale. Our approach to capturing the dynamics of cellular processes involves cryo-vitrifying samples after known time delays following stimulation using custom cryo- plunging and high-pressure freezing instruments. To achieve multicolor electron imaging, we are exploring the property of cathodoluminescence — optical emission induced by the electron beam. We are developing nanoprobes (“cathodophores”) that will be used as luminescent protein tags in electron microscopy. We are applying these new methods to study G-protein- coupled receptor signaling and to visualize the formation of biomolecular condensates.


    This seminar will be held in person and on Zoom. For more information on how to join, please see: https://cmsa.fas.harvard.edu/event/active-matter-seminar

     

  • SEMINARS: Algebraic Dynamics: Stable Algebraic Families of Rational Maps

    Speaker: Rafael Saavedra – Harvard University

    4:00 PM-6:00 PM
    November 17, 2022

    McMullen proved that a stable family of rational maps is either trivial or all its members are Lattès. His proof relies on Thurston’s theorem on postcritically finite maps, which uses Teichmüller theory. I will discuss a recent new proof of McMullen’s theorem due to Zhuchao Ji and Junyi Xie which instead uses Berkovich dynamics over the complex Levi-Civita field.


    For more information, please see:  Algebraic Dynamics Seminar at Harvard

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  • CMSA EVENT: CMSA Quantum Matter in Mathematics and Physics: 3d gravity and gravitational entanglement entropy

    Speaker: Gabriel Wong – Harvard CMSA

    9:30 AM-11:00 AM
    November 22, 2022

    Recent progress in AdS/CFT has provided a good understanding of how the bulk spacetime is encoded in the entanglement structure of the boundary CFT. However, little is known about how spacetime emerges directly from the bulk quantum theory. We address this question in an effective 3d quantum theory of pure gravity, which describes the high temperature regime of a holographic CFT.  This theory can be viewed as a $q$-deformation and dimensional uplift of JT gravity.  Using this model, we show that the Bekenstein-Hawking entropy of a two-sided black hole equals the bulk entanglement entropy of gravitational edge modes.  These edge modes transform under a quantum group, which defines the data associated to an extended topological quantum field theory  Our calculation suggests an effective description of bulk microstates in terms of collective, anyonic degrees of freedom whose entanglement leads to the emergence of the bulk spacetime.  Finally, we give a proposal for obtaining the Ryu Takayanagi formula using the same quantum group edge modes.

    For information on how to join, please see: https://cmsa.fas.harvard.edu/event_category/quantum-matter-seminar/

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  • CMSA EVENT: CMSA Topological Quantum Matter: Continuum field theory of graphene bilayer system

    Speaker: Jian Kang – School of Physical Science and Technology, ShanghaiTech University, Shanghai, China

    9:00 AM-10:00 AM
    November 23, 2022
    20 Garden Street, Cambridge, MA 02138

    The Bistritzer-MacDonald (BM) model predicted the existence of the narrow bands in the magic-angle twisted bilayer graphene (MATBG), and nowadays is a starting point for most theoretical works. In this talk, I will briefly review the BM model and then present a continuum field theory [1] for graphene bilayer system allowing any smooth lattice deformation including the small twist angle. With the gradient expansion to the second order, the continuum theory for MATBG [2] produces the spectrum that almost perfectly matches the spectrum of the microscopic model, suggesting the validity of this theory. In the presence of the lattice deformation, the inclusion of the pseudo-vector potential does not destroy but shift the flat band chiral limit to a smaller twist angle. Furthermore, the continuum theory contains another important interlayer tunneling term that was overlooked in all previous works. This term non-negligibly breaks the particle-hole symmetry of the narrow bands and may be related with the experimentally observed particle-hole asymmetry.

    1. O. Vafek and JK, arXiv: 2208.05933.
    2. JK and O. Vafek, arXiv: 2208.05953.

     

    For more information on how to join, please see: https://cmsa.fas.harvard.edu/event_category/topological-quantum-matter-seminar/

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