Related papers: Rational tangles and the modular group
We construct a functor from the category of oriented tangles in R^3 to the category of Hermitian modules and Lagrangian relations over Z[t,t^{-1}]. This functor extends the Burau representations of the braid groups and its generalization to…
In this paper we study different Hamiltonian systems with polynomial and rational Hamiltonians associated with the generic third Painlev\'e equation and present explicit birational transformations relating them.
The moduli space of parabolic bundles with fixed determinant over a smooth curve of genus greater than one is proved to be rational whenever one of the multiplicities associated to the quasi-parabolic structure is equal to one. It follows…
We determine couples of singular moduli which have rational products
For an orbifold M we define a homology group, called t-singular homology group t-H_q(M), which depends not only on the topological structure of the underlying space of M, but also on the orbifold structure of M. We prove that it is a…
A powerful way to study groups is via their actions on suitable spaces. Classifying spaces for families of subgroups are a type of these spaces, obtained by imposing some strict conditions on the fixed-point sets. We show how in the…
We investigate the Brauer class of the endomorphism algebra of the motive attached to a non-CM form. The ramification of the algebra is shown in many cases to be controlled by the normalized slopes of the form.
This expository paper explores the interaction of group ordering with topological questions, especially in dimensions 2 and 3. Among the topics considered are surfaces, braid groups, 3-manifolds and their structures such as foliations and…
We extend the ramified geometric Satake equivalence due to Zhu (for tamely ramified groups) and the third named author (in full generality) from rational coefficients to include modular and integral coefficients.
The present text surveys some relevant situations and results where basic Module Theory interacts with computational aspects of operator algebras. We tried to keep a balance between constructive and algebraic aspects.
We provide a treatment of isomorphism within a set-theoretic formulation of dependent type theory. Type expressions are assigned their natural set-theoretic compositional meaning. Types are divided into small and large types --- sets and…
A simple and self-contained proof is presented of the well-known fact that the fundamental group of SO(3) is $Z_2$, using a relationship between closed paths in SO(3) and braids.
We define a general notion of centrally $\Gamma$-graded sets and groups and of their graded products, and prove some basic results about the corresponding categories: most importantly, they form braided monoidal categories. Here, $\Gamma$…
Two observations in support of the thesis that trusses are inherent in ring theory are made. First, it is shown that every equivalence class of a congruence relation on a ring or, equivalently, any element of the quotient of a ring $R$ by…
We study modular ortholattices in the variety generated by the finite dimensional ones from an equational and geometric point of view. We relate this to coordinatization results.
In this paper, we develop the theory of symmetric triads with multiplicities. First, we classify abstract symmetric triads with multiplicities. Second, we determine the symmetric triads with multiplicities corresponding to commutative…
We give a simple combinatorial proof that the rotation number for each element in Thompson's group ${\bf T}$ is rational.
We establish a novel connection between algebraic number theory and knot theory. We show that the number of equivalence classes of integral binary quadratic forms of discriminant $t^2 - 4$ (for $t\neq \pm 2$) is equal to the number of…
We give an example of infinite order rational transformation that leaves a linear differential equation covariant. This example can be seen as a non-trivial but still simple illustration of an exact representation of the renormalization…
A simple observation, showing that every groupoid becomes an inverse semigroup after adding one element. In such inverse semigroups all idempotents are mutually orthogonal. This fact implies that every C*-algebra of a discrete groupoid is a…