09:00 AM  09:15 AM


Welcome

 Location
 MSRI: Simons Auditorium
 Video


 Abstract
 
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09:15 AM  10:15 AM


Galois covers in positive characteristic
Rachel Pries (Colorado State University)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
 There are some interactions between model theory and ArtinSchreier extensions of fields. In this talk, I will discuss wildly ramified covers of curves in positive characteristic p.
The presence of an automorphism of order p leads to significant constraints on arithmetic invariants of the Jacobian. I will also discuss results about deformations of wildly ramified covers of curves in positive characteristic.
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10:15 AM  10:45 AM


Tea

 Location
 MSRI: Atrium
 Video


 Abstract
 
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10:45 AM  11:45 AM


Hilbert's Tenth Problem and Mazur's conjectures in large subrings of number fields
Kirsten Eisentraeger (Pennsylvania State University)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
Hilbert's Tenth Problem in its original form was to find an algorithm to decide, given a multivariate polynomial equation with integer coefficients, whether it has a solution over the integers. In 1970 Matiyasevich, building on work by Davis, Putnam and Robinson, proved that no such algorithm exists, i.e. Hilbert's Tenth Problem is undecidable. In this talk we will consider generalizations of Hilbert's Tenth Problem and Mazur's conjectures for large subrings of number fields. We will show that Hilbert's Tenth Problem is undecidable for large complementary subrings of number fields and that the analogues of Mazur's conjectures do not hold in these rings
 Supplements


11:45 AM  12:30 PM


A universal firstorder formula for the ring of integers inside a number field
Jennifer Park (Ohio State University)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
Hilbert's tenth problem over the rational numbers Q (or, any number field K) asks the following: given a polynomial in several variables with coefficients in Q (resp. K), is there a general algorithm that decides whether this polynomial has a solution in Q (resp. K)? Unlike the classical Hilbert's tenth problem over Z, this problem is still unresolved. To reduce this problem to the classical problem, we need a definition of Z in Q (resp. ring of integers in K) using only an existential quantifier. This problem is still open. I will present a definition of the ring of integers in a number field, which uses only one universal quantifier, which is, in a sense, the simplest logical description that we can hope for. This is a generalization of Koenigsmann's work, which defines Z in Q using one universal quantifier.
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12:30 PM  02:00 PM


Lunch

 Location
 MSRI: Atrium
 Video


 Abstract
 
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02:00 PM  03:00 PM


Lehmer problem and applications
Maria Carrizosa (Universite Lyon 1)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
Lehmer conjecture predicts a lower bound for the height of a non zero algebraic integer wich is not a root of unity in terms of its degree. In this talk, we explain how the relative Lehmer problem is related to the ZilberPink conjecture. The idea comes from the rst article of Bombieri, Masser and Zannier on the subject : they used the lower bounds for the height given by the Lehmer problem to proove that the bounded height subset of Gn m that they were interested in, was nite. Indeed, Lehmer type bounds are used to proove that such a subset has bounded degree and by Northcott property they conlcuded niteness. We'll explain how this argument works in the context of abelian varieties.
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03:00 PM  03:30 PM


Tea

 Location
 MSRI: Atrium
 Video


 Abstract
 
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03:30 PM  04:15 PM


Counting algebraic points on definable sets
Margaret Thomas (Purdue University)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
The counting theorem of Pila and Wilkie (2006) has led to a lively interaction between ominimality and diophantine geometry in recent years. Part of this interaction has focussed further on the problem of bounding the density of rational and algebraic points lying on transcendental sets, seeking to sharpen the PilaWilkie bound in certain cases. We shall survey some results in this area, focussing on a conjecture of Wilkie which concerns the real exponential field. Time permitting, we shall also look at some more recent results (by, amongst others, Besson, Boxall, Jones, Masser, and myself) which concern a variety of classical functions, including the Riemann zeta function, Weierstrass zeta functions and Euler's gamma function
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04:15 PM  05:15 PM


Panel Discussion

 Location
 MSRI: Simons Auditorium
 Video


 Abstract
 
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06:30 PM  08:30 PM


Dinner at Taste of the Himalayas

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