In this thesis, we provide new insights into the theory of cascade feedback linearization of control systems. In particular, we present a new explicit class of cascade feedback linearizable control systems, as well as a new obstruction to the existence of a cascade feedback linearization for a given invariant control system. These theorems are presented in Chapter 4, where truncated versions of operators from the calculus of variations are introduced and explored to prove these new results. This connection reveals new geometry behind cascade feedback linearization and establishes a foundation for future exciting work on the subject with important consequences for dynamic feedback linearization.
Geometry of Cascade Feedback Linearizable Control Systems
We’ve all heard the hype: all you have to do is forget about “excluded middle” and then BOOM, your logic is constructive (in a certain sense). But like… why would you do that? And in what sense is it more constructive?
If you aren’t aware, the Law (ha!) of Excluded Middle is the assertion that, for any particular sentence, either that sentence or its negation holds in your logic. In this talk we’ll look at what kind of logic you get when you drop excluded middle and discuss various motivations and interpretations of this, from the philosophical hooey of Brouwer to the structural interpretations arising from Gentzen’s methods. We will also look at a sample of some of the differences in practice and theory that arise when you don’t allow yourself the luxury of excluded middle in your proofs, in particular the interesting effects this has on the theory of real analysis!
Noo don’t exclude the middle you’re so constructive aha
Sep. 24, 2020 1pm (Zoom (vir…
Rep Theory
Jinwei Yang (University of Alberta)
X
Tensor categories of vertex operator algebras play an important role in the study of vertex operator algebras and conformal field theories. A central problem of tensor category theory of Huang-Lepowsky-Zhang is the existence of the vertex tensor category structure. We develop a few general methods to establish the existence of tensor structure on module categories for vertex operator algebras, especially for non-rational and non-C_2 cofinite vertex operator algebras. As applications, we obtain the tensor structure of affine Lie algebras at various levels, affine Lie superalgebra gl(1|1), the Virasoro algebra at all central charges as well as the singlet algebras.
We also study important properties, including constructions of projective covers, fusion rules and rigidities of these tensor categories.
This talk is based on joint work with T. Creutzig, Y.-Z. Huang, F. Orosz Hunziker, C. Jiang, R. McRae and D. Ridout.
Coxeter groups are groups with a certain kind of presentation. Associated to each such group is an R-algebra called the Hecke algebra, and, in some cases, quotients of these algebras are isomorphic to a class of diagram algebras called Generalized Temperley-Lieb algebras. A diagram algebra is an algebra whose basis can be represented as pretty pictures, where multiplication is performed by "pasting pictures together" and following certain rules. Were this talk in person, I would undoubtedly bring construction paper, crayons, and Elmer's glue so we could all do calculations in these algebras together; alas, we will have to settle for looking at pretty pictures on my screen instead. At any rate, we will explore the connection between Coxeter groups and these diagram algebras and discuss some results and open conjectures.
Coxeter groups and diagram algebras (pretty pictures included)
I will discuss what it means for the zero set of a polynomial in several variables to be rational, as well as an open problem regarding cubic polynomials.
Rationality of cubic hypersurfaces
Sep. 29, 2020 1pm (Zoom)
Peter Mayr (Boulder)
X
We review properties of varieties generated by finite simple algebras, their subvarieties and free objects. In particular we show that every congruence modular variety that is generated by a finite nonabelian simple algebra has a unique maximal subvariety.
We review properties of varieties generated by finite simple algebras, their subvarieties and free objects. In particular we show that every congruence modular variety that is generated by a finite nonabelian simple algebra has a unique maximal subvariety.
Geometry of Cascade Feedback Linearizable Control Systems