##### Department of Mathematics,

University of California San Diego

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### Math 243 - Functional Analysis Seminar

## Daniel Hoff

#### UCLA

## Rigid Components of s-Malleable Deformations

##### Abstract:

In the theory of von Neumann algebras, fundamental unsolved problems going back to the 1930s have seen remarkable progress in the last two decades due to Sorin Popa's breakthrough deformation/rigidity theory. Popa's discovery hinges on the fact that, just as stirring a soup allows one to locate its most rigid (and desirable) hidden components, 'deformability' of an algebra $M$ allows one to precisely locate 'rigid' subalgebras known to exist only via a supposed isomorphism $M \cong N$. This talk will focus on joint work with Rolando de Santiago, Ben Hayes, and Thomas Sinclair, which shows that any diffuse subalgebra which is rigid with respect to a mixing $s$-malleable deformation is in fact contained in subalgebra which is uniquely maximal with respect to that rigidity. In particular, an algebra generated by a family of rigid subalgebras which intersect diffusely must itself be rigid with respect to that deformation. The case where this family has two members answers a question of Jesse Peterson asked at the American Institute of Mathematics (AIM), but the result is most striking when the family is infinite.

Host: Adrian Ioana

### December 4, 2018

### 10:00 AM

### AP&M 6402

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