A deep result of Furstenberg from 1967 states that if Γ is a lattice in a semisimple Lie group G, then there exists a measure on Γ with finite first moment such that the corresponding harmonic measure on the Furstenberg boundary of G is absolutely continuous. I will discuss some of the history of this result and some recent generalizations.
I'll give an overview of recent progress in the structure and classification of simple amenable C*-algebras, making parallels to the Connes-Haagerup classification of amenable von Neumann algebras and drawing examples from group actions.
Can we reconstruct a function by knowing only a subset of its values and a subset of the values of the function's Fourier transform?
How many values do we need to know for such a reconstruction? Can we interpolate a given subset of values? What are the possible applications of such interpolation? In this series of lectures, we will try to answer these questions.
In the first lecture, we will speak about the Cohn-Elkies linear programming bound for the sphere packing and how this bound's analysis led to the discovery of a Fourier interpolation formula. The second lecture will discuss explicit constructions of Fourier uniqueness sets and Fourier interpolation formulas. The third lecture will focus on analytic approaches to Fourier uniqueness and interpolation.
Motivated by a discovery by Radchenko and Viazovska and by a work by Ramos and Sousa, we find conditions sufficient for a pair of discrete subsets of the real axis to be a uniqueness or a non-uniqueness pair for the Fourier transform. These conditions are not too far from each other. The uniqueness theorem can be upgraded to the frame bound and an interpolation formula, which in turn produce an abundance of Poisson-like formulas (a.k.a. 'crystalline measures').
Many notions in classical Fourier analysis have beautiful extensions to discrete quantum theory. We give an overview of this rapidly developing area.
This is a 35-lecture course, with each lecture being an hour, given by George Elliott. Note that the 32nd lecture was not recorded. The first 31 lectures are still of great interest, but this needs to be known.
The theory of operator algebras was begun by John von Neumann eighty years ago. In one of the most important innovations of this theory, von Neumann and Murray introduced a notion of equivalence of projections in a self-adjoint algebra (*-algebra) of Hilbert space operators that was compatible with addition of orthogonal projections (also in matrix algebras over the algebra), and so gave rise to an abelian semigroup, now referred to as the Murray-von Neumann semigroup.
Later, Grothendieck in geometry, Atiyah and Hirzebruch in topology, and Serre in the setting of arbitrary rings (pertinent for instance for number theory), considered similar constructions. The enveloping group of the semigroup considered in each of these settings is now referred to as the K-group (Grothendieck's terminology), or as the Grothendieck group.
Among the many indications of the depth of this construction was the discovery of Atiyah and Hirzebruch that Bott periodicity could be expressed in a simple way using the K-group. Also, Atiyah and Singer famously showed that K-theory was important in connection with the Fredholm index. Partly because of these developments, K-theory very soon became important again in the theory of operator algebras. (And in turn, operator algebras became increasingly important in other branches of mathematics.)
The purpose of this course is to give a general, elementary, introduction to the ideas of K-theory in the operator algebra context. (Very briefly, K-theory generalizes the notion of dimension of a vector space.)
The course will begin with a description of the method (K-theoretical in spirit) used by Murray and von Neumann to give a rough initial classication of von Neumann algebras (into types I, II, and III). It will centre around the relatively recent use of K-theory to study Bratteli's approximately finite-dimensional C*-algebras, both to classify them (a result that can be formulated and proved purely algebraically), and to prove that the class of these C*-algebras - what Bratteli called AF algebras - is closed under passing to extensions (a result that uses the Bott periodicity feature of K-theory).
We introduce conformal Bach flow and establish its well-posedness on closed manifolds. We also obtain its backward uniqueness. To give an attempt to study the long-time behaviour of conformal Bach flow, assuming that the curvature and the pressure function are bounded, global and local Shi's type L2-estimate of derivatives of curvatures are derived. To make the talk more accessible, we will spend some time to survey on high-order parabolic curvature flow.
In this talk I will introduce the notion of self-similarity for compact quantum groups. I will start by looking at the quantum automorphism group of an infinite homogeneous rooted tree. Self-similar quantum groups are then certain quantum subgroups of these quantum automorphisms. I will then look at a class of examples called finitely-constrained self-similar quantum groups, and I will describe a subclass as quantum wreath products by subgroups of the quantum permutation group.
Many combinatorial objects (or similar) give rise to a C-algebra with a distinguished C-subalgebra. Examples of such which have been studied a lot in the past decade are étale groupoids which give rise to a so-called Cartan pair, consisting of the groupoid C-algebra and a commutative C-subalgebra. Other examples include discrete groups acting on compact Hausdorff spaces, for which the reduced crossed product C-algebra contains the reduced group C-algebra as a C- subalgebra. I will talk about invariants and properties for such inclusions of C- algebras. This talk will be somewhat non-standard (for me, at least) and will only contain few (if even any) theorems, but should more be thought of as a conversation starter/brainstorm for what invariants one can associate to certain mathematical objects.
The category of Hilbert spaces and bounded linear functions forms the mathematical basis for quantum theory. But why? What physical principles enforce this mathematical structure? This category is also where C*-algebra theory lives, as it's the universal C*-category. But why? What properties does it have that accommodate this mathematical structure? As a first answer, this talk provides axioms that guarantee a category is equivalent to that of Hilbert spaces and bounded linear functions. The axioms are purely categorical and do not presuppose any analytical structure such as complex numbers, continuity, dimension, convexity, probabilities, etc. We will also discuss variations, such as linear contractions, finite-dimensional Hilbert spaces, and Hilbert C*-modules.
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