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SUMMARY:Sebastian Eterovic (UC Berkeley)
DTSTART;VALUE=DATE-TIME:20200609T181500Z
DTEND;VALUE=DATE-TIME:20200609T193000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080028Z
UID:BerkeleyModelTheory/1
DESCRIPTION:Title: Differential existential closedness for the $j$-function\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;VALUE=DATE-TIME:20200616T181500Z
DTEND;VALUE=DATE-TIME:20200616T193000Z
DTSTAMP;VALUE=DATE-TIME:20241016T080028Z
UID:BerkeleyModelTheory/2
DESCRIPTION:Title: Blurrings of the $j$-Function\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 Remmertâ€™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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