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SUMMARY:Sebastian Eterovic (UC Berkeley)
DTSTART:20200609T181500Z
DTEND:20200609T193000Z
DTSTAMP:20260422T212554Z
UID:BerkeleyModelTheory/1
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/BerkeleyMode
 lTheory/1/">Differential existential closedness for the $j$-function</a>\n
 by Sebastian Eterovic (UC Berkeley) as part of Berkeley model theory semin
 ar\n\n\nAbstract\nI will give a proof of the Existential Closedness conjec
 ture for the differential equation of the $j$-function and its derivatives
 . It states that in a differentially closed field certain equations involv
 ing the differential equation of the $j$-function have solutions. Its cons
 equences include a complete axiomatisation of $j$-reducts of differentiall
 y closed fields\, a dichotomy result for strongly minimal sets in those re
 ducts\, and a functional analogue of the Modular Zilber-Pink with Derivati
 ves conjecture. This is joint work with Vahagn Aslanyan and Jonathan Kirby
 .\n
LOCATION:https://researchseminars.org/talk/BerkeleyModelTheory/1/
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SUMMARY:Vahagn Aslanyan (University of East Anglia)
DTSTART:20200616T181500Z
DTEND:20200616T193000Z
DTSTAMP:20260422T212554Z
UID:BerkeleyModelTheory/2
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/BerkeleyMode
 lTheory/2/">Blurrings of the $j$-Function</a>\nby Vahagn Aslanyan (Univers
 ity of East Anglia) as part of Berkeley model theory seminar\n\n\nAbstract
 \nI will define blurred variants of the $j$-function and its derivatives\,
  where blurring is given by the action of a subgroup of $\\GL_2(\\C)$. For
  a dense subgroup (in the complex topology) I will prove an Existential Cl
 osedness theorem which states that all systems of equations in terms of th
 e corresponding blurred $j$ with derivatives have complex solutions\,  exc
 ept where there is a functional transcendence reason why they should not. 
 The proof is based on the Ax-Schanuel theorem and Remmert’s open mapping
  theorem from complex geometry. For the $j$-function without derivatives a
  stronger theorem holds\, namely\, Existential Closedness for $j$ blurred 
 by the action of a subgroup which is dense in $\\GL_2^+(\\R)$\, but not ne
 cessarily in $\\GL_2(\\C)$. In this case apart from the Ax-Schanuel theore
 m and some basic complex geometry we also use o-minimality in the proof. I
 f time permits\, I will also discuss some model theoretic properties of th
 e blurred $j$-function such as stability and quasiminimality. This is join
 t work with Jonathan Kirby.\n
LOCATION:https://researchseminars.org/talk/BerkeleyModelTheory/2/
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