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UID:20260624T082129EDT-7825kHUJsC@132.216.98.100
DTSTAMP:20260624T122129Z
DESCRIPTION:Title: Towards higher dimensional Gromov compactness in $G_2$ a
 nd $Spin(7)$ manifolds\n	Abstract: Let $(M\, \omega)$ be a compact symplect
 ic manifold. If we choose a compatible almost complex structure $J$ (which
  in general is not integrable) then we can study the space of $J$-holomorp
 hic maps $u : \Sigma \to (M\, J)$ from a compact Riemann surface into $M$.
  By appropriately “compactifying” the space of such maps\, one can obtain 
 powerful global symplectic invariants of $M$. At the heart of such a compa
 ctification procedure is understanding the ways in which sequences of such
  maps can degenerate\, or develop singularities. Crucial ingredients are c
 onformal invariance and an energy identity\, which lead to to a plethora o
 f analytic consequences\, including: (i) a mean value inequality\, (ii) in
 terior regularity\, (iii) a removable singularity theorem\, (iv) an energy
  gap\, and (v) compactness modulo bubbling. Riemannian manifolds with clos
 ed $G_2$ or $Spin(7)$ structures share many similar properties to such alm
 ost Kahler manifolds. In particular\, they admit analogues of $J$-holomorp
 hic curves\, called associative and Cayley submanifolds\, respectively\, w
 hich are calibrated and hence homologically volume-minimizing. A programme
  initiated by Donaldson-Thomas and Donaldson-Segal aims to construct simil
 ar such “counting invariants” in these cases. In 2011\, a somewhat overloo
 ked preprint of Aaron Smith demonstrated that such submanifolds can be exh
 ibited as images of a class of maps $u : \Sigma \to M$ satisfying a confor
 mally invariant first order nonlinear PDE analogous to the Cauchy-Riemann 
 equation\, which admits an energy identity involving the integral of highe
 r powers of the pointwise norm $|du|$. I will discuss joint work with Da R
 ong Cheng (Chicago) and Jesse Madnick (McMaster) in which we establish the
  analogous analytic results of (i)-(v) in this setting. arXiv:1909.03512\n
 	\n	\n	 \n
DTSTART:20191106T183000Z
DTEND:20191106T193000Z
LOCATION:Room 920\, Burnside Hall\, CA\, QC\, Montreal\, H3A 0B9\, 805 rue 
 Sherbrooke Ouest
SUMMARY:Spiro Karigiannis (University of Waterloo)
URL:/mathstat/channels/event/spiro-karigiannis-univers
 ity-waterloo-302207
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