Logic Seminar
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Thu, 29/04/2010 17:00 |
Franz-Viktor Kuhlmann (Saskatoon) |
Logic Seminar  |
L3 |
| Elimination of wild ramification is used in the structure theory of valued function fields, with applications in areas such as local uniformization (i.e., local resolution of singularities) and the model theory of valued fields. I will give a survey on the role that Artin-Schreier extensions play in the elimination of wild ramification, and corresponding main theorems on the structure of valued function fields. I will show what these results tell us about local uniformization. I have shown that local uniformization is always possible after a separable extension of the function field of the algebraic variety (separable "alteration"). This was extended to the arithmetic case in joint work with Hagen Knaf. Recently, Michael Temkin has proved local uniformization by purely inseparable alteration. Further, I will describe a classification of Artin-Schreier extensions with non-trivial defect. It can be used to improve one of the above mentioned main theorems ("Henselian Rationality"). This could be a key for a purely valuation theoretical proof of Temkin's result. On the other hand, the classification shows that separable alteration and purely inseparable alteration are just two ways to eliminate the critical defects. So the existence of these two seamingly "orthogonal" local uniformization results does not necessarily indicate that local uniformization without alteration is possible. | |||
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Thu, 06/05/2010 17:00 |
Luc Belair (Montreal/Paris) |
Logic Seminar |
L3 |
| We consider valued fields with a distinguished isometry or contractive derivation, as valued modules over the Ore ring of difference operators. This amounts to study linear difference/differential equations with respect to the distinguished isometry/derivation. Under certain assumptions on the residue field, but in all characteristics, we obtain quantifier elimination in natural languages, and the absence of the independence property. We will consider other operators of interest. | |||
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Thu, 13/05/2010 16:00 |
Philip Habegger - (JOINT WITH NUMBER THEORY SEMINAR) (ETH Zurich) |
Logic Seminar |
L3 |
| Jointly with Number Theory Consider a family of abelian varieties whose base is an algebraic variety. The union of all torsion groups over all fibers of the family will be called the set of torsion points of the family. If the base variety is a point then the family is just an abelian variety. In this case the Manin-Mumford Conjecture, a theorem of Raynaud, implies that a subvariety of the abelian variety contains a Zariski dense set of torsion points if and only if it is itself essentially an abelian subvariety. This talk is on possible extensions to certain families where the base is a curve. Conjectures of André and Pink suggest considering "special points": these are torsion points whose corresponding fibers satisfy an additional arithmetic property. One possible property is for the fiber to have complex multiplication; another is for the fiber to be isogenous to an abelian variety fixed in advance. We discuss some new results on the distribution of such "special points" on the subvarieties of certain families of abelian varieties. One important aspect of the proof is the interplay of two height functions. I will give a brief introduction to the theory of heights in the talk. | |||
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Thu, 20/05/2010 17:00 |
Dana Scott (Carnegie Mellon/Oxford) |
Logic Seminar |
L3 |
| As everyone knows, one popular notion of a (Scott-Ersov) domain is defined as a bounded complete algebraic cpo. These are closely related to algebraic lattices: (i) A domain becomes an algebraic lattice with the adjunction of an (isolated) top element. (ii) Every non-empty Scott-closed subset of an algebraic lattice is a domain. Moreover, the isolated (= compact) elements of an algebraic lattice form a semilattice (under join). This semilattice has a zero element, and, provided the top element is isolated, it also has a unit element. The algebraic lattice itself may be regarded as the ideal completion of the semilattice of isolated elements. This is all well known. What is not so clear is that there is an easy-to-construct domain of countable semilattices giving isomorphic copies of all countably based domains. This approach seems to have advantages over both the so-called "information systems" or more abstract lattice formulations, and it makes definitions of solutions to domain equations very elementary to justify. The "domain of domains" also has a natural computable structure | |||
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Thu, 27/05/2010 00:00 |
tba |
Logic Seminar |
L3 |
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Thu, 27/05/2010 17:00 |
Philip Scowcroft (Wesleyan/Oxford) |
Logic Seminar |
L3 |
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One may answer a question of Macintyre by showing that there are recursive existentially closed dimension groups. One may build such groups having most of the currently known special properties of finitely generic dimension groups, though no finitely generic dimension group is arithmetic.
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Thu, 03/06/2010 17:00 |
Andreas Doering (Oxford) |
Logic Seminar |
L3 |
| Standard quantum logic, as intitiated by Birkhoff and von Neumann, suffers from severe problems which relate quite directly to interpretational issues in the foundations of quantum theory. In this talk, I will present some aspects of the so-called topos approach to quantum theory, as initiated by Isham and Butterfield, which aims at a mathematical reformulation of quantum theory and provides a new, well-behaved form of quantum logic that is based upon the internal logic of a certain (pre)sheaf topos. | |||
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Thu, 10/06/2010 17:00 |
Katrin Tent (Muenster) |
Logic Seminar |
L3 |
| Simple groups of Lie type have a purely group theoretic characterization in terms of subgroup configurations. We here show that for certain Fraisse limits, the automorphism group is a simple group. | |||
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Thu, 17/06/2010 17:00 |
Martin Hills (Paris) |
Logic Seminar |
L3 |
| The n-amalgamation property has recently been explored in connection with generalised imaginaries (groupoid imaginaries) by Hrushovski. This property is useful when studying models of a stable theory together with a generic automorphism, e.g. elimination of imaginaries (e.i.) in ACFA may be seen as a consequence of 4-amalgamation (and e.i.) in ACF. The talk is centered around 4-amalgamation of stably dominated types in algebraically closed valued fields. I will show that 4-amalgamation holds in equicharacteristic 0, even for systems with 1 vertex non stably dominated. This is proved using a reduction to the stable part, where 4-amalgamation holds by a result of Hrushovski. On the other hand, I will exhibit an NIP (even metastable) theory with 4-amalgamation for stable types but in which stably dominated types may not be 4-amalgamated. | |||
