The topological group of homeomorphisms of d-dimensional Euclidean space is a basic object in geometric topology, closely related to understanding the difference between diffeomorphisms and homeomorphisms of all d-dimensional manifolds (except when d=4). Over the last few years a great deal of progress has been made in understanding the algebraic topology of this group. I will report on some of the methods involved, and an emerging conjectural picture.
One of the most famous – and still not fully understood – objects in mathematics is the Mandelbrot set. It is defined as the set of complex numbers c for which the polynomial fc(z)=z2+c has a connected Julia set. But the Mandelbrot set turns out to be related to many different areas of mathematics. Inspired by recent results in arithmetic geometry, I will describe how the tools of arithmetic intersection theory can be applied in the setting of these complex dynamical systems to give new information about the Mandelbrot set.
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).
A graph is said to be a (1-dimensional) expander if the second eigenvalue of its adjacency matrix is bounded away from 1, or almost-equivalently, if it has no sparse vertex cuts. There are several natural ways to generalize the notion of expansion to hypergraphs/simplicial complexes, but one such way is 2-dimensional spectral expansion, in which the local expansion of vertex links/neighborhoods (remarkably) witnesses global expansion. While 1-dimensional expansion is known to be achieved by, e.g., random regular graphs, very few examples of sparse 2-dimensional expanders are known, and at present all are algebraic. It is an open question whether sparse 2-dimensional expanders are natural and "abundant" or "rare." In this talk, we'll give some evidence towards abundance: we show that the set of triangles in a random geometric graph on a high-dimensional sphere yields an expanding simplicial complex of arbitrarily small polynomial degree.
We present buildings as universal covers of certain infinite families of CW-complexes of arbitrary dimension. We will show several different constructions of new families of k-rank graphs and C*-algebras based on these complexes, for arbitrary k. The underlying building structure allows explicit computation of the K-theory as well as the explicit spectra computation for the k-graphs.
The dynamics of iterating a rational function exhibits complicated and interesting behaviour when restricted to points in its Julia set. Kajiwara and Watatani constructed a C*-algebra from a rational function restricted to its Julia set in order to study its dynamics from an operator algebra perspective. They showed the C*-algebras are Kirchberg algebras that satisfy the UCT, and are therefore classified by K-theory. The K-theory groups of these algebras have been computed in some special cases, for instance by Nekrashevych in the case of a hyperbolic and post-critically finite rational function. We compute the K-theory groups for a general rational function using methods different to those used before. In this talk, we discuss our methods and results.
Given a set ℬ of natural numbers, not containing 1, we denote by ℱℬ the set of ℬ-free numbers, that is, ℱℬ = ℤ \ ⋃b∈ℬ bℤ. Let Xη be the ℬ-free subshifts, that is the subshift induced by η, where η denotes the characteristic function of ℱℬ . That means, Xη is the closure of the set of all shifts of η in the space {0, 1}ℤ equipped with the product topology. We are interested in the case when B is a Behrend set, that is, when the set of ℬ-free numbers has zero density. It turns out that this is the case precisely when Xη is proximal and has zero entropy. We prove that the complexity of Xη, with ℬ being a Behrend set, can achieve any subexponential growth. Together with the ℬ-free shift we investigate the ℬ-admissible shift Xℬ and we show that it is transitive if and only if the set B is pairwise coprime, which allows one to characterize dynamically the subshifts generated by the Erdős sets (infinite, coprime and not Behrend). We also estimate the complexity for some classical subshifts (the subshift of primes or semi-primes). The lower estimates are obtained conditionally on Hardy-Littlewood Conjecture or Dickson’s Conjecture. We remark on a recent result of Tao and Ziegler (not assuming the conjectures) that the shift of primes is uncountable.
Several limits of quantum theory can be understood from a simple framework in which some of the basic features of these limits can be established by abstract general results. Often the limit may have a very different structure from its approximants, so one cannot simply let some parameter go to a limiting value. We focus on the unified definition of the limit theory with its states and observables, its dynamics and its equilibrium states. Examples that will be mentioned are (1) the mean field with or without tagged particles, (2) The classical limit (ħ → 0), (3) the limit of infinite lattice systems, (4) Some continuum limits/renormalization theories.
It is shown that the operator space generated by peripheral eigenvectors of a unital completely positive map on a von Neumann algebra has a C*-algebra structure. This extends the notion of non-commutative Poisson boundary by including the point spectrum of the map contained in the unit circle. The main ingredient is dilation theory. This theory provides a simple formula for the new product. The notion has implications to our understanding of quantum dynamics. For instance, it is shown that the peripheral Poisson boundary remains invariant in discrete quantum dynamics.
In recent joint work with Yidong Chen, we discovered spectral gap estimates and concentration inequalities for for dynamics with few generators. Some of these estimates are dimension free and then can be used to feed in the recent theory of complexity initiated by Lloyd and Jaffe, and adapted more recently for specific resources. The goal is to find a viable theory of complexity which holds in type II1 and III1 von Neumann algebras, both of which come naturally in quantum field theory and Witten's take on black holes.
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