Geometry/Topology Seminar

from Friday
June 01, 2018 to Monday
December 31, 2018
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Tuesday
September 25, 2018

4:00 PM - 5:30 PM
Math Tower P-131
Benjamin McMillan, Stony Brook University
Conservation laws and Monge-Ampère parabolic equations

In 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 Monge-Ampère parabolic equations is very non-generic, but I will nonetheless describe how the only parabolic equations with at least one conservation law are those of Monge-Ampere type.


Tuesday
October 02, 2018

4:00 PM - 5:30 PM
Math Tower P-131
Eric Woolgar, University of Alberta
Formal power series solutions of the Bach equation

Conformal 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 4-dimensional Bach equation, but even if we choose to break conformal invariance by imposing a constant-scalar-curvature condition, the so-called 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 well-suited to the Fefferman-Graham method. We choose a well-suited 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 P-131
Xujia Chen, Stony Brook University
Kontsevich-type recursions for counts of real curves

Kontsevich's recursion, proved by Ruan-Tian 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 Kontsevich-type recursions for them in 2007, along with an outline for adapting Ruan-Tian'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 re-interpreting his disk counts as degrees of relatively oriented pseudocycles from moduli spaces of stable real maps and lifting cobordisms from Deligne-Mumford moduli spaces of stable real curves.


Tuesday
October 30, 2018

4:00 PM - 5:30 PM
Math Tower P-131
Yang Li, Imperial College
Dirichlet problem for maximal graphs of higher codimension

Maximal 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 P-131
Ruobing Zhang, Stony Brook University
Compactness and bubbling phenomena of Bach-flat manifolds

This talk centers on the regularity and structure theory of Bach-flat 4-manifolds. We will introduce some recent study of the $σ_2$-curvature equation on Bach-flat 4-manifolds. Specifically, we are interested in the limiting behavior of the solutions and also characterizing the bubble limits of the blowing-up solutions. This is joint work with Alice Chang and Paul Yang.


Tuesday
November 13, 2018

4:00 PM - 5:30 PM
Math Tower P-131
Demetre Kazaras, Stony Brook University
PSC bordism and the periodic eta-invariant

In this talk, we will study spin manifolds equipped with metrics of positive scalar curvature (psc) and infinite-cyclic covers. Considering the Dirac operator on the cover, one may define a periodic eta-invariant. By establishing an index theorem for end-periodic operators, Mrowka-Ruberman-Saveliev 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 eta-invariant can distinguish psc-bordism classes in dimensions less than 7.


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