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Mini-courses on arithmetic geometry
Basel, June 1-3, 2015
The school will be held in Basel from June 1 to 3, 2015.
Mini-courses
There will be three mini-courses of 4 hours (4 times one hour each).
Pietro CORVAJA (Udine)
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Torsion points |
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Ulrich DERENTHAL (Hannover) |
Cox rings and rational points
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Philipp HABEGGER (Basel) |
Diophantine Problems and o-Minimal Structures |
We will also have some exercise sessions.
Schedule Talks: In Basel, Spiegelgasse 1 (ground floor for the talks, 6th floor for coffee break / welcome)
Monday
June 1
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Tuesday
June 2
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Wednesday
June 3 |
9h30
welcome
10h15-11h15
P. Habegger
11h30-12h30
P. Corvaja
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9h-10h
P. Corvaja
coffee break
10h30-11h30
U. Derenthal
11h45-12h45
P. Habegger
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9h-10h
P. Habegger
coffee break
10h30-11h30
P. Corvaja
11h45-12h45
U. Derenthal
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14h-15h
U. Derenthal
15h15-16h15
P. Habegger
coffee break
16h45-17h45
Exercises
18h30
Apéritif |
14h-15
P. Corvaja
15h15-16h15
U. Derenthal
coffee break
16h45-17h45
Exercises
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14h-15
Exercises
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Mini-courses - titles and abstracts
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Pietro CORVAJA - Torsion points |
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Several problems in mathematics, having diverse origins, amount to detecting
when some element of an algebraic group has finite order, or more generally
to finding torsion elements in a given variety. Well-known finiteness
statements were conjectured by Manin, Mumford and Lang for torsion points on jacobians, and eventually proved.
The mini-course will treat several problems related to torsion points on varieties, and show many connections with seemingly unrelated topics.
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Ulrich DERENTHAL - Cox rings and rational points |
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Fano varieties (cubic surfaces, for example) often contain infinitely many rational points. To obtain more precise quantitative information, one may consider the heights of the rational points and ask for the asymptotic behavior of the number of rational points of bounded height, as the bound tends to infinity. This is predicted precisely by Manin's conjecture. We will focus on the approach to Manin's conjecture via universal torsors, which can be made explicit using Cox rings. Time permitting, we will discuss some recent developments, namely the universal torsor method over number fields beyond the rational numbers, and the construction of Cox rings over nonsplit varieties.
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Philipp HABEGGER - Diophantine Problems and o-Minimal Structures |
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Important solved and unsolved problems in diophantine geometry involve describing the distribution of "special points" on a given ambient variety.
This ambient variety can be a certain type of commutative algebraic
group and in which case the special points are those of finite order.
One basic incarnation is to describe all points (x,y) where x and y are
roots of unity with x+y=1. While this particular problem is
solvable using elementary geometry, its natural generalization, the
Manin-Mumford Conjecture, requires more work.
A few years ago, Zannier devised a new strategy to attack such problems
using the theory of o-minimal structures. Originally introduced by model-theorist
in mathematical logic, o-minimal structures provide a powerful new
tool to solve problems in diophantine geometry. They made possible
a new proof of the Manin-Mumford Conjecture by Pila
and Zannier. Pila and others later solved new cases of the
André-Oort Conjecture.
In this course I will explain the diophantine problems mentioned above and
give an introduction to o-minimal structures. A particularly important
result is the theorem of Pila and Wilkie which bridges arithmetic and
model theory. If time permits we will also treat more recent
developments on conjectures on unlikely intersections.
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Participants
Giula Bianco (Neuchâtel)
Jérémy Blanc (Basel)
Martino Borello (EPFL)
Jung-Kyu Canci (Basel)
Gabriel Dill (Basel)
Linda Frey (Basel)
Michele Giacomini (Udine)
Isac Héden (Basel)
Mattias Hemmig (Basel)
Mathieu Huruguen (EPFL)
Nikita Kalinin (Geneva)
Pierre-Marie Poloni (Basel)
Maria Fernanda Robayo (Basel)
Alberto Ravagnani (Neuchâtel)
Andriy Regeta (Basel)
Harry Schmidt (Basel)
Stefan Schmid (Basel)
Mikhail Shkolnikov (Geneva)
Amos Turchet (Göteborg)
Christian Urech (Basel)
Solomon Vishkautsan (Pisa)
Susanna Zimmermann (Basel)
Send an email to Jeremy  Blanc  unibas ch if you would like to participate.
All students of the swiss doctoral program will be supported by the program. Other participants need to find support by another way, or can ask the organisers (some limited support is available).
Organisers
Philipp Habegger (Basel)
Jérémy Blanc (Basel)
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