Calendar

< 2021 >
February 28 - March 06
  • 28
    February 28, 2021
    No events
  • 01
    March 1, 2021

    CMSA Mathematical Physics Seminar: Mathematical supergravity and its applications to differential geometry

    10:00 AM-11:00 AM
    March 1, 2021

    I will discuss the recent developments in the mathematical theory of supergravity that lay the mathematical foundations of the universal bosonic sector of four-dimensional ungauged supergravity and its Killing spinor equations in a differential-geometric framework.  I will provide the necessary context and background. explaining the results pedagogically from scratch and highlighting several open mathematical problems which arise in the mathematical theory of supergravity, as well as some of its potential mathematical applications. Work in collaboration with Vicente Cortés and Calin Lazaroiu.

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

    Special Colloquium

    3:00 PM-4:00 PM
    March 1, 2021

    Title: Robustness Meets Algorithms

    Abstract: Starting from the seminal works of Tukey (1960) and Huber (1964), the field of robust statistics asks: Are there estimators that probably work in the presence of noise? The trouble is that all known provably robust estimators are also hard to compute in high-dimensions.

    Here, we study a fundamental problem in robust statistics, posed in various forms in the above works. Given corrupted samples from a high-dimensional Gaussian, are there efficient algorithms to accurately estimate its parameters? We give the first algorithm that is able to tolerate a constant fraction of corruptions that is independent of the dimension. Moreover, we give a general recipe for detecting and correcting corruptions based on tensor-spectral techniques that are applicable to many other problems.

    I will also discuss how this work fits into the broader agenda of developing mathematical and algorithmic foundations for modern machine learning.

    Registration is required to receive the Zoom information

    Register here to attend

  • 02
    March 2, 2021

    Disc potential functions of Quadrics

    8:00 AM-9:00 AM
    March 2, 2021

    A disc potential function plays an important role in studying a symplectic manifold and its Lagrangian submanifolds. In this talk, I will explain how to compute the disc potential function of quadrics. The potential function provides the Landau—Ginzburg mirror, which agrees with Przyjalkowski’s mirror and a cluster chart of Pech—Rietsch—Williams’ mirror

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

    Integrability of Liouville Theory

    10:00 AM-11:00 AM
    March 2, 2021

    Polyakov introduced Liouville Conformal Field theory (LCFT) in 1981 as a way to put a naturalmeasure on the set of Riemannian metrics over a two dimensional manifold. Ever since, the work of Polyakov has echoed in various branches of physics and mathematics, ranging from string theory to probability theory and geometry. In the context of 2D quantum gravity models, LCFT is related through the Knizhnik-Polyakov-Zamolodchikov relationsto the scaling limit of Random Planar Maps and through the Alday-Gaiotto-Tachikava correspondence LCFT is conjecturally related to certain 4D Yang-Mills theories. Through the work of Dorn, Otto, Zamolodchikov and Zamolodchikov and Teschner LCFT is believed to be to a certain extent integrable. I will review a probabilistic construction of LCFT and recent proofs concerning the integrability of LCFT developed together with F. David, C. Guillarmou, R. Rhodes and V. Vargas.

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

    CMSA Computer Science for Mathematicians: Randomized Dimensionality Reduction for Clustering

    11:30 AM-12:30 PM
    March 2, 2021

    Random dimensionality reduction is a versatile tool for speeding up algorithms for high-dimensional problems. We study its application to two clustering problems: the facility location problem, and the single-link hierarchical clustering problem, which is equivalent to computing the minimum spanning tree. We show that if we project the input pointset $X$ onto a random $d = O(d_X)$-dimensional subspace (where $d_X$ is the doubling dimension of $X$), then the optimum facility location cost in the projected space approximates the original cost up to a constant factor. We show an analogous statement for minimum spanning tree, but with the dimension $d$ having an extra $\log \log n$ term and the approximation factor being arbitrarily close to $1$. Furthermore, we extend these results to approximating solutions instead of just their costs. Lastly, we provide experimental results to validate the quality of solutions and the speedup due to the dimensionality reduction.

    Unlike several previous papers studying this approach in the context of $k$-means and $k$-medians, our dimension bound does not depend on the number of clusters but only on the intrinsic dimensionality of $X$.

    Joint work with Shyam Narayanan, Piotr Indyk, Or Zamir.

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

    Decomposition theorem for semisimple local systems

    3:00 PM-4:00 PM
    March 2, 2021

    In complex algebraic geometry, the decomposition theorem asserts that semisimple geometric objects remain semisimple after taking direct images under proper algebraic maps. This was conjectured by Kashiwara and is proved by Mochizuki and Sabbah in a series of long papers via harmonic analysis and D-modules. In this talk, I would like to explain a more geometric/topological approach in the case of semisimple local systems adapting de Cataldo-Migliorini. As a byproduct, we can recover a weak form of Saito’s decomposition theorem for variations of Hodge structures. Joint work in progress with Chuanhao Wei.

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

  • 03
    March 3, 2021

    CMSA Quantum Matter in Mathematics and Physics: Symmetry-protected sign problem and magic in quantum phases of matter

    10:30 AM-12:00 PM
    March 3, 2021

    We introduce the concepts of a symmetry-protected sign problem and symmetry-protected magic, defined by the inability of symmetric finite-depth quantum circuits to transform a state into a nonnegative real wave function and a stabilizer state, respectively. We show that certain symmetry protected topological (SPT) phases have these properties, as a result of their anomalous symmetry action at a boundary. For example, one-dimensional Z2 × Z2 SPT states (e.g. cluster state) have a symmetry-protected sign problem, and two-dimensional Z2 SPT states (e.g. Levin-Gu state) have both a symmetry-protected sign problem and magic. We also comment on the relation of a symmetry-protected sign problem to the computational wire property of one-dimensional SPT states and the greater implications of our results for measurement based quantum computing.

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

    CMSA New Technologies in Mathematics: Neural Theorem Proving in Lean using Proof Artifact Co-training and Language Models

    3:00 PM-4:00 PM
    March 3, 2021

    Labeled data for imitation learning of theorem proving in large libraries of formalized mathematics is scarce as such libraries require years of concentrated effort by human specialists to be built. This is particularly challenging when applying large Transformer language models to tactic prediction, because the scaling of performance with respect to model size is quickly disrupted in the data-scarce, easily-overfitted regime. We propose PACT (Proof Artifact Co-Training), a general methodology for extracting abundant self-supervised data from kernel-level proof terms for co-training alongside the usual tactic prediction objective. We apply this methodology to Lean, an interactive proof assistant which hosts some of the most sophisticated formalized mathematics to date. We instrument Lean with a neural theorem prover driven by a Transformer language model and show that PACT improves theorem proving success rate on a held-out suite of test theorems from 32\% to 48\%.

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

    Equidistribution and Uniformity in Families of Curves

    3:00 PM-4:00 PM
    March 3, 2021

    In the talk, I will present an equidistribution result for families of (non-degenerate) subvarieties in a (general) family of abelian varieties. This extends a result of DeMarco and Mavraki for curves in fibered products of elliptic surfaces. Using this result, one can deduce a uniform version of the classical Bogomolov conjecture for curves embedded in their Jacobians, namely that the number of torsion points lying on them is uniformly bounded in the genus of the curve. This has been previously only known in a few select cases by work of David–Philippon and DeMarco–Krieger–Ye. Finally, one can obtain a rather uniform version of the Mordell-Lang conjecture as well by complementing a result of Dimitrov–Gao–Habegger: The number of rational points on a smooth algebraic curve defined over a number field can be bounded solely in terms of its genus and the Mordell-Weil rank of its Jacobian. Again, this was previously known only under additional assumptions (Stoll, Katz–Rabinoff–Zureick-Brown).

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

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

    Save the Pilgrim!

    4:30 PM-5:30 PM
    March 3, 2021

    An evil mathematician has kidnapped the Harvard Pilgrim! To win his freedom, a group of undergrads must each find their name in a row of boxes. The odds look dire—but we’ll use some probability theory and combinatorics to find a strategy that dramatically improves our chances. Can you help save our hapless mascot?

    Please go to the College Calendar to register.

  • 04
    March 4, 2021

    CMSA Quantum Matter in Mathematics and Physics: Generalized 't Hooft anomalies in vector-like theories

    10:30 AM-12:00 PM
    March 4, 2021

    ‘t Hooft anomalies provide a unique handle to study the nonperturbative infrared dynamics of strongly-coupled theories.  Recently, it has been realized that higher-form global symmetries can also become anomalous, leading to further constraints on the infrared dynamics.  In this talk, I show how one can turn on ‘t Hooft twists in the color, flavor, and baryon number directions in vector-like asymptotically-free gauge theories, which can be used to find new generalized ‘t Hooft anomalies. I give examples of the constraints the generalized anomalies impose on strongly-coupled gauge theories. Then, I argue that the anomaly inflow can explain a non-trivial intertwining that takes place between the light and heavy degrees of freedom on axion domain walls, which leads to the deconfinement of quarks on the walls.  This phenomenon can be explicitly seen in a weakly-coupled model of QCD compactified on a small circle.

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

  • 05
    March 5, 2021

    Irreducibility of periodic curves of cubic polynomials

    10:00 AM-12:00 PM
    March 5, 2021

    In the moduli space of one variable complex cubic polynomials with a marked critical point, given any $p \ge 1$, we prove that the loci formed by polynomials with the marked critical point periodic of period $p$ is an irreducible curve.  Our methods rely on techniques used to study one-complex-dimensional parameter spaces.  This is joint work with Matthieu Arfeux.

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

  • 06
    March 6, 2021
    No events