In many recent works, analysis and number theory go beyond working side by side and team up in an interconnected back and forth interplay to become a powerful force. Here I describe two distinct meetings of the pair, which result in sharp counts for equilateral triangles in Euclidean space and statistics for how often a random polynomial has Galois group not isomorphic to the full symmetric group.
Morrey’s conjecture arose from a rather innocent-looking question in 1952: is there a local condition characterizing 'ellipticity' in the calculus of variations? Morrey was not able to answer the question, and indeed, it took 40 years until first progress was made with V. Sverak’s ingenious counterexample. Nevertheless, the case pertaining to planar maps remains open despite much progress, and has fascinated many through its interesting connections to complex analysis, geometric function theory, harmonic analysis, probability and martingales, differential inclusions and the geometry of matrix space. In the talk, I will give an overview of some of these connections and some of the recent progress.
This is a 24-lecture course, with each lecture being around 80 minutes, given by Robert McCann. It gives an introduction to optimal transport.
This course is an introduction to the active research areas surrounding optimal transportation and its deep connections to problems in geometry, physics, nonlinear partial differential equations, and machine learning. The basic problem is to find the most efficient structure linking two or more continuous distributions of mass; think of pairing a cloud of electrons with a cloud of positrons so as to minimize average distance to annihilation.
Applications include existence, uniqueness, and regularity of surfaces with prescribed Gauss curvature (the underlying PDE is Monge-Ampère), geometric inequalities with sharp constants, image processing, optimal decision making, long time asymptotics of dissipative systems, and the geometry of fluid motion (Euler's equation and approximations appropriate to atmospheric, oceanic, damped and porous medium flows). The course builds on a background in analysis, including measure theory, but will develop elements as needed from the calculus of variations, game theory, differential equations, fluid mechanics, physics, economics, and geometry. A particular goal will be to expose the developing theories of curvature and dimension in metric-measure geometry, which provide a framework for adapting powerful ideas from Riemannian and Lorentzian geometry to non-smooth settings which arise both naturally in applications, and as limits of smooth problems.
In the absence of measures fully invariant with respect to a group action, this role can be to a certain extent played by the measures "invariant on average", with respect to a certain fixed distribution on the group. These measures are called stationary, and they naturally arise as harmonic measures of random walks. I will provide several partial answers to the general question about the dependence of harmonic measures on the underlying step distributions on the group and discuss counterexamples related to the Minkowski and Denjoy measure classes on the boundary of the classical modular group.
In the seventies, Feldman and Moore studied Cartan pairs of von Neumann algebras. These pairs consist of an algebra A and a maximal commutative subalgebra B with B sitting "nicely" inside of A. They showed that all such pairs of algebras come from twisted groupoid algebras of quite special groupoids (in the measure theoretic category) and their commutative subalgebras of functions on the unit space, and that moreover the groupoid and twist were uniquely determined (up to equivalence). Kumjian and Renault developed the C*-algebra theory of Cartan pairs. Again, in this setting all Cartan pairs arise as twisted groupoid algebras, this time of effective etale groupoids, and again the groupoid and twist are unique (up to equivalence).
In recent years, Matsumoto and Matui exploited this to give C*-algebraic characterizations of continuous orbit equivalence and flow equivalence of shifts of finite type using graph C*-algebras and their commutative subalgebras of continuous functions on the shift space (which form a Cartan pair under mild assumptions on the graph). The key point was translating these dynamical conditions into groupoid language. The ring theoretic analogue of graph C*-algebras are Leavitt path algebras. Leavitt path algebras are also connected to Thompson's group V and some related simple groups considered by Matui and others. Since the Leavitt path algebra associated to a graph is the "Steinberg" algebra of the same groupoid (a ring theoretic version of groupoid C*-algebras whose study was initiated by the speaker), this led people to wonder whether these dynamical invariants can be read off the pair consisting of the Leavitt path algebra and its subalgebra of locally constant maps on the shift space. The answer is yes, and it turns out in the algebraic setting one doesn’t even need any conditions on the graph. Initially work was focused on recovering a groupoid from the pair consisting of its "Steinberg" algebra and the algebra of locally constant functions on the unit space. But no abstract theory of Cartan pairs existed and twists had not yet been considered. Our work develops the complete picture.
It turns out that a twist on a groupoid gives rise to a Cartan pair when the algebra satisfies a groupoid analogue of the Kaplansky unit conjecture. In particular, if the groupoid has a dense set of objects whose isotropy groups satisfy the Kaplansky unit conjecture (e.g., are unique product property groups or left orderable), then the groupoid gives rise to a Cartan pair. This is what happens in the case of Leavitt path algebras where the isotropy groups are either trivial or infinite cyclic and hence left orderable.
A conjecture of George Elliott dating back to the early 1990s asks if separable, simple, nuclear C*-algebras are determined up to isomorphism by their K-theoretic and tracial data. Restricting to purely infinite algebras, this is the famous Kirchberg-Phillips Theorem. The stably finite setting proved to be much more subtle and has been a driving force in research in C*-algebras over the last 30 years. A series of breakthroughs were made in 2015 through the classification results of Elliott, Gong, Lin, and Niu and the quasidiagonality theorem of Tikuisis, White, and Winter. Today, the classification conjecture is now a theorem under two additional regularity assumptions: Z -stability and the UCT. In my recent joint work with José Carrión, Jamie Gabe, Aaron Tikuisis, and Stuart White a much shorter and more conceptual proof of the classification theorem in the stably finite setting was provided. I hope to give an overview of the classification problem for C*-algebras and discuss some of the new techniques that led to the new proof.
An online lecture course by the University of Münster in Von Neumann algebras.
Lecture 1: We'll briskly review basic properties of semi-finite von Neumann algebras. The standard representation, completely positive maps, group von Neumann algebras, the group-measure space construction, and some characterizations of the hyperfinite II1 factor.
Lecture 2: We discuss some approximation properties that are common in "rank 1" groups: Weak amenability and biexactness.
Lecture 3: We discuss properly proximal groups as defined by Boutonnet, Ioana, and myself, and give some applications to group von Neumann algebras associated to higher-rank groups.
Lecture 4: We’ll introduce measure equivalence (ME), W*-equivalence (W*E), and von Neumann equivalence (VNE). We’ll give examples and discuss invariants.
An approximate unitary representation of a group G is a function f from G to U(n) such that f(gh) is close to f(g)f(h) for all g,h. Is every approximate unitary representation just a slight deformation of a unitary representation? The answer depends on G and on the norm on U(n). If G is amenable, the answer is positive for the operator norm on U(n) (Kazhdan '82). The answer remains positive if we use the normalized Hilbert-Schmidt norm and allow a slight change in the dimension n (Gowers-Hatami '15, De Chiffre-Ozawa-Thom '17). For both norms, the answer is negative if G is a non-abelian free group (or a non-elementary word-hyperbolic group). In this talk we shall discuss a similar notion where U(n) is replaced by Sym(n) with the normalized Hamming metric. We study the cases where G is either free, amenable or equal to SLr(ℤ), r ≥ 3. When G is finite, a slight variation of our main theorem provides an efficient probabilistic algorithm to determine whether a function f from G to Sym(n) is close to a homomorphism when |G| and n are both large.
This is a 22-lecture course, with each lecture being around 90 minutes, given by Ilijas Farah.
The route to understanding separable C*-algebras frequently involves a detour via non-separable C*-algebras, such as the Calkin algebra, the asymptotic sequence algebras, ultrapowers, and ultraproducts. Some basic ideas from logic can be used to analyse these massive C*-algebras. Among other things, we will see that the existence of outer automorphisms of the Calkin algebra depends on the set-theoretic axioms.
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