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        "date_end":"2026-01-23",
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        "event_title":"A reflection principle for nonintersecting paths and lozenge tilings with free boundaries (Combinatorics seminar)",
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        "combined_title":"A reflection principle for nonintersecting paths and lozenge tilings with free boundaries (Combinatorics seminar): Seok Hyun Byun (Amherst College)",
        "event_subtitle":"Seok Hyun Byun (Amherst College)",
        "event_type":"Workshop \/ Seminar",
        "event_type_id":"21",
        "description":"Okada and Stembridge's Pfaffian formula for the enumeration of families of nonintersecting paths with fixed starting points and unfixed ending points has been widely used to resolve many challenging problems in enumerative combinatorics. In this talk, we present a new formula that complements Okada and Stembridge's Pfaffian formula. The combinatorial interpretation of the new formula gives a reflection principle for nonintersecting paths. It implies that the enumeration of families of nonintersecting paths with unfixed ending points can be resolved by enumerating families of nonintersecting paths with fixed ending points instead. Using this formula, we also show that the enumeration of lozenge tilings of a large family of regions with free boundaries can be deduced from those without free boundaries and present several applications of this result.",
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                "group_name":"Combinatorics Seminar - Department of Mathematics",
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    {
        "datetime_modified":"20260113T104609",
        "datetime_start":"20260123T150000",
        "datetime_end":"20260123T160000",
        "has_end_time":1,
        "date_start":"2026-01-23",
        "date_end":"2026-01-23",
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        "time_zone":"America\/Detroit",
        "event_title":"AIM Seminar:  Relativistic Initial Data Sets with Prescribed Asymptotics",
        "occurrence_title":"",
        "combined_title":"AIM Seminar:  Relativistic Initial Data Sets with Prescribed Asymptotics: James Wheeler (University of Michigan)",
        "event_subtitle":"James Wheeler (University of Michigan)",
        "event_type":"Lecture \/ Discussion",
        "event_type_id":"13",
        "description":"Abstract:  In General Relativity, Einstein's equation may be viewed as a second-order system of nonlinear PDE's for the Lorentzian metric on (n+1)-dimensional spacetime that admits an initial value formulation. The required initial data is comprised of a (usually complete) n-dimensional Riemannian manifold equipped with a symmetric (0,2)-tensor, respectively representing a spatially global \"instant in time\" and the \"initial velocity\" of the metric. The Gauss-Codazzi equations imply that not every such pair is admissible, however, and a working initial data set must satisfy certain nonlinear elliptic PDE\u2019s dubbed the constraint equations. In this talk, I will discuss recent work with several collaborators on the construction of working asymptotically flat initial data sets in which various asymptotic features of physical interest can be prescribed.\r\n\r\nContact:  AIM Seminar Organizers",
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