Is there a vector space whose dimension is the golden ratio? Of course not--the golden ratio
is not an integer! But this can happen for generalizations of vector spaces--objects of a …

Over 50 years ago, Lovász proved that two graphs are isomorphic if and only if they admit
the same number of homomorphisms from any graph. Other equivalence relations on …

We show that under suitable assumptions, we have a one-to-one correspondence between
classical groups and free quantum groups, in the compact orthogonal case. We classify the …

We introduce a notion of quantum function and develop a compositional framework for finite
quantum set theory based on a 2-category of quantum sets and quantum functions. We use …

We show that the classical de Finetti theorem has a canonical noncommutative counterpart if
we strengthen “exchangeability”(ie, invariance of the joint distribution of the random …

We study sequences of noncommutative random variables which are invariant under
“quantum transformations” coming from an orthogonal quantum group satisfying the …

Quantum permutation matrices and quantum magic squares are generalizations of
permutation matrices and magic squares, where the entries are no longer numbers but …

This is a transcript of a series of eight lectures, 90 min each, held at IMSc Chennai, India
from 5–24 January 2015. We give basic definitions, properties and examples of compact …

We study the orthogonal quantum groups satisfying the “easiness” assumption axiomatized
in our previous paper, with the construction of some new examples and with some partial …

We study a class of matrices with noncommutative entries, which were first considered by
Yu. I. Manin in 1988 in relation with quantum group theory. They are defined as …