# Abstracts

Abstracts for the Departmental Seminar, fall, 2012:

## Lefschetz Fibrations of Some 4-Manifolds

*Mustafa Topkara, 2012.12.28, 14:30*

In this talk, we will investigate some methods to find Lefschetz fibrations with certain properties on a 4-manifold.

## Computing Galois Groups of Polynomials

*Emrah Çakçak, 2012.12.21, 14:30*

I will talk about principles of algorithms to compute Galois groups of polynomials. The talk will be accessible to students familiar with Galois theory of finite extensions of fields.

## Henselianty for fields and for some other structures

*Gönenç Onay, 2012.12.14, 14:30*

Roughly speaking, Ax-Kochen and Ershov theorem states that the
elementary theory of a valued field of characteristic (0,0) is encoded
it its value group and its residue field. This result implies the
following corrected form of a conjecture made by Artin: For each
positive integer d there is a finite set Y of prime numbers, such that
if p is any prime not in Y then every homogeneous polynomial of degree d
over the p-adic numbers in at least d^{2}+1 variables has a
nontrivial zero.

In this talk, after giving equivalent definitions of a henselian valuation on a field, I will explain how one can elementarily define an adapted notion of henselianty for some other structures. Namely, I will give two examples: valued modules over PID (:not necessarily commutative) and valued difference fields. Then I will expose Ax-Kochen and Ershov type theorems for these structures.

## Epsilon factors of the representations of the Weil group

*Şafak Özden, 2012.12.07, 14:30*

The epsilon factors of the representations of the Weil group of a local field encodes the basic arithmetic information of the local field K and is key factor to define the functional equation of the Artin-Weil L-function. Although it is central, nothing explicit is known about these factors, except for the case of representations of dimension 1. The aim of this talk is to discuss the epsilon factors attached to the representations of the absolute Weil group of a local field K.

## Generically transitive actions in the finite Morley rank setting

*Ayşe Berkman, 2012.11.30, 14:30*

Sharply multiply transitive actions are rare in the finite setting, and non-existent in the infinite setting. However, when you look from the model theoretic point of view, you see a more natural concept of multiple transitivity, which yields many examples. I shall try to explain this point of view and mention some classification results obtained jointly with Alexandre Borovik.

## The topology of model theory

*David Pierce, 2012.11.16–23, 14:30*

Model theory is a foundational subject. Like set theory and category theory, it provides a common language for talking about what different mathematicians do. Conversely, other parts of mathematics can illuminate features of model theory. This talk aims to present model theory in both ways: as foundational for mathematics, and as illuminated by other parts of mathematics.

One aim is then to present material that other model theorists should be able to assume in their own talks. This includes the notion of Morley rank, which can be seen as a special case of the topological notion of Cantor–Bendixson rank.

Another aim is to show how the compactness theorem of logic is really that a certain topological space is compact. As a consequence of the compactness theorem, this topological space is the full Stone space of a certain Boolean algebra (and is thus an example of the spectrum of a ring). Some model theory texts seem to suggest the converse, that the compactness theorem is a consequence of the compactness of Stone spaces; but this is incorrect.

## The Diagonal Property on Toric Surfaces

*Özer Öztürk, 2012.10.19, 14:30*

Let *X* be a smooth variety of dimension *n*. *X* is
said to satisfy the diagonal property if there exists a vector bundle
*E* of rank *n* on *X×X* and a section of *E*
such that the image of the diagonal embedding of *X* into
*X×X* is the zero scheme of *s*. We shall discuss the
diagonal property on toric surfaces after an itroduction to toric
varieties. This is a joint work with A.U.Ö. Kişisel.

## On generalized fixed point free automorphisms

*Kıvanç Ersoy,
2012.10.12, 14:30*

Thompson proved that a finite group with a fixed point free automorphism is solvable. In this talk we will define some generalizations of fixed point free automorphisms and prove some results. Moreover, we will discuss about the following question of Mazurov:

Let

Gbe an infinite periodic group with a fixed point free automorphism of prime order. CanGcontain an element of orderp?

## Enriching fields by multiplicative subgroups

*Ayhan Günaydın,
2012.09.28, 14:30, and 2012.10.05, 14:30*

After a quick introduction to model theoretic concepts needed in the rest of the talk, I will talk about the model theoretic properties (such as the structure of definable sets and stability) of expansions of a field by a ‘small’ subset. Here smallness is quite a technical (and temporary) term; however, it is implied by some natural examples arising from number theory, one example being the case of finitely generated multiplicative groups. I will mostly concentrate on the case of algebraically closed fields, but if time allows I will briefly mention the analogous work for the field of real numbers.