We study the norm topology on the pure state space of a C*-algebra A. This is a rather strong topology but it is nonetheless interesting because it corresponds to the topology of the projective Hilbert space of any irreducible representation A. We prove that any continuous map into this space is locally equivalent to composition of a fixed pure state with a norm-continuous family of inner automorphisms of A. This follows as a corollary of our main result: the operators obtained from the Kadison transitivity theorem can be chosen to depend continuously on the initial data. Finally, we will discuss how the aforementioned inner automorphisms can be used to construct local trivializations giving the unitary group of A the structure of a principal bundle over any path component of the pure state space.

The seminar will be held in hybrid mode. Join Zoom Meeting https://cuboulder.zoom.us/j/94002553301 Passcode 790356

In general, it is hard to find solutions to differential equations. How can we obtain information without being able to solve them explicitly? We will talk about some geometric and topological methods that can help us knowing more about their behavior. These ideas are very visual and somewhat simple but applying them can become a tricky business!

I know that it exists, but I donâ€™t know what it is!

Wed, Apr. 6 5pm (MATH 350)

Grad Student Seminar

Eli Orvis (CU Boulder)

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Number theorists are interested in solving equations over specified rings (Z, for example), while geometers are interested in the shape of solution sets. Deep insights during the 20th century brought these fields together and initiated the field of "arithmetic geometry." In this talk, we'll give an introduction to the circle of ideas that relates algebraic number theory, commutative algebra, and algebraic geometry. In particular, we will motivate the group of divisors of an algebraic curve by recognizing it as the parallel of a natural object in the number-theoretic setting.

P.S. Courses in all three subjects listed in the penultimate sentence will (hopefully) be offered next year, so I encourage anyone interested in these subjects to attend this talk as a "warm-up"!