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  • March 8, 2021
    3:00 pm
    Virtually

    COLLOQUIUMS

    Speaker: Melissa (Chiu-Chu) Liu - Columbia University   Title: Special Colloquium
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  • CMSA EVENT: 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

  • COLLOQUIUMS

    COLLOQUIUMS
    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

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  • DIFFERENTIAL GEOMETRY SEMINAR

    DIFFERENTIAL GEOMETRY SEMINAR
    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

  • MATHEMATICAL PICTURE LANGUAGE SEMINAR

    MATHEMATICAL PICTURE LANGUAGE SEMINAR
    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 EVENT: 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

  • HARVARD-MIT ALGEBRAIC GEOMETRY SEMINAR

    HARVARD-MIT ALGEBRAIC GEOMETRY SEMINAR
    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

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  • CMSA EVENT: 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 EVENT: 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

  • NUMBER THEORY SEMINAR
    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.

  • OPEN NEIGHBORHOOD SEMINAR

    OPEN NEIGHBORHOOD SEMINAR
    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.

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  • CMSA EVENT: CMSA Mathematical Physics Seminar: Virasoro constraints for stable pairs
    10:00 AM-11:00 AM
    March 8, 2021

    The theory of stable pairs (PT) with descendents, defined on a 3-fold X, is a sheaf theoretical curve counting theory. Conjecturally, it is equivalent to the Gromov-Witten (GW) theory of X via a universal (but intricate) transformation, so we can expect that the Virasoro conjecture on the GW side should have a parallel in the PT world. In joint work with A. Oblomkov, A. Okounkov, and R. Pandharipande, we formulated such a conjecture and proved it for toric 3-folds in the stationary case. The Hilbert scheme of points on a surface S might be regarded as a component of the moduli space of stable pairs on S x P1, and the Virasoro conjecture predicts a new set of relations satisfied by tautological classes on S[n] which can be proven by reduction to the toric case.

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

  • COLLOQUIUMS

    COLLOQUIUMS
    Special Colloquium

    3:00 PM-4:00 PM
    March 8, 2021

    Title: Topological Recursion and Enumerative Geometry

    Abstract: Given a holomorphic curve in the complex 2-plane together with a suitably normalized symmetric meromorphic bilinear differential, the Chekhov-Eynard-Orantin Topological Recursion defines an infinite sequence of symmetric meromorphic multilinear differentials W_{g,n} on the curve. In many examples, the invariants W_{g,n} provide answers to enumerative problems. I will describe Topological Recursion and present several examples in which the answers are Hodge integrals (which are intersection numbers on moduli of curves) or Gromov-Witten invariants (which are virtual counts of holomorphic maps from Riemann surfaces to a Kahler manifold).

    Registration is required to receive the Zoom information

    Register here to attend

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  • COLLOQUIUMS

    COLLOQUIUMS
    Special Colloquium

    3:00 PM-4:00 PM
    March 16, 2021

    Title: Recent progress on random field Ising model

    Abstract: Random field Ising model is a canonical example to study the effect of disorder on long range order. In 70’s, Imry-Ma predicted that in the presence of weak disorder, the long-range order persists at low temperatures in three dimensions and above but disappears in two dimensions. In this talk, I will review mathematical development surrounding this prediction, and I will focus on recent progress on exponential decay and on correlation length in two dimensions. The talk is based on a joint work with Jiaming Xia and a joint work with Mateo Wirth.

    Registration is required to receive the Zoom information

    Register here to attend

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  • ALGEBRAIC DYNAMICS SEMINAR
    10:00 AM-12:00 PM
    March 19, 2021

    Let K be a complete and algebraically closed field, such as C or the p-adic field C_p, and let f\in K(z) be a rational function of degree d\geq 2. The map f is said to be hyperbolic if there is some metric on its Julia set with respect to which it is expanding. A celebrated 1983 theorem of Mane, Sad, and Sullivan shows that for K=C, hyperbolic maps are J-stable, meaning that nearby maps in moduli space have topologically conjugate dynamics on their Julia sets. In this talk, we show that if K is non-archimedean, an a priori weaker bounded-contraction condition also yields J-stability. This project is joint work with Junghun Lee.

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

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