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Big Data Conference 2024
September 6, 2024 - September 7, 2024      9:00 am
https://cmsa.fas.harvard.edu/event/bigdata_2024/   On  September 6-7, 2024, the CMSA will host the tenth annual Conference on Big Data. The Big Data Conference features speakers from the...
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< 2018 >
October
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  • GAUGE-TOPOLOGY-SYMPLECTIC SEMINAR: Construction of symplectic 4-manifolds and Lefschetz fibrations via Luttinger surgery

    GAUGE-TOPOLOGY-SYMPLECTIC SEMINAR
    Construction of symplectic 4-manifolds and Lefschetz fibrations via Luttinger surgery

    Speaker: Anar Akhmedov – University of Minnesota, visiting scholar @Harvard

    3:30 PM-4:30 PM
    October 5, 2018
    Science Center 507
    1 Oxford Street, Cambridge, MA 02138 USA
    Abstract:

    Luttinger surgery is a certain type of Dehn surgery along a Lagrangian torus in a symplectic 4-manifold. The surgery was introduced by Karl Luttinger in 1995, who used it to study Lagrangian tori in R^4. Luttinger’s surgery has been very effective tool recently for constructing exotic smooth and symplectic structures on 4-manifolds. In this talk, using Luttinger surgery, I will present the constructions of smallest known closed exotic simply connected symplectic 4-manifolds and exotic Lefschetz fibrations over 2-sphere whose total spaces have arbitrary finitely presented group G as the fundamental group. If time permits, we will also discuss some applications to the geography of symplectic 4-manifolds and Lefschetz fibrations. Part of these are (separate) joint works with B. Doug Park, Burak Ozbagci, and Naoyuki Monden.

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  • GAUGE-TOPOLOGY-SYMPLECTIC SEMINAR: Lojasiewicz inequalities and Morse-Bott functions

    GAUGE-TOPOLOGY-SYMPLECTIC SEMINAR
    Lojasiewicz inequalities and Morse-Bott functions

    Speaker: Paul Feehan – Rutgers

    3:30 PM-4:30 PM
    October 12, 2018
    Science Center 507
    1 Oxford Street, Cambridge, MA 02138 USA
    Abstract:

    The Lojasiewicz gradient and distance inequalities for real analytic functions on Euclidean space were first proved by Stanislaw Lojasiewicz (1965) using methods of semianalytic and subanalytic sets, arguments later simplified by Bierstone and Milman. We shall first describe a more direct proof of the Lojasiewicz gradient inequality that uses resolution of singularities for real analytic varieties to reduce to the case of functions with simple normal crossings, where the Lojasiewicz exponent may be computed explicitly — thus giving insight into its geometric meaning. It is well-known and easy to prove that if a function on a Banach space is Morse-Bott, then its Lojasiewicz exponent is 1/2. We show that the less obvious converse is also true: if the Lojasiewicz exponent of an analytic function on a Banach space is 1/2 at a critical point, then the function is Morse-Bott on a neighborhood of that point. We illustrate these phenomena with applications of Lojasiewicz inequalities to the Yang-Mills energy function near the critical set of flat connections on a principal G-bundle over a closed Riemannian manifold.

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  • GAUGE-TOPOLOGY-SYMPLECTIC SEMINAR: Equivariant symplectic capacities

    GAUGE-TOPOLOGY-SYMPLECTIC SEMINAR
    Equivariant symplectic capacities

    Speaker: Michael Hutchings – Berkeley

    3:30 PM-4:30 PM
    October 26, 2018
    Science Center 507
    1 Oxford Street, Cambridge, MA 02138 USA
    Abstract:

    We define a sequence of symplectic capacities using positive S^1-equivariant symplectic homology. These capacities are conjecturally equal to the Ekeland-Hofer capacities. However they satisfy axioms which allow them to be computed for many examples, such as convex toric domains. They also give sharp obstructions to symplectic embeddings of cubes into convex toric domains. The asymptotics of these capacities are conjecturally related to Lagrangian embeddings. This is a joint work with Jean Gutt.

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