We introduce a family of extremal polynomials associated with the prolongation of a stratified nilpotent Lie algebra. These polynomials are related to a new algebraic characterization of abnormal subriemannian geodesics in stratified nilpotent Lie groups. They satisfy a set of remarkable structure relations that are used to integrate the adjoint equations.

Let G(K) be the group of K-rational points of a connected adjoint simple algebraic group over a nonarchimedean local field K. In this paper we classify the unipotent representations of G(K) in terms of the geometry of the Langlands dual group. This was known earlier in the special case where G(K) is an inner form of a split group.

Abstract Let G be a simple algebraic group of adjoint type over an algebraically closed field k bad characteristic. We show that its sheets conjugacy classes are parametrized by -conjugacy pairs $(M,{\mathcal O})$ where M is the identity component centralizer semisimple element in and ${\mathcal O}$ rigid unipotent class , analogy with good characteristic case.

It is shown that projectivized irreducible components of nilpotent cones of complex symmetric spaces are projective self-dual algebraic varieties. Other properties equivalent to their projective self-duality are found. 1. Let g be a semisimple complex Lie algebra, let G be the adjoint group of g, and let θ ∈ Aut g be an element of order 2. We set k := {x ∈ g | θ(x) = x}, p := {x ∈ g | θ(x) = −x...

These notes are based on lectures in algebraic topology taught by Peter May and Henry Chan at the 2016 University of Chicago Math REU. They are loosely chronological, having been reorganized for my benefit and significantly annotated by my personal exposition, plus solutions to in-class/HW exercises, plus content from readings (from May’s Finite Book), books (e.g. May’s Concise Course, Munkres’...

This expository paper introduces the concept of monads and explores some of its connections to algebraic structures. With an emphasis on the adjoint functors that naturally participate in our conclusions, we justify how monads give us a distinct, ‘categorical’ way of discussing common structures such as groups and rings. In the final section, we consider a key example of how looking at structur...

The author’s algebraic theory of boundary value problems has permitted systematizing Trefftz method and expanding its scope. The concept of TH-completeness has played a key role for such developments. This paper is devoted to revise the present state of these matters. Starting from the basic concepts of the algebraic theory, Green–Herrera formulas are presented and Localized Adjoint Method (LAM...

This paper presents a review and summary of recent research on the boundary value problems for linear ordinary and partial differential equations, with special attention to the investigations of the current authors emphasizing the applications of complex symplectic spaces. In the first part of the previous century, Stone and von Neumann formulated the theory of self-adjoint extensions of symmet...

We apply the algebraic method to Bateman Hamiltonian and obtain its natural frequencies ladder operators from adjoint or regular matrix representation of that operator. Present analysis shows eigenfunctions compatible with complex eigenvalues obtained earlier by other authors are not square integrable. In addition this, annihilate an infinite number such eigenfunctions.

In a previous paper, we have introduced a general approach for connecting two many-sorted theories through connection functions that behave like homomorphisms on the shared signature, and have shown that, under appropriate algebraic conditions, decidability of the validity of universal formulae in the component theories transfers to their connection. This work generalizes decidability transfer ...