Let $\alpha$ be a real irrational number. Almost all information about the rational approximation to $\alpha$ may be found in the irrationality measure function ${\psi}_{\alpha}(t)=m\phantom{\rule{0}{0ex}}i\phantom{\rule{0}{0ex}}{n}_{q\in {\mathbb{Z}}_{+},q\le t}||q\phantom{\rule{0}{0ex}}\alpha ||$ (here || · || stands for the distance to the nearest integer). By Lagrange theorem we know that the jumps of the function ${\psi}_{\alpha}(t)$ correspond to the convergents of the continued fraction expansion of $\alpha$. However, there are various different "irrationality measure functions". We will discuss properties some of these functions, Diophantine spectra and relatively new results on indeterminacy and quantization.
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Commutators and derived notions like nilpotence, solvability ... have been generalized from groups to universal algebras in the 1970s. The more recently developed higher commutators yield stronger concepts like supernilpotence. We compare these two versions of nilpotence and pose some open questions about them.
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The first in a two-part series on the Lean proof assistant, this lecture introduces the mathematics of functional programming, namely, the lambda calculus. The second lecture will give a basic introduction to Lean, a relatively new addition to the growing family of functional programming languages. Lean improves upon earlier proof assistants by making it more natural and less painful to formalize mathematical proofs.
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Topological T-duality -- first investigated by Bouwknegt-Evslin-Mathai, Bunke-Schick, and others -- concerns a surprising duality isomorphism between twisted cohomology theories on different spaces. It has traditionally been studied for twisted K-theory and deRham cohomology, owing to a physical origin where these objects can be considered as spaces of fields in a certain field theory. In joint work with John Lind and Hisham Sati, we have established a general context in which duality theorems of this sort can be proven. This context is naturally related to Cartier duality of finite Hopf algebras, and gives a host of new examples, particularly arising from chromatic homotopy theory.
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In the study of quadratic moment maps there appears to be a dichotomy between large and small representations. In the large case the moment map exhibits some features of regularity. This enables one to make qualitative and quantitative statements about the symplectic quotients. In the talk I will report on recent joint work with Gerald Schwarz and Christopher Seaton which proves that for 2-large representations the appropriately defined complex symplectic quotient has symplectic singularities. This in particular entails that the symplectic quotient is graded Gorenstein domain and is a normal variety with rational singularities. It is expected that this result generalizes to small representations as well, even though here the moment map can have all sorts of pathologies. For this conjecture it is crucial to use the definition of the symplectic quotient that involves the real radical of the ideal generated by the moment map.