We consider a family of nonlinear maps that are generated from the continued fraction expansion of a function on a hyperelliptic curve of genus g, as originally described by van der Poorten. Using the connection with the classical theory of J-fractions and orthogonal polynomials, we show that in the elliptic case g=1 this provides a straightforward derivation of Hankel determinant formulae for the terms of a general Somos-4 sequence, which were found in particular cases by Chang, Hu, and Xin. We extend these formulae to the higher genus case, and find analogues of Chebyshev polynomials (which correspond to the case g=0). Moreover, for all g>0 we show that the iteration for the continued fraction expansion is equivalent to a discrete Lax pair with a natural Poisson structure, and the associated nonlinear map is a discrete integrable system, connected with solutions of the infinite Toda lattice.
Continued fractions, hyperelliptic curves and higher genus Chebyshev polynomials
A clonoid from an algebra A to an algebra B is a set of functions from finite powers of A into B that is closed under composition with term operations of A on the domain side and under composition with term operations of B on the codomain. We investigate clonoids from one module to another. When A and B are finite modules of coprime order (and the submodule lattice of A is distributive) we show that each clonoid from A to B is finitely generated and the lattice of clonoids from A to B is finite. We then extend these results to finite abelian Mal’cev algebras. We show how a 2-nilpotent Mal’cev algebra can be classified with the help of an associated clonoid between abelian Mal’cev algebras. This is joint work with Peter Mayr.
Clonoids and nilpotent algebras
Oct. 24, 2023 2:30pm (Math 3…
Lie Theory
Shashank Kanade (University of Denver)
X
I will give a gentle introduction to the combinatorial Rogers--Ramanujan identities. While these identities are over a century old, and have many proofs, the first representation-theoretic proof was given by Lepowsky and Wilson about four decades ago. Now-a-days, these identities are ubiquitous in several areas of mathematics and physics. I will mention how these identities arise from affine Lie algebras and quantum invariants of knots.
The world of Rogers--Ramanujan identities Sponsored by the Meyer Fund
Oct. 24, 2023 3:30pm (MATH 3…
Topology
David Chan (Michigan State University)
X
The algebraic K theory of a manifold M is a space whose homotopy groups record interesting geometric properties. Some of these invariants, like the Wall finiteness obstruction and Whitehead torsion, depend only on the fundamental group G of M and can also be defined in terms of the algebraic K theory of the group ring Z[G]. The relationship between the algebraic K theory of M and the algebraic K theory of Z[G] is made precise by a comparison map called the linearization. In this talk we will review these ideas and discuss some recent work on a refinement of the linearization map to the genuine equivariant algebraic K theory of spaces. This is joint work with Maxine Calle and Andres Mejia.
A linearization map for equivariant algebraic K theory
Continued fractions, hyperelliptic curves and higher genus Chebyshev polynomials