Одељење за математику, 3. децембар 2021.

• 01. Децембар, 2021
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Наредни састанак Семинара биће одржан онлајн у петак, 3. децембра 2021. У оквиру састанка биће одржана два предавања.

Прво предавање биће одржано са почетком у 14.15 часова.

Предавач: Krzysztof Krupinski, University of Wroclaw

Наслов предавања: ON STABLE QUOTIENTS

Апстракт: By a hyper definable set we mean a quotient of a a type-definable set by a type-definable equivalence relation. There is a natural definition of stability of such an object (generalizing stable  type-definable sets). In [HP], Haskel and Pillay studied stable quotients of a 0-type-definable group G, assuming NIP. Their main result says that, under NIP, there always exists a smallest type-definable subgroup G^st with G/G^st stable. It is easy to see that G^st is normal and 0-type-definable.

We solve two problems from [HP]. Our first result says that if G is a 0-type-definable group in a distal theory, then G^st=G^00 (where G^00 is the smallest ⋀-definable subgroup of bounded index). In order to get it, we prove that distality is preserved under passing from T to the hyper imaginary expansion T^heq. The second result is an example of a group G definable in a non-distal, NIP theory for which G=G^00 but G^st is not an intersection of definable groups. Our example is a saturated extension of (R,+,[0,1]). Moreover, we make some observations on whether there is such an example which is a group of finite exponent. We also take the opportunity and give several characterizations of stability of hyper definable sets, involving continuous logic, but this part will not be discussed during the seminar, except stating a few equivalent definitions of stability which do not refer directly to continuous logic. This is a joint work with Adrian Portillo.

[HP] M. Haskel, A. Pillay, ''On maximal stable quotients of definable groups in NIP theories'', JSL 83 (2018), 117-122.
[KP] K. Krupinski, A Portillo, ''On stable quotients'', preprint, 2021.

Друго предавање биће одржано са почетком у 15.15 часова.

Предавач: Бобан Величковић, Université Paris Diderot

Наслов предавања: NON VANISHING HIGHER DERIVED LIMITS

Апстракт: In the study of strong homology Mardesic and Prasolov isolated a certain inverse system of abelian groups A indexed by functions from ω to ω. They showed that if strong homology is additive on a class of spaces containing closed subsets of Euclidean spaces then the higher derived limits lim^n A must vanish for n >0. They also proved that under the Continuum Hypothesis lim^1 A does not vanish. On the other hand Dow, Simon and Vaughan showed that under PFA lim^1 A=0. The question whether lim^n A vanishes for higher n has attracted considerable attention recently. First, Bergfalk shows that it was consistent lim^2 A does not vanish. Later Bergfalk and Lambie-Hanson showed that, assuming modest large cardinal axioms, lim^n A vanishes for all n. The large cardinal assumption was later removed by Bergfalk, Hrusak and Lambie-Hanson. We complete the picture by showing that, for any n>0, it is relatively consistent with ZFC that lim^n A is non zero. This is a joint work with Alessandro Vignati.

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