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  • DIFFERENTIAL GEOMETRY SEMINAR

    DIFFERENTIAL GEOMETRY SEMINAR
    Quasimodular forms from Betti numbers

    8:00 AM-9:00 AM
    April 6, 2021

    This talk will be about refined curve counting on local P^2, the noncompact Calabi-Yau 3-fold total space of the canonical line bundle of the projective plane. I will explain how to construct quasimodular forms starting from Betti numbers of moduli spaces of dimension 1 coherent sheaves on P^2. This gives a proof of some stringy predictions about the refined topological string theory of local P^2 in the Nekrasov-Shatashvili limit. Partly based on work with Honglu Fan, Shuai Guo, and Longting Wu.

    Zoom: https://harvard.zoom.us/j/96709211410?pwd=SHJyUUc4NzU5Y1d0N2FKVzIwcmEzdz09

  • CMSA EVENT: CMSA Math Science Literature Lecture Series
    9:00 AM-10:30 AM
    April 6, 2021

    TITLE: Isadore Singer’s Work on Analytic Torsion

    ABSTRACT: I will review two famous papers of Ray and Singer on analytic torsion written approximately half a century ago. Then I will sketch the influence of analytic torsion in a variety of areas of physics including anomalies, topological field theory, and string theory.

    Talk chair: Cumrun Vafa

    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.
  • MATHEMATICAL PICTURE LANGUAGE SEMINAR
    10:00 AM-11:00 AM
    April 6, 2021

    The space spanned by homotopy classes of free oriented loops on a 2-manifold carries an interesting algebraic structure (a Lie bialgebra structure) due to Goldman and Turaev. This structure is defined in terms of intersections and self-intersections of planar curves. In the talk, we will explain a surprising link between the Gaoldman-Turaev theory and the Kashiwara-Vergne problem on properties of the Baker-Campbell-Hausdorff series. Important tools in establishing this link are the non-commutative divergence cocycle and a novel characterization of conjugacy classes in free Lie algebras in terms of cyclic words. The talk is based on joint works with N. Kawazumi, Y. Kuno and F. Naef.

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

  • MATHEMATICAL PICTURE LANGUAGE SEMINAR
    10:00 AM-11:00 AM
    April 6, 2021

    The space spanned by homotopy classes of free oriented loops on a 2-manifold carries an interesting algebraic structure (a Lie bialgebra structure) due to Goldman and Turaev. This structure is defined in terms of intersections and self-intersections of planar curves. In the talk, we will explain a surprising link between the Gaoldman-Turaev theory and the Kashiwara-Vergne problem on properties of the Baker-Campbell-Hausdorff series. Important tools in establishing this link are the non-commutative divergence cocycle and a novel characterization of conjugacy classes in free Lie algebras in terms of cyclic words. The talk is based on joint works with N. Kawazumi, Y. Kuno and F. Naef.

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

  • CMSA EVENT: CMSA Computer Science for Mathematicians: Confidence-Budget Matching for Sequential Budgeted Learning
    11:30 AM-12:30 PM
    April 6, 2021

    A core element in decision-making under uncertainty is the feedback on the quality of the performed actions. However, in many applications, such feedback is restricted. For example, in recommendation systems, repeatedly asking the user to provide feedback on the quality of recommendations will annoy them. In this work, we formalize decision-making problems with querying budget, where there is a (possibly time-dependent) hard limit on the number of reward queries allowed. Specifically, we consider multi-armed bandits, linear bandits, and reinforcement learning problems. We start by analyzing the performance of `greedy’ algorithms that query a reward whenever they can. We show that in fully stochastic settings, doing so performs surprisingly well, but in the presence of any adversity, this might lead to linear regret. To overcome this issue, we propose the Confidence-Budget Matching (CBM) principle that queries rewards when the confidence intervals are wider than the inverse square root of the available budget. We analyze the performance of CBM based algorithms in different settings and show that they perform well in the presence of adversity in the contexts, initial states, and budgets.

    Joint work with Yonathan Efroni, Aadirupa Saha and Shie Mannor.

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

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  • CMSA EVENT: CMSA Quantum Matter in Mathematics and Physics: Higher Form Symmetries in string/M-theory
    10:30 AM-12:00 PM
    April 7, 2021

    In this talk, I will give an overview of recent developments in geometric constructions of field theory in string/M-theory and identifying higher form symmetries. The main focus will be on d>= 4 constructed from string/M-theory. I will also discuss realization in terms of holographic models in string theory. In the talk next week Lakshya Bhardwaj will speak about 1-form symmetries in class S, N=1 deformations thereof and the relation to confinement.

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

  • RANDOM MATRIX SEMINAR
    2:00 PM-3:00 PM
    April 7, 2021

    In many problems in statistical learning, random matrix theory, and statistical physics, one needs to simulate dynamics on random matrix ensembles. A classical example is to use iterative methods to compute the extremal eigenvalues/eigenvectors of a (spiked) random matrix. Other examples include approximate message passing on dense random graphs, and gradient descent algorithms for solving learning and estimation problems with random initialization. We will show that all such dynamics can be simulated by an efficient matrix-free scheme, if the random matrix is drawn from an ensemble with translation-invariant properties. Examples of such ensembles include the i.i.d. Gaussian (i.e. the rectangular Ginibre) ensemble, the Haar-distributed random orthogonal ensemble, the Gaussian orthogonal ensemble, and their complex-valued counterparts.

    A “direct” approach to the simulation, where one first generates a dense n × n matrix from the ensemble, requires at least O(n^2) resource in space and time. The new algorithm, named Householder Dice (HD), overcomes this O(n^2) bottleneck by using the principle of deferred decisions: rather than fixing the entire random matrix in advance, it lets the randomness unfold with the dynamics. At the heart of this matrix-free algorithm is an adaptive and recursive construction of (random) Householder reflectors. These orthogonal transformations exploit the group symmetry of the matrix ensembles, while simultaneously maintaining the statistical correlations induced by the dynamics. The memory and computation costs of the HD algorithm are O(nT) and O(n T^2), respectively, with T being the number of iterations. When T ≪ n, which is nearly always the case in practice, the new algorithm leads to significant reductions in runtime and memory footprint.

    Finally, the HD algorithm is not just a computational trick. I will show how its construction can serve as a simple proof technique for several problems in high-dimensional estimation.

    Zoom: https://harvard.zoom.us/j/99333938108

  • NUMBER THEORY SEMINAR

    NUMBER THEORY SEMINAR
    The motivic Satake equivalence

    3:00 PM-4:00 PM
    April 7, 2021

    The geometric Satake equivalence due to Lusztig, Drinfeld, Ginzburg, Mirković and Vilonen is an indispensable tool in the Langlands program. Versions of this equivalence are known for different cohomology theories such as Betti cohomology or algebraic D-modules over characteristic zero fields and $\ell$-adic cohomology over arbitrary fields. In this talk, I explain how to apply the theory of motivic complexes as developed by Voevodsky, Ayoub, Cisinski-Déglise and many others to the construction of a motivic Satake equivalence. Under suitable cycle class maps, it recovers the classical equivalence. As dual group, one obtains a certain extension of the Langlands dual group by a one dimensional torus. A key step in the proof is the construction of intersection motives on affine Grassmannians. A direct consequence of their existence is an unconditional construction of IC-Chow groups of moduli stacks of shtukas. My hope is to obtain on the long run independence-of-$\ell$ results in the work of V. Lafforgue on the Langlands correspondence for function fields. This is ongoing joint work with Jakob Scholbach from Münster.

    Zoom: https://harvard.zoom.us/j/99334398740

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

  • CMSA EVENT: CMSA New Technologies in Mathematics: Type Theory from the Perspective of Artificial Intelligence
    3:00 PM-4:00 PM
    April 7, 2021

    This talk will discuss dependent type theory from the perspective of artificial intelligence and cognitive science.  From an artificial intelligence perspective it will be argued that type theory is central to defining the “game” of mathematics — an action space and reward structure for pure mathematics. From a cognitive science perspective type theory provides a model of the grammar of the colloquial (natural) language of mathematics.  Of particular interest is the notion of a signature-axiom structure class and the three fundamental notions of equality in mathematics — set-theoretic equality between structure elements, isomorphism between structures, and Birkoff and Rota’s notion of cryptomorphism between structure classes.  This talk will present a version of type theory based on set-theoretic semantics and the 1930’s notion of structure and isomorphism given by the Bourbaki group of mathematicians.  It will be argued that this “Bourbaki type theory” (BTT) is more natural and accessible to classically trained mathematicians than Martin-Löf type theory (MLTT). BTT avoids the Curry-Howard isomorphism and axiom J of MLTT.  The talk will also discuss BTT as a model of MLTT.  The BTT model is similar to the groupoid model in that propositional equality is interpreted as isomorphism but different in various details.  The talk will also briefly mention initial thoughts in defining an action space and reward structure for a game of mathematics.

    Zoom: https://harvard.zoom.us/j/99018808011?pwd=SjRlbWFwVms5YVcwWURVN3R3S2tCUT09

  • OPEN NEIGHBORHOOD SEMINAR

    OPEN NEIGHBORHOOD SEMINAR
    Outward-facing mathematics

    4:30 PM-5:30 PM
    April 7, 2021

    I will talk, pretty casually, about random walks, which I first learned about as part of my Harvard undergrad thesis in finite group theory, and which turn out to be at the heart of the mathematical analysis of gerrymandering; along the way I will talk about the project of doing mathematics in a way that engages with the world outside the math department walls.

    Please go to the College Calendar to register.

    Zoom: https://calendar.college.harvard.edu/event/open_neighborhood_seminar

    Website: https://math.harvard.edu/ons

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  • ALGEBRAIC DYNAMICS SEMINAR
    10:00 AM-12:00 PM
    April 9, 2021

    Canonical heights are a standard tool of arithmetic dynamics over global fields. When studying families of dynamical systems, or systems over larger fields, however, there are significant geometric obstacles to constructing canonical heights. Either Northcott fails to hold, or the construction requires a model for the family with such strict properties (good reduction, minimality, extensions of rational maps,…) that such a model is unlikely to exist outside of very special settings. Instead, I’ll show how to resolve this problem using vector-valued heights, first introduced in characteristic zero by Yuan and Zhang. These generalize R-valued heights, produce canonical heights for any polarized dynamical system, and exhibit Northcott finiteness conditional on a strong non-isotriviality condition, generalizing work of Lang-Neron, Baker, and Chatzidakis-Hrushovski. This is achieved by working simultaneously over a system of models with much weaker requirements. As part of ongoing work, I’ll show how these arithmetic methods can produce results that hold over any field, and discuss how this can extend to quasi-projective varieties as well as projective varieties.

    Go to http://people.math.harvard.edu/~demarco/AlgebraicDynamics/ for Zoom information.

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  • CMSA EVENT: CMSA Mathematical Physics Seminar: Networks and quantization
    10:00 AM-11:00 AM
    April 12, 2021

    I will describe two quantization scenarios. The first scenario involves the construction of a quantum trace map computing a link “invariant” (with possible wall-crossing behavior) for links L in a 3-manifold M, where M is a Riemann surface C times a real line. This construction unifies the computation of familiar link invariant with the refined counting of framed BPS states for line defects in 4d N=2 theories of class S. Certain networks on C play an important role in the construction. The second scenario concerns the study of Schroedinger equations and their higher order analogues, which could arise in the quantization of Seiberg-Witten curves in 4d N=2 theories. Here similarly certain networks play an important part in the exact WKB analysis for these Schroedinger-like equations. At the end of my talk I will also try to sketch a possibility to bridge these two scenarios.

    Zoom: https://harvard.zoom.us/j/91780604388?pwd=d3BqazFwbDZLQng0cEREclFqWkN4UT09

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  • CMSA EVENT: CMSA Math Science Literature Lecture Series
    9:00 AM-10:30 AM
    April 13, 2021

    TITLE: K-theory and characteristic classes in topology and complex geometry (a tribute to Atiyah and Hirzebruch)

    ABSTRACT: We will discuss the K-theory of complex vector bundles on
    topological spaces and of holomorphic vector bundles on complex
    manifolds. A central question is the relationship between K-theory
    and cohomology. This is done in topology by constructing
    characteristic classes, but other constructions appear in the

    holomorphic or algebraic context. We will discuss the Hirzebruch-
    Riemann-Roch formula, the Atiyah-Hirzebruch spectral sequence, the

    role of complex cobordism, and other tools developed later on, like
    the Bloch-Ogus spectral sequence.

    Talk chair: Baohua Fu

    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.
  • MATHEMATICAL PICTURE LANGUAGE SEMINAR

    MATHEMATICAL PICTURE LANGUAGE SEMINAR
    Aspects of M Theory

    10:00 AM-11:00 AM
    April 13, 2021

    After giving a brief introduction to Membrane Theory and its matrix regularization, commenting on an inherent dynamical symmetry for all M-branes (the related “reconstruction-algebra” for M=1, strings, being the Virasoro algebra), I will explain some very recent work, including the observation that super-symmetrizable systems canonically (i.e. more or less automatically) have a Lax-pair formulation, with calculable r-matrix, – the appearance of infinite-dimensional CKL-algebras naturally entering the double bracket equations of Quantum Minimal Surfaces (IKKT model) and the (“BFFS”) membrane matrix model. 

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

  • HARVARD-MIT ALGEBRAIC GEOMETRY SEMINAR
    3:00 PM-4:00 PM
    April 13, 2021

    I will work on a  Calabi-Yau 3-fold X which satisfies the Bogomolov-Gieseker conjecture of Bayer-Macr\`i-Toda for weak stability conditions, such as the quintic threefold. I will explain how wall-crossing with respect to weak stability conditions gives an expression of Joyce’s generalised Donaldson-Thomas invariants counting Gieseker semistable sheaves of any rank greater than or equal to one on X in terms of those counting sheaves of rank 0 and pure dimension 2.  This is joint work with Richard Thomas.

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

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  • CMSA EVENT: CMSA Quantum Matter in Mathematics and Physics: Confinement and 1-form Symmetries in 4d from 6d (2,0)
    10:30 AM-12:00 PM
    April 14, 2021

    In this talk, I will discuss several issues related to thermoelectric transport, with application to topological invariants of chiral topological phases in two dimensions. In the first part of the talk, I will argue in several different ways that the only topological invariants associated with anomalous edge transport are the Hall conductance and the thermal Hall conductance. Thermoelectric coefficients are shown to vanish at zero temperature and do not give rise to topological invariants. In the second part of the talk, I will describe microscopic formulas for transport coefficients (Kubo formulas) which are valid for arbitrary interacting lattice systems. I will show that in general “textbook” Kubo formulas require corrections. This is true even for some dissipative transport coefficients, such as Seebeck and Peltier coefficients. I will also make a few remarks about “matching” (in the sense of Effective Field Theory) between microscopic descriptions of transport and hydrodynamics.

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

  • CMSA EVENT: CMSA New Technologies in Mathematics: A Bayesian neural network predicts the dissolution of compact planetary systems
    3:00 PM-4:00 PM
    April 14, 2021

    Despite over three hundred years of effort, no solutions exist for predicting when a general planetary configuration will become unstable. I will discuss our deep learning architecture (arxiv:2101.04117) which pushes forward this problem for compact systems. While current machine learning algorithms in this area rely on scientist-derived instability metrics, our new technique learns its own metrics from scratch, enabled by a novel internal structure inspired from dynamics theory. The Bayesian neural network model can accurately predict not only if, but also when a compact planetary system with three or more planets will go unstable. Our model, trained directly from short N-body time series of raw orbital elements, is more than two orders of magnitude more accurate at predicting instability times than analytical estimators, while also reducing the bias of existing machine learning algorithms by nearly a factor of three. Despite being trained on three-planet configurations, the model demonstrates robust generalization to five-planet systems, even outperforming models designed for that specific set of integrations. I will also discuss some work on recovering symbolic representations of such models using arxiv:2006.11287.

    Zoom: https://harvard.zoom.us/j/99018808011?pwd=SjRlbWFwVms5YVcwWURVN3R3S2tCUT09

  • NUMBER THEORY SEMINAR
    3:00 PM-4:00 PM
    April 14, 2021

    We will describe the notion of affine schemes and their modifications in the context of Arakelov geometry. Using geometry of numbers in infinite rank, we will study their cohomological properties. Concretely, given an affine scheme X over Z and a compact subset K of the set of complex points of X, we will investigate the geometry of those proper arithmetic curves in X whose complex points lie in K. This is joint work with Jean-Benoît Bost.

    Zoom: https://harvard.zoom.us/j/99334398740

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

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  • CMSA EVENT: CMSA Math Science Literature Lecture Series
    1:00 PM-2:30 PM
    April 16, 2021

    TITLE: Deep Networks from First Principles

    ABSTRACT: In this talk, we offer an entirely “white box’’ interpretation of deep (convolution) networks from the perspective of data compression (and group invariance). In particular, we show how modern deep layered architectures, linear (convolution) operators and nonlinear activations, and even all parameters can be derived from the principle of maximizing rate reduction (with group invariance). All layers, operators, and parameters of the network are explicitly constructed via forward propagation, instead of learned via back propagation. All components of so-obtained network, called ReduNet, have precise optimization, geometric, and statistical interpretation. There are also several nice surprises from this principled approach: it reveals a fundamental tradeoff between invariance and sparsity for class separability; it reveals a fundamental connection between deep networks and Fourier transform for group invariance – the computational advantage in the spectral domain (why spiking neurons?); this approach also clarifies the mathematical role of forward propagation (optimization) and backward propagation (variation). In particular, the so-obtained ReduNet is amenable to fine-tuning via both forward and backward (stochastic) propagation, both for optimizing the same objective. This is joint work with students Yaodong Yu, Ryan Chan, Haozhi Qi of Berkeley, Dr. Chong You now at Google Research, and Professor John Wright of Columbia University.

    Talk chair: Harry Shum

    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 Math Science Literature Lecture Series
    9:00 AM-10:30 AM
    April 20, 2021

    TITLE: The Atiyah-Singer Index Theorem

    ABSTRACT: The story of the index theorem ties together the Gang of Four— Atiyah, Bott, Hirzebruch, and Singer—and lies at the intersection of analysis, geometry, and topology. In the first part of the talk I will recount high points in the early developments. Then I turn to subsequent variations and applications. Throughout I emphasize the role of the Dirac operator.

    Talk chair: Cumrun Vafa

    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.
  • MATHEMATICAL PICTURE LANGUAGE SEMINAR

    MATHEMATICAL PICTURE LANGUAGE SEMINAR
    Emerging frontiers in nuclear magnetic resonance

    10:00 AM-11:00 AM
    April 20, 2021

    Nuclear Magnetic Resonance (NMR) is a powerful spectroscopic technique that provides information about matter at an atomic resolution. One of the applications of NMR is to decipher the molecular architecture of biomolecules including nucleic acids and proteins. In addition to providing information on the structure of biomolecules, NMR also provides information on the dynamics of these molecule machines. The seminar will introduce some basics of NMR, discuss some of the current limitations, and present new methods to push the frontiers of NMR.

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

  • CMSA EVENT: CMSA Computer Science for Mathematicians: EigenGame: SVD as a Nash Equilibrium
    11:30 AM-12:30 PM
    April 20, 2021

    We present a novel view on singular value decomposition (SVD) as a competitive game in which each approximate singular vector is controlled by a player whose goal is to maximize their own utility function. We analyze the properties of this EigenGame and the behavior of its gradient based updates. The resulting algorithm — which combines elements from Oja’s rule with a generalized Gram-Schmidt orthogonalization — is naturally decentralized and hence parallelizable through message passing. EigenGame’s updates are biased if computed using minibatches of data, which hinders convergence and more sophisticated parallelism in the stochastic setting. Therefore, in follow-up work, we propose an unbiased stochastic update that is asymptotically equivalent to EigenGame, enjoys greater parallelism allowing computation on datasets of larger sample sizes, and outperforms the original EigenGame in experiments. We demonstrate the a) scalability of the algorithm by conducting principal component analyses of large image datasets, language datasets, and neural network activations and b) generality by reusing the same algorithm to perform spectral clustering of a social network. We discuss how this new view of SVD as a differentiable game can lead to further algorithmic developments and insights.

    This talk is based on two recent works, both joint work with Brian McWilliams, Claire Vernade, and Thore Graepel —

    https://arxiv.org/abs/2010.00554 (EigenGame – ICLR ‘21)

    https://arxiv.org/abs/2102.04152 (EigenGame Unloaded – ICML ‘21 submission)

    — and will focus in detail on some of the more interesting mathematical properties of the game.

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

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