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$q$-deformed rationals and irrationals
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abstract
The concept of $q$-deformation, or ``$q$-analogue'' arises in many areas of mathematics. In algebra and representation theory, it is the origin of quantum groups; $q$-deformations are important for knot invariants, combinatorial enumeration, discrete geometry, analysis, and many other parts of mathematics. In mathematical physics, $q$-deformations are often understood as ``quantizations''. The recently introduced notion of a $q$-deformed real number is based on the geometric idea of invariance by a modular group action. The goal of this lecture is to explain what is a $q$-rational and a $q$-irrational, demonstrate beautiful properties of these objects, and describe their relations to many different areas. We also tried to describe some applications of $q$-numbers.
Forward citations
Cited by 2 Pith papers
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Coefficients of $q$-real numbers: their combinatorial meaning and growth
Every coefficient of every q-real number between 1 and 2 is bounded in absolute value by the corresponding coefficient of the q-deformed golden ratio, resolving the radius-of-convergence conjecture.
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Dimers, filters, and $q$-deformed real numbers
Every positive real x gets a snake-graph dimer model whose distinguished-edge odds define a new q-deformation [[x]]_q, equal to q[x]_q for rational x.
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