Department of

# Mathematics

Seminar Calendar
for events the day of Thursday, December 17, 2015.

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events for the
events containing

Questions regarding events or the calendar should be directed to Tori Corkery.
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1  2  3  4  5  6  7          1  2  3  4  5                   1  2
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Thursday, December 17, 2015

1:00 pm in 345 Altgeld Hall,Thursday, December 17, 2015

#### Knot Complement Commensurability

###### Neil Hoffman (Melbourne)

Abstract: Two 3-manifolds are commensurable if they share a common finite sheeted cover. Commensurability partitions the set of hyperbolic 3-manifolds into equivalence classes, called commensurability classes. Hyperbolic knot complements appear to be rare in commensurability class, in fact Reid and Walsh have conjectured that there are at most three hyperbolic knot complements in a commensurability class. I will discuss recent progress on Reid and Walsh's conjecture with a focus on the open problems in this area.

4:30 pm in 245 Altgeld Hall,Thursday, December 17, 2015

#### Rigidity in von Neumann algebras

###### Ionut Chifan (University of Iowa)

Abstract: Back in the 30's F. Murray and J. von Neumann discovered a natural way to associate a von Neumann algebra, $L^\infty(X)\rtimes \Gamma$, to any measure preserving action of a countable group on a probability space $\Gamma \curvearrowright X$. The general paradigm of classifying $L^\infty(X)\rtimes \Gamma$ in terms of $\Gamma \curvearrowright X$ has emerged overtime as an interesting yet very challenging theme of study as these algebras tend to have very limited "memory" of the initial data. This is best illustrated by A. Connes' celebrated result which asserts that all free ergodic actions of amenable groups on probability spaces give rise to isomorphic algebras. The rigidity phenomena in this subject, i.e. instances when $L^\infty(X)\rtimes \Gamma$ remembers well $\Gamma \curvearrowright X$, emerged only over the last decade via S. Popa's deformation/rigidity theory and represent a very active area of research in operator algebras. In the first part of my talk I will survey these developments, commenting on the methods involved and their importance and also pointing out my contributions to the subject. The second part of the talk focuses on my recent results regarding the classification of ultrapower II$_1$ factors. Precisely, I will show that the group factors introduced by D. McDuff in '69 have non-isomorphic ultrapowers with respect to any ultrafilter on any set. This entails the existence of a continuum of pairwise non-elementarily equivalent II_1 factors, settling a well-known open problem in continuous model theory. view talk at https://youtu.be/iOLdoSwyQAw