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UID:e09778b67270ec68b085326bf1db0cf2
CATEGORIES:Geometric Analysis Seminar
CREATED:20220425T092719
SUMMARY:Variations of the Z2-Dirac operator
LOCATION:Zoom
DESCRIPTION:Abstract: It is a classical result that on a compact 3-manifold $Y$, the se
 t of metrics for which there exists a harmonic spinor of the spin Dirac ope
 rator is a codimension 1 subset in the space of all metrics. In this talk, 
 I will discuss an extension of this result to the case of the $mathbb Z_2$-
 Dirac operator, which is defined as the Dirac operator on the complement of
  a codimension 2 submanifold $mathcal Zsubseteq Y$ twisted by a flat connec
 tion on $Y-mathcal Z$ whose holonomy lies in $mathbb Z_2$. The deformation 
 problem for solutions of this operator carries an infinite-dimensional obst
 ruction for a fixed $mathcal Z$. Coupling the operator to the geometry of $
 mathcal Z$  by considering the infinite-dimensional family of Dirac operato
 rs parameterized by embedded submanifolds gives rise to a Fredholm problem 
 up to a loss of regularity phenomenon. The proof of the result then require
 s the use of the Nash-Moser Implicit Function Theorem or related techniques
 .\n
X-ALT-DESC;FMTTYPE=text/html:<p>Abstract: It is a classical result that on a compact 3-manifold $Y$, the
  set of metrics for which there exists a harmonic spinor of the spin Dirac 
 operator is a codimension 1 subset in the space of all metrics. In this tal
 k, I will discuss an extension of this result to the case of the $mathbb Z_
 2$-Dirac operator, which is defined as the Dirac operator on the complement
  of a codimension 2 submanifold $mathcal Zsubseteq Y$ twisted by a flat con
 nection on $Y-mathcal Z$ whose holonomy lies in $mathbb Z_2$. The deformati
 on problem for solutions of this operator carries an infinite-dimensional o
 bstruction for a fixed $mathcal Z$. Coupling the operator to the geometry o
 f $mathcal Z$&nbsp; by considering the infinite-dimensional family of Dirac
  operators parameterized by embedded submanifolds gives rise to a Fredholm 
 problem up to a loss of regularity phenomenon. The proof of the result then
  requires the use of the Nash-Moser Implicit Function Theorem or related te
 chniques.</p>
CONTACT:Gregory Jacob Parker (MIT)
X-EXTRAINFO:Zoom link: https://rutgers.zoom.us/j/97953490430?pwd=RmxtS0pnVEFmaEc5U2tlY3
 NDdW92Zz09\nMeeting ID: 979 5349 0430\nPassword: 515087
DTSTAMP:20260828T163205
DTSTART;TZID=America/New_York:20220426T145000
DTEND;TZID=America/New_York:20220426T155000
SEQUENCE:0
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