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Math Community Web Page

Following up on the #shutdownSTEM discussions, the Department of Mathematics has launched a community web page, with evolving content to be created through community effort.
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#shutdownSTEM–10 June 2020

In partnership with the Harvard Division of Science, the Mathematics Department used the call to #shutdownSTEM on 10 June to initiate a discussion of actions...
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< 2019 >
November
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  • Special loci of rank one local systems
    3:00 PM-4:00 PM
    November 5, 2019
    1 Oxford Street, Cambridge, MA 02138 USA

    Joint with Moritz Kerz. We show a structure theorem on those loci which enables us to deduce several consequences, one of which being Hard Lefschetz in rank one in positive characteristic.

  • The Gauss curvature flow – the shape of worn stones
    4:15 PM-5:15 PM
    November 5, 2019
    1 Oxford Street, Cambridge, MA 02138 USA

    The Gauss curvature flow was introduced to model the shape of worn stones. It is a parabolic Monge-Ampere type flow for convex hypersurfaces, involving regularity issues for fully nonlinear equations and free boundary problems, and involving several entropy. In this talk, we will discuss the free boundary problem concerning worn stone with flat sides, and the optimal regularity of non-concave fully nonlinear equations. When time permits, we also talk about the singularity analysis, especially about the relations to the affine geometry at the critical case, to fully nonlinear equations at the super-critical case, and to convex geometry and spectral analysis at the sub-critical case.

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  • Scalar Curvature and Intrinsic Flat Convergence
    10:30 AM-11:30 AM
    November 6, 2019

    After reviewing the definition of intrinsic flat convergence, I will present open problems on the almost rigidity or stability of classic rigidity theorems for manifolds with scalar curvature bounds and survey partial results towards these conjectures.

  • Purity for flat cohomology
    3:00 PM-4:00 PM
    November 6, 2019
    1 Oxford Street, Cambridge, MA 02138 USA

    The absolute cohomological purity for étale cohomology of Gabber–Thomasson implies that an étale cohomology class on a regular scheme extends uniquely over a closed subscheme of large codimension. I will discuss the corresponding phenomenon for flat cohomology. The talk is based on joint work with Peter Scholze.

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  • Renyi entanglement entropy in quantum many-body systems
    3:30 PM-4:30 PM
    November 12, 2019
    17 Oxford Street, Cambridge, MA 02138 USA

    We present two concrete examples where the Renyi rather than just the von Neumann entanglement entropy is necessary in order to obtain certain insights into quantum many-body systems.

    In the first example, we consider systems supporting ballistic information propagation and diffusive transport. It is well known that the linear-in-time growth of the von Neumann entanglement entropy (starting from a product state) is a probe of the former. Perhaps surprisingly, we show that the Renyi entanglement entropy (with Renyi index greater than 1) grows diffusively (i.e., as a square root of time) and is consequently a probe of the latter.

    In the second example, we study the problem of approximating local properties of a quantum many-body state using matrix product and projected entangled pair representations in one and two dimensions, respectively. We prove that area laws for the Renyi entanglement entropy (with Renyi index less than 1) lead to nontrivial upper bounds on the bond dimension. The bounds only depend on the accuracy of the desired approximation but not the system size.

    References:

    https://arxiv.org/abs/1902.00977

    https://arxiv.org/abs/1903.10048

  • On the SYZ Mirror Symmetry of Log Calabi-Yau Surfaces
    4:15 PM-5:15 PM
    November 12, 2019
    1 Oxford Street, Cambridge, MA 02138 USA

    The Strominger-Yau-Zaslow conjecture predicts the existence of special Lagrangians fibration in Calabi-Yau manifolds near large complex structure limit. The SYZ conjecture has been an important guiding principle for mirror symmetry and many of the implications are verified. In this talk, I will report on the recent progress on the SYZ fibration on certain log Calabi-Yau surfaces using the Lagrangian mean curvature flow and the theory of J-holomorphic curves. As a bi-product, we produce many new special Lagrangian submanifolds. I will also explain the applications in mirror symmetry, including the tropical/holomorphic correspondence for log Calabi-Yau surfaces and a mathematical realization of renormalization process of Hori-Vafa. Part of the talk is based on the joint work with T. Collins, A. Jacob and S-C. Lau, T-J. Lee.

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  • Some congruences and consequences in number theory and beyond
    4:00 PM-6:00 PM
    November 14, 2019
    1 Oxford Street, Cambridge, MA 02138 USA

    In the mid-1800s, Kummer observed some striking congruences between certain values of the Riemann zeta function, which have important consequences in algebraic number theory, in particular for unique factorization in certain rings. In spite of its potential, this topic lay mostly dormant for nearly a century until it was revived by Iwasawa in the mid-1950s. Since then, advances in arithmetic geometry and number theory (in particular, for modular forms, certain analytic functions that play a central role in number theory) have enabled substantial extension to congruences in the context of other arithmetically significant data, and this has remained an active area of research. In this talk, I will survey old and new tools for studying such congruences. I will conclude by introducing some unexpected challenges that arise when one tries to take what would seem like immediate next steps beyond the current state of the art.

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  • Current Developments in Mathematics 2019
    All day
    November 22, 2019-November 23, 2019
    1 Oxford Street, Cambridge, MA 02138 USA
    Organized By:

    David Jerison, Paul Seidel, Nike Sun (MIT)
    Denis Auroux, Mark Kisin, Lauren Williams, Horng-Tzer Yau, Shing-Tung Yau (Harvard)

    Registration

    Speakers:

    Svetlana Jitomirskaya (University of California)
    Subash Khot (NYU)
    Jun Li (Stanford)
    Andre Neves (University of Chicago)
    Geordie Williamson (University of Sydney)

    Talks on Youtube

    Dates

    Friday, November 22, 2019 Lecture Hall C
    Saturday, November 23, 2019 Lecture Hall C

    Travel

    For directions click here.

    Support

    Limited funding to help defray travel expenses is available for graduate students and recent PhDs. Please check later for more information.
    1. A letter indicating your name, address, current status, university affiliation, citizenship, and area of study. F1 visa holders are eligible to apply for support.
    2. If you are a graduate student, please send a brief letter of recommendation from a faculty member to explain the relevance of the conference to your studies or research.

    Sponsor

    The event is sponsored by the National Science Foundation NSF, Harvard University and MIT

    Hotels

    For travel information, click here.
    Sheraton Commander
    16 Garden Street, Cambridge MA 02138
    telephone: (617) 547-4800
    hotel-Harvard campus: 5-10 minute walk
    Irving House
    24 Irving St., Cambridge MA 02138
    telephone: (617) 547-4600,
    hotel-Harvard campus: 5-10 minute walk
    Friendly Inn(B + B)
    1673 Cambridge St., Cambridge, MA 02138
    telephone: (617) 547-7851
    hotel-Harvard campus: 5-10 minute walk
    Courtyard Marriott Cambridge
    777 Memorial Drive
    Cambridge, Massachusetts 02139 USA
    617-492-7777, 1-800-321-2211
    15-minute walk to Harvard Square,
    Free shuttle to Harvard
    Hotel Tria
    220 Alewife Brook Pkwy
    Cambridge, Massachusetts, 02138-1102
    (617) 491-8000, outside Harvard Square.
    10 minute walk to public transportation
    Free Shuttle bus to Harvard

    Information

    Please address any questions to Maureen Armstrong at cdm@math.harvard.edu

    Last update: Last update: August 17, 2019.

    “Harvard University is committed to maintaining a safe and healthy educational and work environment in which no member of the University community is, on the basis of sex, sexual orientation, or gender identity, excluded from participation in, denied the benefits of, or subjected to discrimination in any University program or activity. More information can be found here“.

  • Universal signatures of a black hole’s photon ring
    10:30 AM-11:30 AM
    November 22, 2019

    The Event Horizon Telescope image of the supermassive black hole in the galaxy M87 is dominated by a bright, unresolved ring. General relativity predicts that embedded within this image lies a thin “photon ring,” which is composed of an infinite sequence of self-similar subrings that are indexed by the number of photon orbits around the black hole. The subrings approach the edge of the black hole “shadow,” becoming exponentially narrower but weaker with increasing orbit number, with seemingly negligible contributions from high order subrings. In the talk, I will discuss the structure of the photon ring, starting with non-rotating black holes, and then proceeding to the complex patterns that emerge when rotation is taken into account. Subsequently I will argue that the subrings produce strong and universal signatures on long interferometric baselines. These signatures offer the possibility of precise measurements of black hole mass and spin, as well as tests of general relativity, using only a sparse interferometric array.

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  • Current Developments in Mathematics 2019
    All day
    November 23, 2019-November 23, 2019
    1 Oxford Street, Cambridge, MA 02138 USA
    Organized By:

    David Jerison, Paul Seidel, Nike Sun (MIT)
    Denis Auroux, Mark Kisin, Lauren Williams, Horng-Tzer Yau, Shing-Tung Yau (Harvard)

    Registration

    Speakers:

    Svetlana Jitomirskaya (University of California)
    Subash Khot (NYU)
    Jun Li (Stanford)
    Andre Neves (University of Chicago)
    Geordie Williamson (University of Sydney)

    Talks on Youtube

    Dates

    Friday, November 22, 2019 Lecture Hall C
    Saturday, November 23, 2019 Lecture Hall C

    Travel

    For directions click here.

    Support

    Limited funding to help defray travel expenses is available for graduate students and recent PhDs. Please check later for more information.
    1. A letter indicating your name, address, current status, university affiliation, citizenship, and area of study. F1 visa holders are eligible to apply for support.
    2. If you are a graduate student, please send a brief letter of recommendation from a faculty member to explain the relevance of the conference to your studies or research.

    Sponsor

    The event is sponsored by the National Science Foundation NSF, Harvard University and MIT

    Hotels

    For travel information, click here.
    Sheraton Commander
    16 Garden Street, Cambridge MA 02138
    telephone: (617) 547-4800
    hotel-Harvard campus: 5-10 minute walk
    Irving House
    24 Irving St., Cambridge MA 02138
    telephone: (617) 547-4600,
    hotel-Harvard campus: 5-10 minute walk
    Friendly Inn(B + B)
    1673 Cambridge St., Cambridge, MA 02138
    telephone: (617) 547-7851
    hotel-Harvard campus: 5-10 minute walk
    Courtyard Marriott Cambridge
    777 Memorial Drive
    Cambridge, Massachusetts 02139 USA
    617-492-7777, 1-800-321-2211
    15-minute walk to Harvard Square,
    Free shuttle to Harvard
    Hotel Tria
    220 Alewife Brook Pkwy
    Cambridge, Massachusetts, 02138-1102
    (617) 491-8000, outside Harvard Square.
    10 minute walk to public transportation
    Free Shuttle bus to Harvard

    Information

    Please address any questions to Maureen Armstrong at cdm@math.harvard.edu

    Last update: Last update: August 17, 2019.

    “Harvard University is committed to maintaining a safe and healthy educational and work environment in which no member of the University community is, on the basis of sex, sexual orientation, or gender identity, excluded from participation in, denied the benefits of, or subjected to discrimination in any University program or activity. More information can be found here“.

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  • Exact bosonization in higher dimensions and the duality between supercohomology fermionic SPT and higher-group bosonic SPT phases
    10:30 AM-11:30 AM
    November 25, 2019
    1 Oxford Street, Cambridge, MA 02138 USA

    The first part of this talk will introduce generalized Jordan–Wigner transformation on arbitrary triangulation of any simply connected manifold in 2d, 3d and general dimensions. This gives a duality between all fermionic systems and a new class of  lattice gauge theories. This map preserves the locality and has an explicit dependence on the second Stiefel–Whitney class and a choice of spin structure on the manifold. In the Euclidean picture, this mapping is exactly equivalent to introducing topological terms (Chern-Simon term in 2d or the Steenrod square term in general) to the Euclidean action. We can increase the code distance of this mapping, such that this mapping can correct all 1-qubit and 2-qubits errors and is useful for the simulation of fermions on the quantum computer. The second part of my talk is about SPT phases. By the boson-fermion duality, we are able to show the equivalent between any supercohomology fermionic SPT and some higher-group bosonic SPT phases. Particularly in (3+1)D, we have constructed a unitary quantum circuit for any supercohomology fermionic SPT state with gapped boundary construction. This fermionic SPT state is derived by gauging higher-form symmetry in the higher-group bosonic SPT and ungauging the fermion parity. The bulk-boundary correspondence in (3+1)D fermion SPT phases will also be briefly discussed.

  • From para-Hermitian geometry to mirror symmetry
    12:00 PM-1:00 PM
    November 25, 2019
    1 Oxford Street, Cambridge, MA 02138 USA

    In recent years, para-Hermitian geometry has been used to describe T-duality covariant spacetimes for string theory. In my talk, I will present applications of para-Hermitian geometry to 2D (2,2) SUSY sigma models and show that this geometry gives rise to a new, yet unexplored, notion of mirror symmetry.

  • From Quantum Field Theory to Geometric Representation Theory
    3:00 PM-4:00 PM
    November 25, 2019
    1 Oxford Street, Cambridge, MA 02138 USA

    Quantum Field Theory is a framework of fundamental physics, which in particular has played important roles in the modern development of various subjects in mathematics, including enumerative geometry, knot theory, and low-dimensional topology. On the other hand, Geometric Representation Theory is a subject in mathematics that studies a linear model of various types of symmetries using powerful techniques of algebraic geometry. In recent years, there has been much progress relating the two subjects, enriching the subject of Geometric Representation Theory. In this colloquium style talk, we will review some recent advancements on the topic. No prior knowledge of either Quantum Field Theory or Geometric Representation Theory will be assumed.

  • Communication Complexity of Randomness Manipulation
    4:30 PM-5:30 PM
    November 25, 2019
    1 Oxford Street, Cambridge, MA 02138 USA
    The task of manipulating randomness has been a subject of intense investigation in the theory of computer science. The classical definition of this task consider a single processor massaging random samples from an unknown source and trying to convert it into a sequence of uniform independent bits.
    In this talk I will talk about a less studied setting where randomness is distributed among different players who would like to convert this randomness to others forms with relatively little communication. For instance players may be given access to a source of biased correlated bits, and their goal may be to get a common random bit out of this source. Even in the setting where the source is known this can lead to some interesting questions that have been explored since the 70s with striking constructions and some surprisingly hard questions. After giving some background, I will describe a recent work which explores the task of extracting common randomness from correlated sources with bounds on the number of rounds of interaction.
    Based on joint works with Mitali Bafna (Harvard), Badih Ghazi (Google) and Noah Golowich (Harvard).
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  • Ancient solutions to the Mean Curvature Flow and applications
    4:15 PM-5:15 PM
    November 26, 2019
    1 Oxford Street, Cambridge, MA 02138 USA

    In the last 35 years, geometric flows have proven to be a powerful tool in geometry and topology. The Mean Curvature Flow is, in many ways, the most natural flow for hypersurfaces in Euclidean space. In this talk, which will assume no prior knowledge, I will present recent progress in classifying ancient solutions to the mean curvature flow (including joints work with Kyeongsu Choi, Robert Haslhofer and Brian White). I will also explain how this classification assists in answering fundamental questions regarding the singularity formation of the flow, and describe what are the remaining challenges in converting the mean curvature flow into the powerful tool we hope it can become.

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