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Diving Into Math with Emmy Noether
September 10, 2022      4:30 pm
Diving Into Math with Emmy Noether A theatre performance about the life of one of history’s most influential mathematicians. When: Saturday, September 10, 2022 Panel...
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February
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  • SEMINARS
    9:00 AM-10:00 AM
    March 1, 2022

    Eugene Wachspress introduced polypols as real bounded semialgebraic sets in the plane that generalize polygons. He aimed to generalize barycentric coordinates from triangles to arbitrary polygons and further to polypols. For this, he defined the adjoint curve of a rational polypol. In the study of scattering amplitudes in physics, positive geometries are real semialgebraic sets together with a rational canonical form. We combine these two worlds by providing an explicit formula for the canonical form of a rational polypol in terms of defining equations of the adjoint curve and the facets of the polypol. For the special case of polygons, we show that the adjoint curve is hyperbolic and provide an explicit description of its nested ovals. Finally, we discuss the map that associates the adjoint curve to a given rational polypol, in particular the cases where this map is finite. For instance, using monodromy we find that a general quartic curve is the adjoint of 864 heptagons.
    This talk is based on joint work with R. Piene, K. Ranestad, F. Rydell, B. Shapiro, R. Sinn,  M.-S. Sorea, and S. Telen.

    https://harvard.zoom.us/j/91799784675?pwd=MS9LV25DWk9RcmJoRVM0K3RGWkFRdz09

  • 9:30 AM-10:30 AM
    March 1, 2022

    This is a report about work in progress with: Adeel Khan, Aloysha Latyntsev, Hyeonjun Park and Charanya Ravi. We will describe a virtual Atiyah-Bott formula for Artin stacks.  In the Deligne-Mumford case our methods allow us to remove the global resolution hypothesis for the virtual normal bundle.


    https://harvard.zoom.us/j/97335783449?pwd=S3U0eVdyODFEdzNaRXVEUTF3R3NwZz09

  • SEMINARS
    9:30 AM-10:30 AM
    March 1, 2022

    We will present some rigorous results about entropy and  relative entropy in QFT, motivated in part by recent physicists’ work which however depends on heuristic arguments such as introducing cut off and using path integrals.


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

  • HARVARD-MIT ALGEBRAIC GEOMETRY SEMINAR

    HARVARD-MIT ALGEBRAIC GEOMETRY SEMINAR
    The Rank of Syzygies

    3:00 PM-4:00 PM
    March 1, 2022

    I will explain a notion of rank for the relations amongst the equations of a projective variety, generalizing the classical notion of rank of a quadric. I will then explain a result telling us that, for a general canonical curve, all syzygies are linear combinations of syzygies of minimal rank. This is a sweeping generalization of old results of Andreotti-Mayer, Harris-Arbarello and Green, which tell us that canonical curves are defined by quadrics of rank four. As a special case, this perspective gives us a new, and simpler, proof of Green’s conjecture for general curves.


    https://harvard.zoom.us/j/91082896381?pwd=TmJCanRPVlM4VGR2L1RzZGVGbHRVQT09

  • SEMINARS
    8:30 PM-9:30 PM
    March 1, 2022

    The SYZ proposal suggests that mirror symmetry is T-duality. It is a folklore that locally free sheaves are mirror to a Lagrangian multi-section of the SYZ fibration. In this talk, I will introduce the notion of tropical Lagrangian multi-sections and discuss how to obtain from such object to a class of locally free sheaves on the log Calabi-Yau spaces that Gross-Siebert have considered. I will also discuss a joint work with Kwokwai Chan and Ziming Ma, where we proved the smoothability of a class of locally free sheaves on some log Calabi-Yau surfaces by using combinatorial data obtained from tropical Lagrangian multi-sections.


    Zoom Link: https://cuhk.zoom.us/j/95804207557

    Meeting ID: 958 0420 7557; Passcode: 20220302

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  • SEMINARS
    9:30 AM-10:30 AM
    March 2, 2022

    We consider SL_n-local systems on graphs on surfaces and show how the associated Kasteleyn matrix can be used to compute probabilities of various topological events involving the overlay of n independent dimer covers (or “n-webs”).
    This is joint work with Dan Douglas and  Haolin Shi.
  • SEMINARS
    10:30 AM-12:00 PM
    March 2, 2022

    We construct infinitely many new exactly solvable local commuting projector lattice Hamiltonian models for general bosonic beyond group cohomology invertible topological phases of order two and four in any spacetime dimensions, whose boundaries are characterized by gravitational anomalies. Examples include the beyond group cohomology invertible phase “w2w3” in (4+1)D that has an anomalous boundary topological order with fermionic particle and fermionic loop excitations that have mutual statistics. Finally, we will demonstrate a few examples of fermionic loop excitations.


    https://harvard.zoom.us/j/977347126
    Password: cmsa

  • NUMBER THEORY SEMINAR
    2:00 PM-3:00 PM
    March 2, 2022

    Scaling laws and associated downstream trends can be used as an organizing principle when thinking about current and future ML progress.  I will briefly review scaling laws for generative models in a number of domains, emphasizing language modeling.  Then I will discuss scaling results for transfer from natural language to code, and results on python programming performance from “codex” and other models.  If there’s time I’ll discuss prospects for the future — limitations from dataset sizes, and prospects for RL and other techniques.


    https://harvard.zoom.us/j/99651364593?pwd=Q1R0RTMrZ2N ZQjg1U1ZOaUYzSE02QT09

  • NUMBER THEORY SEMINAR
    3:00 PM-4:00 PM
    March 2, 2022
    1 Oxford Street, Cambridge, MA 02138 USA

    Inspired by an analogy between torsion and preperiodic points, Zhang has proposed a dynamical generalization of the classical Manin-Mumford and Bogomolov conjectures. A special case of these conjectures, for `split’ maps, has recently been established by Nguyen, Ghioca and Ye. In particular, they show that two rational maps have at most finitely many common preperiodic points, unless they are `related’. Recent breakthroughs by Dimitrov, Gao, Habegger and Kühne have established that the classical Bogomolov conjecture holds uniformly across curves of given genus.

    In this talk we discuss uniform versions of the dynamical Bogomolov conjecture across 1-parameter families of split maps and curves. To this end, we establish instances of a ‘relative dynamical Bogomolov conjecture’. This is joint work with Harry Schmidt (University of Basel).


     

  • HARVARD-MIT COMBINATORICS SEMINAR

    HARVARD-MIT COMBINATORICS SEMINAR
    Newell-Littlewood numbers

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

    The Newell-Littlewood numbers are defined in terms of the  Littlewood-Richardson coefficients. Both arise as tensor product  multiplicities for a classical Lie group. A. Klyachko connected  eigenvalues of sums of Hermitian matrices to the saturated LR-cone and  established defining linear inequalities. We prove analogues for the  saturated NL-cone. This is based on work with Gidon Orelowitz, Nicolas  Ressayre and Alexander Yong.


     

  • OPEN NEIGHBORHOOD SEMINAR

    OPEN NEIGHBORHOOD SEMINAR
    Counting shapes

    4:30 PM-5:30 PM
    March 2, 2022
    1 Oxford Street, Cambridge, MA 02138 USA

    We will explore the fascinating history of counting shapes, starting with counting dots (you can have no dots or one dot or two dots…) and then moving on to counting shapes of higher dimension. Hopefully you will see some connections to interesting questions in geometry, physics, homotopy theory, and number theory.


     

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  • SEMINARS
    9:00 AM-10:00 AM
    March 3, 2022

    While over-parameterization is widely believed to be crucial for the success of optimization for the neural networks, most existing theories on over-parameterization do not fully explain the reason — they either work in the Neural Tangent Kernel regime where neurons don’t move much, or require an enormous number of neurons. In this talk I will describe our recent works towards understanding training dynamics that go beyond kernel regimes with only polynomially many neurons (mildly overparametrized). In particular, we first give a local convergence result for mildly overparametrized two-layer networks. We then analyze the global training dynamics for a related overparametrized tensor model. For both works, we rely on a key intuition that neurons in overparametrized models work in groups and it’s important to understand the behavior of an average neuron in the group. Based on two works: https://arxiv.org/abs/2102.02410 and https://arxiv.org/abs/2106.06573.


    Professor Rong Ge is Associate Professor of Computer Science at Duke University. He received his Ph.D. from the Computer Science Department of Princeton University, supervised by Sanjeev Arora. He was a post-doc at Microsoft Research, New England. In 2019, he received both a Faculty Early Career Development Award from the National Science Foundation and the prestigious Sloan Research Fellowship. His research interest focus on theoretical computer science and machine learning. Modern machine learning algorithms such as deep learning try to automatically learn useful hidden representations of the data. He is interested in formalizing hidden structures in the data and designing efficient algorithms to find them. His research aims to answer these questions by studying problems that arise in analyzing text, images, and other forms of data, using techniques such as non-convex optimization and tensor decompositions.


    Zoom ID: 950 2372 5230 (Password: cmsa)

  • SEMINARS
    9:30 AM-11:00 AM
    March 3, 2022

    In this talk we will discuss the interaction between magnetic monopoles and massless fermions. In the 1980’s Callan and Rubakov showed that in the simplest example and that fermion-monopole interactions catalyze proton decay in GUT completions of the standard model. Here we will explain how fermions in general representations interact with general spherically symmetric monopoles and classify the types of symmetries that are broken: global symmetries with ABJ-type anomalies.


    https://harvard.zoom.us/j/977347126
    Password: cmsa

4
  • GAUGE-TOPOLOGY-SYMPLECTIC SEMINAR
    3:30 PM-4:30 PM
    March 4, 2022
    1 Oxford Street, Cambridge, MA 02138 USA

    It is well-known that for any finite group G, there exists a closed 3-manifold M with G as a quotient of pi_1(M). However, we can ask more detailed questions about the possible finite quotients of 3-manifold groups, e.g. if G and H are finite groups, does there exist a 3-manifold group with G as a quotient but not H as a quotient?  We give an answer to all such questions.  To prove existence of 3-manifolds with certain finite quotients but not others, we use a probabilistic method, by first proving a formula for the distribution of the (profinite completion of) the fundamental group of a random 3-manifold in the Dunfield-Thurston model of random Heegard splittings as the genus goes to infinity.  This is joint work with Will Sawin.
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  • SEMINARS
    9:00 AM-10:00 AM
    March 8, 2022

    The random greedy algorithm for finding a maximal independent set in a graph has been studied extensively in various settings in combinatorics, probability, computer science, and chemistry. The algorithm builds a maximal independent set by inspecting the graph’s vertices one at a time according to a random order, adding the current vertex to the independent set if it is not connected to any previously added vertex by an edge.

    In this talk, I will present a simple yet general framework for calculating the asymptotics of the proportion of the yielded independent set for sequences of (possibly random) graphs, involving a valuable notion of local convergence. I will demonstrate the applicability of this framework by giving short and straightforward proofs for results on previously studied families of graphs, such as paths and various random graphs, and by providing new results for other models such as random trees.

    If time allows, I will discuss a more delicate (and combinatorial) result, according to which, in expectation, the cardinality of a random greedy independent set in the path is no larger than that in any other tree of the same order.


    Password: 1251442
  • SEMINARS
    9:30 AM-10:30 AM
    March 8, 2022

    The Nishimori line (NL) describes a special one parameter family of disordered spin models which are invariant under a gauge transformation acting jointly on the spins and the disorder. For group valued spins such, as SO(2) and SU(2), we prove that there is long range order in 3 or more dimensions along the NL. The proof relies on work of Abbe, Massoulie, Montanari, Sly and Srivastava on group synchronization. It does not use reflection positivity. This is joint work with Christophe Garban, arXiv:2109.01617v2.


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

9
  • SEMINARS
    3:00 AM-4:00 AM
    March 9, 2022

    In 1998, using arguments from M-theory, Gopakumar and Vafa argued that there are integer BPS invariants of symplectic Calabi-Yau 3-folds. Unfortunately, they did not give a direct mathematical definition of their BPS invariants, but they predicted that they are related to the Gromov-Witten invariants by a transformation of the generating series. The Gopakumar-Vafa conjecture asserts that if one defines the BPS invariants indirectly through this procedure, then they satisfy an integrality and a (genus) finiteness condition.

     

    The integrality conjecture has been resolved by Ionel and Parker. A key innovation of their proof is the introduction of the cluster formalism: an ingenious device to side-step questions regarding multiple covers and super-rigidity. Their argument could not resolve the finiteness conjecture, however. The reason for this is that it relies on Gromov’s compactness theorem for pseudo-holomorphic maps which requires an a priori genus bound. It turns out, however, that rather powerful tools from geometric measure theory imply a compactness theorem for pseudo-holomorphic cycles. This can be used to upgrade Ionel and Parker’s cluster formalism and prove both the integrality and finiteness conjecture.


    Zoom Link: https://cuhk.zoom.us/j/95974708402

    Meeting ID: 959 7470 8402

    Passcode: 20220309

  • SEMINARS
    9:30 AM-10:30 AM
    March 9, 2022

    The  low  dimensional  manifold  hypothesis  posits  that  the  data  found  in many applications, such as those involving natural images, lie (approximately) on low dimensional manifolds embedded in a high dimensional Euclidean space. In this setting, a typical neural network defines a function that takes a finite number of vectors in the embedding space as input.  However, one often needs to  consider  evaluating  the  optimized  network  at  points  outside  the  training distribution.  We analyze the cases where the training data are distributed in a linear subspace of Rd.  We derive estimates on the variation of the learning function, defined by a neural network, in the direction transversal to the subspace.  We study the potential regularization effects associated with the network’s depth and noise in the codimension of the data manifold.


    For information on how to join, please see:  https://cmsa.fas.harvard.edu/seminars-and-colloquium/

  • SEMINARS
    10:30 AM-12:00 PM
    March 9, 2022

    Motivated by a puzzle arising from recent work on staggered lattice fermions we introduce Kaehler-Dirac fermions and describe their connection both to Dirac fermions and staggered fermions. We show that they suffer from a gravitational anomaly that breaks a chiral U(1) symmetry specific to Kaehler-Dirac fermions down to Z_4 in any even dimension. In odd dimensions we show that the effective theory that results from integrating out massive Kaehler-Dirac fermions is a topological gravity theory. Such theories generalize Witten’s construction of (2+1) gravity as a Chern Simons theory. In the presence of a domain wall massless modes appear on the wall which can be consistently coupled to gravity due to anomaly inflow from the bulk gravitational theory. Much of this story parallels the usual discussion of topological insulators. The key difference is that the twisted chiral symmetry and anomaly structure of Kaehler-Dirac theories survives intact under discretization and governs the behavior of the lattice models. $Z_4$ invariant four fermion interactions can be used to gap out states in such theories without breaking symmetries and in flat space yields the known constraints on the number of Majorana fermions needed symmetric mass generation namely eight and sixteen Majorana spinors in two and four dimensions.


    For information on how to join, please see:  https://cmsa.fas.harvard.edu/seminars-and-colloquium/

  • SEMINARS
    2:00 PM-3:00 PM
    March 9, 2022

    We automatically solve, explain, and generate university-level course problems from thirty STEM courses (at MIT, Harvard, and Columbia) for the first time.
    We curate a new dataset of course questions and answers across a dozen departments: Aeronautics and Astronautics, Chemical Engineering, Chemistry, Computer Science, Economics, Electrical Engineering, Materials Science, Mathematics, Mechanical Engineering, Nuclear Science, Physics, and Statistics.
    We generate new questions and use them in a Columbia University course, and perform A/B tests demonstrating that these machine generated questions are indistinguishable from human-written questions and that machine generated explanations are as useful as human-written explanations, again for the first time.
    Our approach consists of five steps:
    (i) Given course questions, turn them into programming tasks;
    (ii) Automatically generate programs from the programming tasks using a Transformer model, OpenAI Codex, pre-trained on text and fine-tuned on code;
    (iii) Execute the programs to obtain and evaluate the answers;
    (iv) Automatically explain the correct solutions using Codex;
    (v) Automatically generate new questions that are qualitatively indistinguishable from human-written questions.
    This work is a significant step forward in applying machine learning for education, automating a considerable part of the work involved in teaching.
    Our approach allows personalization of questions based on difficulty level and student backgrounds, and scales up to a broad range of courses across the schools of engineering and science.
    This is joint work with students and colleagues at MIT, Harvard University, Columbia University, Worcester Polytechnic Institute, and University of Waterloo.


    https://harvard.zoom.us/j/99651364593?pwd=Q1R0RTMrZ2N ZQjg1U1ZOaUYzSE02QT09

  • NUMBER THEORY SEMINAR
    3:00 PM-4:00 PM
    March 9, 2022
    1 Oxford Street, Cambridge, MA 02138 USA

    For a smooth projective curve $C/\mathbb{Q}$, how many field extensions of $\mathbb{Q}$ — of given degree and bounded discriminant — arise from adjoining a point of $C(verline{\mathbb{Q}})$? Can we further count the number of such extensions with a specified Galois group?
    Asymptotic lower bounds for these quantities have been found for elliptic curves by Lemke Oliver and Thorne, for hyperelliptic curves by Keyes, and for superelliptic curves by Beneish and Keyes. We discuss similar asymptotic lower bounds that hold for all smooth plane curves $C$.


     

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  • SEMINARS
    9:00 AM-10:00 AM
    March 10, 2022

    The gig economy provides workers with the benefits of autonomy and flexibility, but it does so at the expense of work identity and co-worker bonds. Among the many reasons why gig workers leave their platforms, an unexplored aspect is the organization identity. In a series of studies, we develop a team formation and inter-team contest at a ride-sharing platform. We employ an end-to-end data science approach, combining methodologies from randomized field experiments, recommender systems, and counterfactual machine learning. Together, our results show that platform designers can leverage team identity and team contests to increase revenue and worker engagement in a gig economy.


    For information on how to join, please see:  https://cmsa.fas.harvard.edu/seminars-and-colloquium/

  • SEMINARS
    9:30 AM-10:30 AM
    March 10, 2022

    We consider the Cauchy problem for the Einstein equations for cosmological spacetimes, i.e. spacetimes with compact spatial hypersurfaces. Various classes of those dynamical spacetimes have been constructed and analyzed using CMC foliations or equivalently the CMC-Einstein flow. We will briefly review the Andersson-Moncrief stability result of negative Einstein metrics under the vacuum Einstein flow and then present various recent generalizations to the nonvacuum case. We will emphasize what difficulties arise in those generalizations, how they can be handled depending on the matter model at hand, and what implications we can draw from these results for cosmology. We then turn to a scenario where the CMC Einstein flow leads to a large data result in 2+1-dimensions.


    For information on how to join, please see:  https://cmsa.fas.harvard.edu/seminars-and-colloquium/

  • SEMINARS
    8:00 PM-9:30 PM
    March 10, 2022

    We present computations of the thermal Hall coefficient of phonons scattering off defects with multiple energy levels. Using a microscopic formulation based on the Kubo formula, we find that the leading contribution perturbative in the phonon-defect coupling is of the `side-jump’ type, which is proportional to the phonon lifetime. This contribution is at resonance when the phonon energy equals a defect level spacing. Our results are obtained for different defect models, and include models of an impurity quantum spin in the presence of quasi-static magnetic order with an isotropic Zeeman coupling to the applied field.
    This work is based on arxiv: 2201.11681


    For information on how to join, please see:  https://cmsa.fas.harvard.edu/seminars-and-colloquium/

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  • SEMINARS
    9:00 AM-10:00 AM
    March 15, 2022

    I will first review the construction of the moduli space of tropical curves (or metric graphs), and its relation to graph complexes. The graph Laplacian may be interpreted as a tropical version of the classical Torelli map and its determinant is the Kirchhoff graph polynomial (also called 1st Symanzik), which is one of the two key components in Feynman integrals in high energy physics.The other component is the so-called 2nd Symanzik polynomial, which is defined for graphs with external half edges and involves particle masses (edge colourings). I will explain how this too may be interpreted as the determinant of a generalised graph Laplacian, and how it leads to a volumetric interpretation of a certain class of Feynman integrals.

    For information on how to join, please see:  https://cmsa.fas.harvard.edu/seminars-and-colloquium/


  • SEMINARS
    9:30 AM-10:30 AM
    March 15, 2022

    It is by now well-known that mirror symmetry may be expressed as an equivalence between categories associated to dual Kahler manifolds. Following a proposal of Teleman, we inaugurate a program to understand 3d mirror symmetry as an equivalence between 2-categories associated to dual holomorphic symplectic stacks. We consider here the abelian case, where our theorem expresses the 2-category of spherical functors as a 2-category of coherent sheaves of categories. Applications include categorifications of hypertoric category O and of many related constructions in representation theory. This is joint work with Justin Hilburn and Aaron Mazel-Gee.


    For information on how to join, please see:  https://cmsa.fas.harvard.edu/seminars-and-colloquium/

  • SEMINARS
    9:30 PM-10:30 PM
    March 15, 2022

    Billiards on an elliptical billiard table are completely integrable: phase space is foliated by invariant submanifolds for the billiard flow. Birkhoff conjectured that ellipses are the only plane domains with integrable

    billiards. Avila-deSimoi- Kaloshin proved the conjecture for ellipses of sufficiently small eccentricity.   Kaloshin-Sorrentino proved local results for all eccentricities.  On the quantum level, the analogous conjecture is that ellipses are uniquely determined by their Dirichlet (or, Neumann) eigenvalues. Using the results on the Birkhoff conjecture, Hamid Hezari and I proved that for ellipses of small eccentricity are indeed uniquely determined by their eigenvalues. Except for disks, which Kac proved to be uniquely determined, these are the only domains for which it is known that one can hear their shape.


    Zoom Link: https://cuhk.zoom.us/j/97001510671

    (Meeting ID: 970 0151 0671; Passcode: 20220316)

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  • SEMINARS
    10:30 AM-12:00 PM
    March 16, 2022

    Some recent work in the quantum gravity literature has considered what happens when the amplitudes of a TQFT are summed over the bordisms between fixed in-going and out-going boundaries. We will comment on these constructions. The total amplitude, that takes into account all in-going and out-going boundaries can be presented in a curious factorized form. This talk reports on work done with Anindya Banerjee and is based on the paper on the e-print arXiv   2201.00903.


    For information on how to join, please see:  https://cmsa.fas.harvard.edu/seminars-and-colloquium/

17
  • SEMINARS
    9:00 AM-10:00 AM
    March 17, 2022

    Deep neural networks have achieved significant empirical success in many fields, including the fields of computer vision and natural language processing. Along with its empirical success, deep learning has been theoretically shown to be attractive in terms of its expressive power. However, the theory of expressive power does not ensure that we can efficiently find an optimal solution in terms of optimization and generalization, during the optimization process. In this talk, I will discuss some mathematical properties of optimization and generalization for deep neural networks.


    For information on how to join, please see:  https://cmsa.fas.harvard.edu/seminars-and-colloquium/

  • SEMINARS
    9:00 AM-10:00 AM
    March 17, 2022

    Deep neural networks have achieved significant empirical success in many fields, including the fields of computer vision and natural language processing. Along with its empirical success, deep learning has been theoretically shown to be attractive in terms of its expressive power. However, the theory of expressive power does not ensure that we can efficiently find an optimal solution in terms of optimization and generalization, during the optimization process. In this talk, I will discuss some mathematical properties of optimization and generalization for deep neural networks.


    For information on how to join, please see:  https://cmsa.fas.harvard.edu/seminars-and-colloquium/

  • SEMINARS
    9:30 AM-11:00 AM
    March 17, 2022

    The Swampland program aims at uncovering the universal implications of quantum gravity at low-energy physics. I will review the basic ideas of the Swampland program, formal and phenomenological implications, and provide a survey of the techniques commonly used in Swampland research including tools from quantum information, holography, supersymmetry, and string theory.


    For information on how to join, please see:  https://cmsa.fas.harvard.edu/seminars-and-colloquium/

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  • SEMINARS
    9:30 AM-10:30 AM
    March 21, 2022

    The AdS/CFT conjecture in physics posits the existence of a  correspondence between gravitational theories in asymptotically Anti-de  Sitter (aAdS) spacetimes and field theories on their conformal boundary.  In this presentation, we prove rigorous mathematical statements toward  this conjecture.In particular, we show there is a one-to-one correspondence between aAdS  solutions of the Einstein-vacuum equations and a suitable space of data  on the conformal boundary (consisting of the boundary metric and the  boundary stress-energy tensor). We also discuss consequences of this  result, as well as the main ingredient behind its proof: a unique  continuation property for wave equations on aAdS spacetimes.


    For information on how to join, please see:  https://cmsa.fas.harvard.edu/seminars-and-colloquium/

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  • SEMINARS
    All day
    March 22, 2022-March 25, 2022

    The study of static Riemannian metrics arises naturally in general relativity and differential geometry. A static metric produces a special Einstein manifold, and it interconnects with scalar curvature deformation and gluing. The well-known Uniqueness Theorem of Static Black Holes says that an asymptotically flat, static metric with black hole boundary must belong to the Schwarzschild family. In the same vein, most efforts have been made to classify static metrics as known exact solutions. In contrast to the rigidity phenomena and classification efforts, Robert Bartnik proposed the Static Vacuum Extension Conjecture (originating from his other conjectures about quasi-local masses in the 80’s) that there is always a unique, asymptotically flat, static vacuum metric with quite arbitrarily prescribed Bartnik boundary data. In this course, I will discuss some recent progress confirming this conjecture for large classes of boundary data. The course is based on joint work with Zhongshan An, and the tentative plan is

    1. The conjecture and an overview of the results
    2. Static regular: a sufficient condition for existence and local uniqueness
    3. Convex boundary, isometric embedding, and static regular
    4. Perturbations of any hypersurface are static regular

     

    March 22 – 25, 2022
    22nd & 23rd, 10:00 am – 11:30am ET
    24th & 25th, 11:00 am – 12:30pm ET

    Location: Hybrid. CMSA main seminar room, G-10. Zoom link will be available.

    All in-person attendees must register online.

    For more information, please see https://cmsa.fas.harvard.edu/gr-program/

     

  • CMSA EVENT: CMSA Combinatorics, Physics and Probability: Flip Processes
    9:30 AM-10:30 AM
    March 22, 2022

    We introduce a class of random graph processes, which we call \emph{flip processes}. Each such process is given by a \emph{rule} which is just a function $\mathcal{R}:\mathcal{H}_k\rightarrow \mathcal{H}_k$ from all labelled $k$-vertex graphs into itself ($k$ is fixed). The process starts with a given $n$-vertex graph $G_0$. In each step, the graph $G_i$ is obtained by sampling $k$ random vertices $v_1,\ldots,v_k$ of $G_{i-1}$ and replacing the induced graph $F:=G_{i-1}[v_1,\ldots,v_k]$ by $\mathcal{R}(F)$. This class contains several previously studied processes including the Erd\H{o}s–R\’enyi random graph process and the triangle removal process.

    Given a flip process with a rule $\mathcal{R}$, we construct time-indexed trajectories $\Phi:\Gra\times [0,\infty)\rightarrow\Gra$ in the space of graphons. We prove that for any $T > 0$ starting with a large finite graph $G_0$ which is close to a graphon $W_0$ in the cut norm, with high probability the flip process will stay in a thin sausage around the trajectory $(\Phi(W_0,t))_{t=0}^T$ (after rescaling the time by the square of the order of the graph).

    These graphon trajectories are then studied from the perspective of dynamical systems. Among others, we study continuity properties of these trajectories with respect to time and the initial graphon, existence and stability of fixed points and speed of convergence (whenever the infinite time limit exists). We give an example of a flip process with a periodic trajectory. This is joint work with Frederik Garbe, Matas \v Sileikis and Fiona Skerman (arXiv:2201.12272).

    We also study several specific families flip processes. This is joint work with Pedro Ara\’ujo, Eng Keat Hng and Matas \v{S}ileikis (in preparation).

    A brief introduction to the necessary bits of the theory of graph limits will be given in the talk.

  • SEMINARS
    9:30 AM-10:30 AM
    March 22, 2022

    Classical machine learning (ML) provides a potentially powerful approach to solving challenging quantum many-body problems in physics and chemistry. However, the advantages of ML over more traditional methods have not been firmly established. In this work, we prove that classical ML algorithms can efficiently predict ground state properties of gapped Hamiltonians in finite spatial dimensions, after learning from data obtained by measuring other Hamiltonians in the same quantum phase of matter. In contrast, under widely accepted complexity theory assumptions, classical algorithms that do not learn from data cannot achieve the same guarantee. We also prove that classical ML algorithms can efficiently classify a wide range of quantum phases of matter. Our arguments are based on the concept of a classical shadow, a succinct classical description of a many-body quantum state that can be constructed in feasible quantum experiments and be used to predict many properties of the state. Extensive numerical experiments corroborate our theoretical results in a variety of scenarios, including Rydberg atom systems, 2D random Heisenberg models, symmetry-protected topological phases, and topologically ordered phases.


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

  • HARVARD-MIT ALGEBRAIC GEOMETRY SEMINAR

    HARVARD-MIT ALGEBRAIC GEOMETRY SEMINAR
    A Conjecture of Mori and Families of Plane Curves

    3:00 PM-4:00 PM
    March 22, 2022
    1 Oxford Street, Cambridge, MA 02138 USA

    Consider a smooth family of hypersurfaces of degree d in P^{n+1}.
    When is every smooth projective limit of this family also a hypersurface?
    While it is easy to construct example of limits that are not
    hypersurfaces when the degree d is composite, Mori conjectured that, if d
    is prime and n>2, every smooth projective limit is indeed a hypersurface.
    However, there are counterexamples when n=1 or 2; for example, one can
    take a family of degree 5 plane curves and degenerate to a smooth
    hyperelliptic (non-planar) curve.  In this talk, we will propose a
    re-formulation of Mori’s conjecture that explains the failure in low
    dimensions, provide results in dimension one, and discuss a general
    approach to the problem using moduli spaces of pairs.  This is joint work
    with David Stapleton.


     

23
  • SEMINARS
    All day
    March 23, 2022-March 25, 2022

    The study of static Riemannian metrics arises naturally in general relativity and differential geometry. A static metric produces a special Einstein manifold, and it interconnects with scalar curvature deformation and gluing. The well-known Uniqueness Theorem of Static Black Holes says that an asymptotically flat, static metric with black hole boundary must belong to the Schwarzschild family. In the same vein, most efforts have been made to classify static metrics as known exact solutions. In contrast to the rigidity phenomena and classification efforts, Robert Bartnik proposed the Static Vacuum Extension Conjecture (originating from his other conjectures about quasi-local masses in the 80’s) that there is always a unique, asymptotically flat, static vacuum metric with quite arbitrarily prescribed Bartnik boundary data. In this course, I will discuss some recent progress confirming this conjecture for large classes of boundary data. The course is based on joint work with Zhongshan An, and the tentative plan is

    1. The conjecture and an overview of the results
    2. Static regular: a sufficient condition for existence and local uniqueness
    3. Convex boundary, isometric embedding, and static regular
    4. Perturbations of any hypersurface are static regular

     

    March 22 – 25, 2022
    22nd & 23rd, 10:00 am – 11:30am ET
    24th & 25th, 11:00 am – 12:30pm ET

    Location: Hybrid. CMSA main seminar room, G-10. Zoom link will be available.

    All in-person attendees must register online.

    For more information, please see https://cmsa.fas.harvard.edu/gr-program/

     

  • CMSA EVENT: CMSA Colloquium: Fluctuation scaling or Taylor’s law of heavy-tailed data, illustrated by U.S. COVID-19 cases and deaths
    9:30 AM-10:30 AM
    March 23, 2022

    Over the last century, ecologists, statisticians, physicists, financial quants, and other scientists discovered that, in many examples, the sample variance approximates a power of the sample mean of each of a set of samples of nonnegative quantities. This power-law relationship of variance to mean is known as a power variance function in statistics, as Taylor’s law in ecology, and as fluctuation scaling in physics and financial mathematics. This survey talk will emphasize ideas, motivations, recent theoretical results, and applications rather than detailed proofs. Many models intended to explain Taylor’s law assume the probability distribution underlying each sample has finite mean and variance. Recently, colleagues and I generalized Taylor’s law to samples from probability distributions with infinite mean or infinite variance and higher moments. For such heavy-tailed distributions, we extended Taylor’s law to higher moments than the mean and variance and to upper and lower semivariances (measures of upside and downside portfolio risk). In unpublished work, we suggest that U.S. COVID-19 cases and deaths illustrate Taylor’s law arising from a distribution with finite mean and infinite variance. This model has practical implications. Collaborators in this work are Mark Brown, Richard A. Davis, Victor de la Peña, Gennady Samorodnitsky, Chuan-Fa Tang, and Sheung Chi Phillip Yam.


    for information on how to join, please go to:

    https://cmsa.fas.harvard.edu/cmsa-colloquium

  • CMSA EVENT: CMSA Quantum Matter in Mathematics and Physics: Non-zero momentum requires long-range entanglement
    10:30 AM-12:00 PM
    March 23, 2022

    I will show that a quantum state in a lattice spin (boson) system must be long-range entangled if it has non-zero lattice momentum, i.e. if it is an eigenstate of the translation symmetry with eigenvalue not equal to 1. Equivalently, any state that can be connected with a non-zero momentum state through a finite-depth local unitary transformation must also be long-range entangled. The statement can also be generalized to fermion systems. I will then present two applications of this result: (1) several different types of Lieb-Schultz-Mattis (LSM) theorems, including a previously unknown version involving only a discrete Z_n symmetry, can be derived in a simple manner; (2) a gapped topological order (in space dimension d>1) must weakly break translation symmetry if one of its ground states on torus has nontrivial momentum – this generalizes the familiar physics of Tao-Thouless in fractional quantum Hall systems.


    For information on how to join, please see:  https://cmsa.fas.harvard.edu/seminars-and-colloquium/

  • SEMINARS
    2:00 PM-3:00 PM
    March 23, 2022

    We explore the use of expert iteration in the context of language modeling applied to formal mathematics. We show that at same compute budget, expert iteration, by which we mean proof search interleaved with learning, dramatically outperforms proof search only. We also observe that when applied to a collection of formal statements of sufficiently varied difficulty, expert iteration is capable of finding and solving a curriculum of increasingly difficult problems, without the need for associated ground-truth proofs. Finally, by applying this expert iteration to a manually curated set of problem statements, we achieve state-of-the-art on the miniF2F benchmark, automatically solving multiple challenging problems drawn from high school olympiads.


    For information on how to join, please see:  https://cmsa.fas.harvard.edu/seminars-and-colloquium/

  • NUMBER THEORY SEMINAR
    3:00 PM-4:00 PM
    March 23, 2022
    1 Oxford Street, Cambridge, MA 02138 USA

    Cohen and Lenstra gave a conjectural distribution for the  class group of a random quadratic number field. Since the class group  is the Galois group of the maximum abelian unramified extension, a  natural generalization would be to give a conjecture for the  distribution of the Galois group of the maximal unramified extension.  Previous work has produced a plausible conjecture in special cases,  with the most general being recent work of Liu, Wood, and Zurieck-Brown.

    There is a deep analogy between number fields and 3-manifolds. Thus,  an analogous question would be to describe the distribution of the  profinite completion of the fundamental group of a random 3-manifold.  In this talk, I will explain how Melanie Wood and I answered this  question for a model of random 3-manifolds defined by Dunfield and  Thurston, and how the techniques we used should allow us, in future  work, to give a more general conjecture in the number field case.


     

  • OPEN NEIGHBORHOOD SEMINAR
    4:30 PM-5:30 PM
    March 23, 2022
    1 Oxford Street, Cambridge, MA 02138 USA

    These are three concepts that examine how quantities vary: invariance and equivariance, from mathematics, and covariance, from statistics. How do they relate to each other?

    We will see how groups and symmetries are at the heart of various problems in statistics. I will also describe approaches to estimate parameters in statistical models using invariant theory, based on joint work with Carlos Amendola, Kathlen Kohn, and Philipp Reichenbach.


     

24
  • SEMINARS
    All day
    March 24, 2022-March 25, 2022

    The study of static Riemannian metrics arises naturally in general relativity and differential geometry. A static metric produces a special Einstein manifold, and it interconnects with scalar curvature deformation and gluing. The well-known Uniqueness Theorem of Static Black Holes says that an asymptotically flat, static metric with black hole boundary must belong to the Schwarzschild family. In the same vein, most efforts have been made to classify static metrics as known exact solutions. In contrast to the rigidity phenomena and classification efforts, Robert Bartnik proposed the Static Vacuum Extension Conjecture (originating from his other conjectures about quasi-local masses in the 80’s) that there is always a unique, asymptotically flat, static vacuum metric with quite arbitrarily prescribed Bartnik boundary data. In this course, I will discuss some recent progress confirming this conjecture for large classes of boundary data. The course is based on joint work with Zhongshan An, and the tentative plan is

    1. The conjecture and an overview of the results
    2. Static regular: a sufficient condition for existence and local uniqueness
    3. Convex boundary, isometric embedding, and static regular
    4. Perturbations of any hypersurface are static regular

     

    March 22 – 25, 2022
    22nd & 23rd, 10:00 am – 11:30am ET
    24th & 25th, 11:00 am – 12:30pm ET

    Location: Hybrid. CMSA main seminar room, G-10. Zoom link will be available.

    All in-person attendees must register online.

    For more information, please see https://cmsa.fas.harvard.edu/gr-program/

     

  • CMSA EVENT: CMSA Interdisciplinary Science Seminar: An operadic structure on supermoduli spaces
    9:00 AM-10:00 AM
    March 24, 2022

    The operadic structure on the moduli spaces of algebraic curves  encodes in a combinatorial way how nodal curves in the boundary can be obtained by glueing smooth curves along marked points. In this talk, I will present a generalization of the operadic structure to moduli spaces of SUSY curves (or super Riemann surfaces). This requires colored graphs and generalized operads in the sense of Borisov-Manin. Based joint work with Yu. I. Manin and Y. Wu. https://arxiv.org/abs/2202.10321

    For information on how to join, please go to:

    https://cmsa.fas.harvard.edu/interdisciplinary-science-seminar

  • CMSA EVENT: CMSA Quantum Matter in Mathematics and Physics: Edge physics at the deconfined transition between a quantum spin Hall insulator and a superconductor
    9:30 AM-11:00 AM
    March 24, 2022

    I will talk about the edge physics of the deconfined quantum phase transition (DQCP) between a spontaneous quantum spin Hall (QSH) insulator and a spin-singlet superconductor (SC). Although the bulk of this transition is in the same universality class as the paradigmatic deconfined Neel to valence-bond-solid transition, the boundary physics has a richer structure due to proximity to a quantum spin Hall state. We use the parton trick to write down an effective field theory for the QSH-SC transition in the presence of a boundary and calculate various edge properties in a large-N limit. We show that the boundary Luttinger liquid in the QSH state survives at the phase transition, but only as fractional degrees of freedom that carry charge but not spin. The physical fermion remains gapless on the edge at the critical point, with a universal jump in the fermion scaling dimension as the system approaches the transition from the QSH side. The critical point could be viewed as a gapless analogue of the QSH state but with the full SU(2) spin rotation symmetry, which cannot be realized if the bulk is gapped. This talk reports on the work done with Liujun Zou and Chong Wang (arxiv:2110.08280).


    For information on how to join, please see:  https://cmsa.fas.harvard.edu/seminars-and-colloquium/

  • SEMINARS
    9:30 AM-10:30 AM
    March 24, 2022

    I will talk about my work on the compressible Euler equations. We prove the local-in-time existence the solution of the compressible Euler equations in $3$-D, for the Cauchy data of the velocity, density and vorticity $(v,\varrho, mega) \in H^s\times H^s\times H^{s’}$, $2<s'<s$.  The result extends the sharp result of Smith-Tataru and Wang, established in the irrotational case, i.e $mega=0$, which is known to be optimal for $s>2$. At the opposite extreme, in the incompressible case, i.e. with a constant density,  the result is known to hold for $mega\in H^s$, $s>3/2$ and fails for $s\le 3/2$, see the work of Bourgain-Li. It is thus natural to conjecture that the optimal result should be  $(v,\varrho, mega) \in H^s\times H^s\times H^{s’}$, $s>2, \, s’>\frac{3}{2}$. We view our work as an important step in proving the conjecture. The main difficulty in establishing sharp well-posedness results for general compressible Euler flow is due to the highly nontrivial interaction between the sound waves, governed by quasilinear wave equations, and vorticity which is transported by the flow. To overcome this difficulty, we separate the dispersive part of a sound wave from the transported part and gain regularity significantly by exploiting the nonlinear structure of the system and the geometric structures of the acoustic spacetime.


    For information on how to join, please see:  https://cmsa.fas.harvard.edu/seminars-and-colloquium/

  • CMSA EVENT: CMSA Active Matter Seminar: Topological defects drive layer formation in gliding bacteria colonies
    1:00 PM-2:00 PM
    March 24, 2022

    The developmental cycle of Myxococcus xanthus involves the coordination of many hundreds of thousands of cells aggregating to form mounds known as fruiting bodies. This aggregation process begins with the sequential formation of more and more cell layers. Using three-dimensional confocal imaging we study this layer formation process by observing the formation of holes and second layers within a base monolayer of M xanthus cells. We find that cells align with each other over the majority of the monolayer forming an active nematic liquid crystal with defect point where cell alignment is undefined. We find that new layers and holes form at positive and negative topological defects respectively. We model the cell layer using hydrodynamic modeling and find that this layer and hole formation process is driven by active nematic forces through cell motility and anisotropic substrate friction.


    For information on how to join, please see:  https://cmsa.fas.harvard.edu/seminars-and-colloquium/

25
  • SEMINARS
    All day
    March 25, 2022-March 25, 2022

    The study of static Riemannian metrics arises naturally in general relativity and differential geometry. A static metric produces a special Einstein manifold, and it interconnects with scalar curvature deformation and gluing. The well-known Uniqueness Theorem of Static Black Holes says that an asymptotically flat, static metric with black hole boundary must belong to the Schwarzschild family. In the same vein, most efforts have been made to classify static metrics as known exact solutions. In contrast to the rigidity phenomena and classification efforts, Robert Bartnik proposed the Static Vacuum Extension Conjecture (originating from his other conjectures about quasi-local masses in the 80’s) that there is always a unique, asymptotically flat, static vacuum metric with quite arbitrarily prescribed Bartnik boundary data. In this course, I will discuss some recent progress confirming this conjecture for large classes of boundary data. The course is based on joint work with Zhongshan An, and the tentative plan is

    1. The conjecture and an overview of the results
    2. Static regular: a sufficient condition for existence and local uniqueness
    3. Convex boundary, isometric embedding, and static regular
    4. Perturbations of any hypersurface are static regular

     

    March 22 – 25, 2022
    22nd & 23rd, 10:00 am – 11:30am ET
    24th & 25th, 11:00 am – 12:30pm ET

    Location: Hybrid. CMSA main seminar room, G-10. Zoom link will be available.

    All in-person attendees must register online.

    For more information, please see https://cmsa.fas.harvard.edu/gr-program/

     

26
27
28
29
  • SEMINARS
    All day
    March 29, 2022-April 1, 2022

    I will present aspects of a theorem, joint with Mihalis Dafermos, Gustav Holzegel and Igor Rodnianski, on the full finite codimension nonlinear asymptotic stability of the Schwarzschild family of black holes.

     

    March 29 – April 1, 2022: 10:00am – 12:00pm ET, each day

    Location: Hybrid. CMSA main seminar room, G-10. Zoom link will be available.

    All in-person attendees must register online.

    For more information, please see https://cmsa.fas.harvard.edu/gr-program/

     

  • SEMINARS
    9:30 AM-10:30 AM
    March 29, 2022

    I will discuss the asymptotics of multimatrix models. In a small parameters (or high temperature) region, their free energy is well known to converge and to be related with the enumeration of maps. The latter can be proved by relating the so-called Dyson-Schwinger equations with Tutte surgery by nice pictures. In general, the convergence of the free energy is unknown, a problem which is closely related with the lack of a full entropy theory in free probability. We will discuss few models that can be analyzed, in particular in the large parameters (or low temperature) region.


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

  • HARVARD-MIT ALGEBRAIC GEOMETRY SEMINAR

    HARVARD-MIT ALGEBRAIC GEOMETRY SEMINAR
    How to count using equivariant cohomology?

    3:00 PM-4:00 PM
    March 29, 2022
    1 Oxford Street, Cambridge, MA 02138 USA

    How often does a general cubic surface appear in a 4-dimensional linear system?  As a slice of a general cubic 3-fold?  I will describe how we can solve problems of this kind using equivariant geometry.  This is joint work with Anand Patel and Dennis Tseng.


     

  • SEMINARS
    9:30 PM-10:30 PM
    March 29, 2022

    We present an effective quantization theory for chiral deformation of two dimensional conformal field theories. We explain a connection between the quantum master equation and the chiral homology for vertex operator algebras. As an application, we construct correlation functions of the curved beta-gamma/b-c system and establish a coupled equation relating to chiral homology groups of chiral differential operators. This can be viewed as the vertex algebra analogue of the trace map in algebraic index theory. The talk is based on the recent work arXiv:2112.14572 [math.QA].


    Zoom link: https://cuhk.zoom.us/j/99612299554

    (Meeting ID: 996 1229 9554; Passcode: 20220330)

  • SEMINARS
    9:30 PM-10:30 PM
    March 29, 2022

    We present an effective quantization theory for chiral deformation of two dimensional conformal field theories. We explain a connection between the quantum master equation and the chiral homology for vertex operator algebras. As an application, we construct correlation functions of the curved beta-gamma/b-c system and establish a coupled equation relating to chiral homology groups of chiral differential operators. This can be viewed as the vertex algebra analogue of the trace map in algebraic index theory. The talk is based on the recent work arXiv:2112.14572 [math.QA].


    For more information, please see: http://www.ims.cuhk.edu.hk/cgi-bin/SeminarAdmin/bin/Web

30
  • SEMINARS
    All day
    March 30, 2022-April 1, 2022

    I will present aspects of a theorem, joint with Mihalis Dafermos, Gustav Holzegel and Igor Rodnianski, on the full finite codimension nonlinear asymptotic stability of the Schwarzschild family of black holes.

     

    March 29 – April 1, 2022: 10:00am – 12:00pm ET, each day

    Location: Hybrid. CMSA main seminar room, G-10. Zoom link will be available.

    All in-person attendees must register online.

    For more information, please see https://cmsa.fas.harvard.edu/gr-program/

     

  • CMSA EVENT: CMSA Colloquium: Edge Modes and Gravity
    9:30 AM-10:30 AM
    March 30, 2022

    In this talk I first review some of the many appearances of localized degrees of freedom — edge modes —  in a variety of physical systems. Edge modes are implicated for example in quantum entanglement and in various topological and holographic dualities. I then review recent work in which it has been realized that a careful treatment of such modes, paying attention to relevant symmetries, is required in order to properly understand such basic physical quantities as Noether charges. From many points of view, it is conjectured that this physics may be pointing at basic properties of quantum spacetimes and gravity.


    For information on how to join, please see:  https://cmsa.fas.harvard.edu/seminars-and-colloquium/

  • CMSA EVENT: CMSA Quantum Matter in Mathematics and Physics: Renormalization group flow as optimal transport
    10:30 AM-12:00 PM
    March 30, 2022

    We show that Polchinski’s equation for exact renormalization group flow is equivalent to the optimal transport gradient flow of a field-theoretic relative entropy.  This gives a surprising information-theoretic formulation of the exact renormalization group, expressed in the language of optimal transport.  We will provide reviews of both the exact renormalization group, as well as the theory of optimal transportation.  Our results allow us to establish a new, non-perturbative RG monotone, and also reformulate RG flow as a variational problem.  The latter enables new numerical techniques and allows us to establish a systematic connection between neural network methods and RG flows of conventional field theories.  Our techniques generalize to other RG flow equations beyond Polchinski’s.


    For information on how to join, please see:  https://cmsa.fas.harvard.edu/seminars-and-colloquium/

  • CMSA EVENT: CMSA New Technologies in Mathematics: Memorizing Transformer
    2:00 PM-3:00 PM
    March 30, 2022

    Language models typically need to be trained or fine-tuned in order to acquire new knowledge, which involves updating their weights. We instead envision language models that can simply read and memorize new data at inference time, thus acquiring new knowledge immediately. In this talk, I will discuss how we extend language models with the ability to memorize the internal representations of past inputs. We demonstrate that an approximate NN lookup into a non-differentiable memory of recent (key, value) pairs improves language modeling across various benchmarks and tasks, including generic webtext (C4), math papers (arXiv), books (PG-19), code (Github), as well as formal theorems (Isabelle). We show that the performance steadily improves when we increase the size of memory up to 262K tokens. We also find that the model is capable of making use of newly defined functions and theorems during test time.


    For information on how to join, please see:  https://cmsa.fas.harvard.edu/seminars-and-colloquium/

  • NUMBER THEORY SEMINAR
    3:00 PM-4:00 PM
    March 30, 2022
    1 Oxford Street, Cambridge, MA 02138 USA

    The unbounded denominators conjecture, first raised by Atkin and Swinnerton-Dyer, asserts that a modular form for a finite index subgroup of SL_2(Z) whose Fourier coefficients have bounded denominators must be a modular form for some congruence subgroup. In this talk, we will give a sketch of the proof of this conjecture based on a new arithmetic algebraization theorem. This is joint work with Frank Calegari and Vesselin Dimitrov.


     

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