Miscellaneous (presentations, translations, interviews, etc)

Duality for Classical Groups, a talk by Chris Jantzen

Chris Jantzen et al. — Mon, 09 Apr 2012 08:33:13 PDT

Finite groups of Lie type, Hecke algebras and p-adic groups all admit an operation on irreducible representations called duality. This operator takes irreducible representations to irreducible representations, often of a much different nature (e.g., the dual of the trivial representation is the Steinberg representation). In this talk, we give a brief history of duality in these contexts, and describe how one might how to explicitly calculate the dual of an irreducible representation for classical p-adic groups.

Automorphic Representations of $GL_n$ and their $L$ functions, a talk by Shuichiro Takeda

Shuichiro Takeda et al. — Mon, 09 Apr 2012 08:33:12 PDT

The concept of automorphic representations, which can be considered as a huge generalization of classical modular forms, are one of the central themes of modern number theory known as the Langlands program. In this talk, I will introduce automorphic representations of GL(n) and their L-functions in a way accessible to non-experts and briefly discus some recent developments.

The Lattice Model of the Weil representation and the Howe duality conjecture, a talk by Shuichiro Takeda

Shuichiro Takeda et al. — Mon, 09 Apr 2012 08:32:11 PDT

The lattice model of the Weil representation over non-archimedean local field of odd residual characteristic has been known for decades, and is used to prove the Howe duality conjecture for unramified dual pairs when the residue characteristic is odd. In this talk, we will talk on how to modify the lattice model of the Weil representation so that it is defined independently of the residue characteristic. Although to define the lattice model alone is not enough to prove the Howe duality conjecture for even residual characteristic, we will propose a couple of conjectural lemmas which imply the Howe duality conjecture for unramified dual pairs independently of the residue characteristic. Also those two lemmas can be proven for certain cases, which allow us to prove (a version of) the Howe duality conjecture for the even-orthogonal-symplectic dual pair of equal rank for a certain class of representations, independently of the residue characteristic.

Tempered Representations (Seminar)

Chris Jantzen — Fri, 18 Feb 2011 05:04:14 PST

Introduction to tempered representations on p-adic classical groups. Recent work developing a classification of these representations along the lines of the Moeglin-Tadic classification of the discrete series.

Tempered Representations (Colloquium)

Chris Jantzen — Thu, 10 Feb 2011 19:42:43 PST

The spectral decomposition of the space L^2(G) as a direct sum or direct integral is discussed in four cases: the compact abelian circle group, the noncompact abelian real line, a nonabelian compact finite group, and the noncompact, nonabelian group of 2 by 2 matrices with real entries and determinant one.

Representations of Finite Groups

Joseph Hundley — Wed, 02 Feb 2011 06:51:13 PST

Notes from a series of lectures given in the Representation Theory Seminar during Fall 2010.

Explicit Feedback Linearization of Control Systems

Issa Amadou Tall — Fri, 03 Sep 2010 11:15:13 PDT

This paper addresses the problem of feedback linearization of nonlinear control systems via state and feedback transformations. Necessary and sufficient geometric conditions were provided in the early eighties but finding the feedback linearizing coordinates is subject to solving a system of partial differential equations and had remained open since then. We will provide in this paper a complete solution to the problem (see the companion paper where the state linearization has been addressed) by defining an algorithm that allows to compute explicitly the linearizing state coordinates and feedback for any nonlinear control system that is truly feedback linearizable. Each algorithm is performed using a maximum of n - 1 steps (n being the dimension of the system) and they are made possible by explicitly solving the Flow-box or straightening theorem. A possible implementation via software like mathematica/ matlab/maple using simple integrations, derivations of functions might be considered.

State Linearization of Control Systems: An Explicit Algorithm

Issa Amadou Tall — Fri, 03 Sep 2010 11:10:41 PDT

In this paper we address the problem of linearization of nonlinear control systems using coordinate transformations. Although necessary and sufficient geometric conditions have been provided in the early eighties, the problem of finding the linearizing coordinates is subject to solving a system of partial differential equations and remained open 30 years later. We will provide here a complete solution to the problem by defining an algorithm allowing to compute explicitly the linearizing state coordinates for any nonlinear control system that is indeed linearizable. Each algorithm is performed using a maximum of n - 1 steps (n being the dimension of the system) and they are made possible by explicitly solving the Flow-box or straightening theorem. The problem of feedback linearization is addressed in a companion paper. A possible implementation via software like mathematica/matlab/maple using simple integrations, derivations of functions might be considered.

Time-Invariant Quadratic Hamiltonians via Generalized Transformations

Issa Amadou Tall — Fri, 03 Sep 2010 11:03:08 PDT

In this paper we give necessary and sufficient conditions for achieving a quadratic positive definite time-invariant Hamiltonian for time-varying generalized Hamiltonian control systems using canonical transformations. Those necessary and sufficient conditions form a system of partial differential equations that reduces to the matching conditions obtained earlier in the literature for time-invariant systems. Their theoretical solvability is proved via the Cauchy-Kowalevskaya theorem and their practical solvability discussed in some particular cases. Systems with time-invariant positive definite Hamiltonian are known to yield a passive input-output map and can be stabilized by unity feedback, which underlines the importance of achieving the positive definiteness and time-invariancy for the Hamiltonian. We illustrate the results with few examples including the rolling coin.

Introduction to Lie groups

Joseph Hundley — Wed, 13 Jan 2010 06:25:52 PST

Short introduction to Lie groups. Definition and basic properties, definition of Lie algebra, etc.

Construction of the Compact Real Form of $G_2$ via the Octonions

Joseph F. Pleso — Mon, 23 Nov 2009 06:57:20 PST

Octonions

Robert W. Fitzgerald — Mon, 02 Nov 2009 12:11:11 PST

We present the basic properties of the octonions and construct the five exceptional simple Lie algebras.

Feedback Linearizability of Strict Feedforward Systems

Witold Respondek et al. — Tue, 14 Apr 2009 16:43:38 PDT

For any strict feedforward system that is feedback linearizable we provide (following our earlier results) an algorithm, along with explicit transformations, that linearizes the system by change of coordinates and feedback in two steps: first, we bring the system to a newly introduced Nonlinear Brunovský canonical form (NBr) and then we go from (NBr) to a linear system. The whole linearization procedure includes diffeo-quadratures (differentiating, integrating, and composing functions) but not solving PDE’s. Application to feedback stabilization of strict feedforward systems is given.

How Many Symmetries Does Admit a Nonlinear Single-Input Control System Around an Equilibrium?

Witold Respondek et al. — Thu, 22 Jan 2009 15:35:56 PST

We describe all symmetries of a single-input nonlinear control system, that is not feedback linearizable and whose first order approximation is controllable, around an equilibrium point. For a system such that a feedback transformation, bringing it to the canonical form, is analytic we prove that the set of all local symmetries of the system is exhausted by exactly two 1-parameter families of symmetries, if the system is odd, and by exactly one 1-parameter family otherwise. We also prove that the form of the set of symmetries is completely described by the canonical form of the system: possessing a nonstationary symmetry, a 1-parameter family of symmetries, or being odd corresponds, respectively, to the fact that the drift vector field of the canonical form is periodic, does not depend on the first variable, or is odd. If the feedback transformation bringing the system to its canonical form is formal, we show an analogous result for an infinitesimal symmetry: its existence is equivalent to the fact that the drift vector field of the formal canonical form does not depend on the first variable. We illustrate our results by studying symmetries of the variable length pendulum.

Normal Forms, Canonical Forms, and Invariants of Single Input Nonlinear Systems Under Feedback

Issa Amadou Tall et al. — Thu, 22 Jan 2009 15:07:08 PST

We study the feedback group action on single-input nonlinear control systems. We follow an approach of Kang and Krener based on analysing, step by step, the action of homogeneous transformations on the homogeneous part of the system. We construct a dual normal form and dual invariants with respect to those obtained by Kang. We also propose a canonical form and show that two systems are equivalent via a formal feedback if and only if their canonical forms coincide. We give an explicit construction of transformations bringing the system to its normal, dual normal, and canonical form.

Weighted Canonical Forms of Nonlinear Single-Input Control Systems with Noncontrollable Linearization

Issa Amadou Tall et al. — Thu, 22 Jan 2009 14:54:00 PST

We propose a weighted canonical form for single-input systems with noncontrollable first order approximation under the action of formal feedback transformations. This weighted canonical form is based on associating different weights to the linearly controllable and linearly noncontrollable parts of the system. We prove that two systems are formally feedback equivalent if and only if their weighted canonical forms coincide up to a diffeomorphism whose restriction to the linearly controllable part is identity.

Strict Feedforward Form and Symmetries of Nonlinear Control Systems

Witold Respondek et al. — Thu, 22 Jan 2009 13:56:53 PST

We establish a relation between strict feedforward form and symmetries of nonlinear control systems. We prove that a system is feedback equivalent to the strict feedforward form if and only if it gives rise to a sequence of systems, such that each element of the sequence, firstly, possesses an infinitesimal symmetry and, secondly, it is the factor system of the preceding one, i.e., is reduced from the preceding one by its symmetry. We also propose a strict feedforward normal form and prove that a smooth strict feedforward system can be smoothly brought to that form.

Normal Forms for Two-Inputs Nonlinear Control Systems

Issa Amadou Tall et al. — Thu, 22 Jan 2009 13:11:55 PST

We study the feedback group action on two-inputs non-linear control systems. We follow an approach proposed by Kang and Krener which consists of analyzing the system and the feedback group step by step. We construct a normal form which generalizes that obtained in the single-input case. We also give homogeneous m-invariants of the action of the group of homogeneous transformations on the homogeneous systems of the same degree. We illustrate our results by analyzing the normal form and invariants of homogeneous systems of degree two.

Smooth and Analytic Normal and Canonical Forms for Strict Feedforward Systems

Issa Amadou Tall et al. — Thu, 22 Jan 2009 10:35:33 PST

Recently we proved that any smooth (resp. analytic) strict feedforward system can be brought into its normal form via a smooth (resp. analytic) feedback transformation. This will allow us to identify a subclass of strict feedforward systems, called systems in special strict feedforward form, shortly (SSFF), possessing a canonical form which is an analytic counterpart of the formal canonical form. For (SSFF)-systems, the step-by-step normalization procedure of Kang and Krener leads to smooth (resp. convergent analytic) normalizing feedback transformations. We illustrate the class of (SSFF)-systems by a model of an inverted pendulum on a cart.

Normal Forms for Nonlinear Discrete Time Control Systems

Boumediene Hamzi et al. — Thu, 22 Jan 2009 10:22:42 PST

We study the feedback classification of discrete-time control systems whose linear approximation around an equilibrium is controllable. We provide a normal form for systems under investigation.