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UID:90607d022d98b96073a22e20f68f6012
CATEGORIES:Algebra Seminar
CREATED:20260429T111826
SUMMARY:Fake Projective Planes: From Existence to Explicit Equations
LOCATION:Hill 705
DESCRIPTION:<p>Since Mumford's 1979 construction proved the existence of fake projectiv
 e planes, a natural problem has been to make these surfaces explicit: how c
 an one actually write down defining equations for a fake projective plane? 
 The history of this question passes through several major stages. First cam
 e Mumford's original existential construction via p-adic uniformization. Th
 is was followed by the arithmetic classification program: results of Klingl
 er and Yeung showed that fake projective planes are arithmetic ball quotien
 ts, while subsequent work of Prasad-Yeung and Cartwright-Steger reduced the
  possibilities to a complete list of 50 cases up to isomorphism of fundamen
 tal groups, or 100 up to bi-holomorphism. A new phase began when Borisov an
 d Keum produced the first explicit equations for a fake projective plane, o
 pening the door to concrete geometric study of these surfaces. Since then, 
 further explicit constructions have been obtained by several authors. In th
 is talk I will survey this development, explain the main ideas behind the p
 assage from existence to equations, and discuss possible interactions with 
 fake quadrics, which may be viewed as a degree-two analogue in the broader 
 landscape of ``fake'' surfaces. A new phase began when Borisov and Keum pro
 duced the first explicit equations for a fake projective plane, opening the
  door to concrete geometric study of these surfaces. Since then, further ex
 plicit constructions have been obtained by several authors. In this talk I 
 will survey this development, explain the main ideas behind the passage fro
 m existence to equations, and discuss possible interactions with fake quadr
 ics, which may be viewed as a degree-two analogue in the broader landscape 
 of ``fake'' surfaces.</p>
CONTACT:Bojue Wang -Rutgers University
DTSTAMP:20260828T012404
DTSTART;TZID=America/New_York:20260429T140000
DTEND;TZID=America/New_York:20260429T150000
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