I considered the first construction in my book, Outer Billiards on Kites [S2]. The arithmetic graphs served as the main tool for understanding the dynamics of outer billiards on kites. See §2 for definitions. The second construction, which is quite general, is based on patterns of oriented lines in the Euclidean plane. See §3 for definitions. The concrete instance of the multigrid construction ...

We prove in HOL that three proof systems for classical rst-order predicate logic, the Hilbertian axiomatization, the system of natural deduction, and a variant of sequent calculus, are isomorphic. The isomorphism is in the sense that provability of a conclusion from hypotheses in one of these proof systems is equivalent to provability of this conclusion from the same hypotheses in the others. P...

3 Groups 12 3.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.2 Subgroups, Cosets and Lagrange’s Theorem . . . . . . . . . . . . . . . . . . 14 3.3 Finitely Generated Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.4 Permutation Groups and Group Actions . . . . . . . . . . . . . . . . . . . . 19 3.5 The Oribit-Stabiliser Theorem . ....

A natural question to ask is: which divisors of degree 0 do not arise from meromorphic functions? The answer is given in a theorem of Abel, which we will present here. Since each divisor up to linear equivalence also corresponds to an isomorphism class of line bundles of degree 0, we will also be able to use Abel’s theorem to classify degree 0 line bundles on X as points of a complex torus call...

What ingredients are necessary to describe all maximal subgroups of the general finite group G? This paper is concerned with providing such an analysis. A good first reduction is to take into account the first isomorphism theorem, which tells us that the maximal subgroups containing a given normal subgroup N of G correspond, under the natural projection, to the maximal subgroups of the quotient...

How can we describe all finite groups? Before we address this question, let’s write down a list of all the finite groups of small orders ≤ 15, up to isomorphism. We have seen almost all of these already. If G is abelian, it is easy to write down all possible G of a given order, using the Fundamental Theorem of Finite Abelian Groups: G must be isomorphic to a direct product of cyclic groups, and...

We reprove the theorem of Feigin and Frenkel relating the center of the critical level enveloping algebra of the Kac-Moody algebra for a semisimple Lie algebra to opers (which are certain de Rham local systems with extra structure) for the Langlands dual group. Our proof incorporates a construction of Beilinson and Drinfeld relating the Feigin-Frenkel isomorphism to (more classical) Langlands d...