Tuesday September 25, 2018 4:00 PM  5:30 PM Math Tower P131
 Benjamin McMillan, Stony Brook University
Conservation laws and MongeAmpère parabolic equationsIn this talk I will describe how the geometry of an arbitrary parabolic second order equation governs the behavior of its conservation laws, and conversely, how the existence of a conservation law puts strong geometric restrictions on a parabolic equation. In particular, the class of MongeAmpère parabolic equations is very nongeneric, but I will nonetheless describe how the only parabolic equations with at least one conservation law are those of MongeAmpere type.

Tuesday October 02, 2018 4:00 PM  5:30 PM Math Tower P131
 Eric Woolgar, University of Alberta
Formal power series solutions of the Bach equationConformal gravity is an alternative to Einstein gravity in 4 dimensions, obtained by replacing the Einstein equation by the Bach equation, which has many more solutions. Maldacena has proposed that the theories are equivalent, provided one imposes certain boundary and physical conditions to remove the additional solutions of the Bach equation. We test this idea. Following the method laid out by Fefferman and Graham for the Einstein equation, we expand asymptotically hyperbolic solutions of the Bach equation in power series about conformal infinity, so as to identify the free data and find those data that yield Einstein metrics. There are infinitely many free data, reflecting the conformal invariance of the 4dimensional Bach equation, but even if we choose to break conformal invariance by imposing a constantscalarcurvature condition, the socalled mass aspect tensor remains freely specifiable.
In dimensions greater than 4, there are many different generalizations of the Bach tensor, most of which are not wellsuited to the FeffermanGraham method. We choose a wellsuited definition and find that the free data separate into two pairs of data, reflecting the separation of data for the Einstein equation into "Dirichlet" and "Neumann" data.
This talk is based on joint work with Aghil Alaee.

Tuesday October 23, 2018 4:00 PM  5:30 PM Math Tower P131
 Xujia Chen, Stony Brook University
Kontsevichtype recursions for counts of real curvesKontsevich's recursion, proved by RuanTian in the early 90s, enumerates rational curves in complex surfaces. Welschinger defined invariant signed counts of real rational curves in real surfaces (complex surfaces with a conjugation) in 2003. Solomon interpreted Welschinger's invariants as holomorphic disk counts in 2006 and proposed Kontsevichtype recursions for them in 2007, along with an outline for adapting RuanTian's homotopy style argument to the real setting. For many symplectic fourfolds, these recursions determine all invariants from basic inputs. We establish Solomon's recursions by reinterpreting his disk counts as degrees of relatively oriented pseudocycles from moduli spaces of stable real maps and lifting cobordisms from DeligneMumford moduli spaces of stable real curves.

Tuesday October 30, 2018 4:00 PM  5:30 PM Math Tower P131
 Yang Li, Imperial College
Dirichlet problem for maximal graphs of higher codimensionMaximal submanifolds in Lorentzian type ambient spaces are the formal analogues of minimal submanifolds in Euclidean spaces, which arise naturally in adiabatic problems for G2 manifolds. We obtain general existence and uniqueness results for the Dirichlet problem of graphical maximal submanifolds in any codimension, which stand in sharp contrast to the analogous problem for graphical minimal submanifolds.

Tuesday November 06, 2018 4:00 PM  5:30 PM Math Tower P131
 Ruobing Zhang, Stony Brook University
Compactness and bubbling phenomena of Bachflat manifoldsThis talk centers on the regularity and structure theory of Bachflat 4manifolds. We will introduce some recent study of the $σ_2$curvature equation on Bachflat 4manifolds. Specifically, we are interested in the limiting behavior of the solutions and also characterizing the bubble limits of the blowingup solutions. This is joint work with Alice Chang and Paul Yang.

Tuesday November 13, 2018 4:00 PM  5:30 PM Math Tower P131
 Demetre Kazaras, Stony Brook University
PSC bordism and the periodic etainvariantIn this talk, we will study spin manifolds equipped with metrics of positive scalar curvature (psc) and infinitecyclic covers. Considering the Dirac operator on the cover, one may define a periodic etainvariant. By establishing an index theorem for endperiodic operators, MrowkaRubermanSaveliev have shown that this invariant can distinguish path components in the moduli space of psc metrics. In this talk we go further, using a minimal hypersurface method to show that the periodic etainvariant can distinguish pscbordism classes in dimensions less than 7.

