[MD-sorular] Matematik Koyu'nde Grup Teori

[Ali Nesin]

28 Eylul – 6 Ekim arasinda (yani Seker Bayrami’nda) Matematik Koyu’nde Grup Teorisi Calistayi yapilacak.

Toplam 9 gun.

Hedef Kitle: Ileri duzeyde lisans ogrencileri, lisansüstü öğrenciler ve arastirmacilar.

Calistayda konusma vermek ya da calistaya katilmak isteyenler bana yazabilirler.

Lisansustu ogrencileri problemlerini ve calismalarini sunabilirler.

Simdilik program asagidaki gibi. Daha da zenginlesecegini umuyoruz.

“Registration fee� yok. Ogrencilere gunde 20 YTL, ucretlilere 30 YTL. kalma ve gunde uc ogun yemek ve her sey dahil

Ali

1) Basic Advanced Group Theory by Ali Nesin: Sylow Theorems. Solvable and Nilpotent groups. Decomposition theorems. Hall’s Theorem. Permutation groups, sharply 2 and 3 transitive groups. Prerequisites: A first year abstract algebra course. 

2) Classical Groups by Ali Nesin. Special linear, symplectic and orthogonal groups. From scratch and as much as the time allows. Prerequisites: Linear algebra and basic group theory. 

3) Kleinian and Fuchsian Groups by Andrei Ratiu: These are discrete subgroups of PSL2(ℂ). Mobius transformations of the extended complex plane. Action of the Mobius transformations on the upper half-space in �3. Types of Mobius transformations. The definition and the main properties of Kleinian groups and Fuchsian groups. Prerequisites: Complex numbers and some rudiments of topology. 

4) Jordan Automorphisms of Some Radical Rings by Feride KuzucuoÄŸlu (one day). 

5) Centralizers in Locally Finite Simple Groups by Mahmut KuzucuoÄŸlu (one day). 

6) Sylow Subgroups of Some Classical Groups over Finite Fields by Nedim Narman (4 days).  

7) Infinite Galois Theory and Profinite Groups by Özlem Beyarslan (3 days). Inverse limits, p-adic numbers and the prüfer group. The absolute Galois group of a finite field. Prerequisites: Basic field theory. 

8) Compact Lie Groups by Selçuk Demir: Topological groups. Lie groups. Compact Lie groups. Tangent space and its Lie algebra structure. Exponential mapping. Maximal tori. Root systems, Weyl group, Dynkin diagram. Haar measure. Peter-Weyl Theorem. 

9) Property (T) Groups: by Talia Fernos (2 days). Property (T) is a property of groups that was introduced by Kazhdan in 1967 with the aim of showing that higher rank lattices are finitely generated. This is still the only proof that works for all such lattices. Simply stated, it means that any unitary representation which is “close'� to having invariant vectors must have invariant vectors. Although it can be seen as an analytical property and is not a quasi-isometric invariant, it has many geometrical consequences. Among these is that property (T) groups do not admit non-trivial actions on many “simple� spaces. Such spaces include trees, the circle (with a sufficiently smooth action), and walled spaces. In these talks we begin with a survey of property (T). We will then prove some nice consequences of property (T) to demonstrate its simplicity and strength. We will show that property (T) is in some sense orthogonal to amenability (although non-abelian free groups are neither) by showing that any amenable group with property (T) is necessarily finite. We will also show that property (T) groups are finitely generated, quotients of finitely presented (T)-groups, and that they don't act on trees. If time, we will also discuss why it is not invariant under quasi-isometry. 

 

 

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