Geometry and Topology

Speaker: Rick Jardine (Western)

"Presheaves of spectra"

Time: 15:30

Room: MC 107

This talk is a basic introduction to the stable homotopy theory of presheaves of spectra, and its applications.

Analysis Seminar

Speaker: Wayne Grey (Western)

"The inclusion problem for mixed norm Lebesgue spaces"

Time: 14:30

Room: MC 107

Mixed norm Lebesque spaces are Banach spaces of multi-variable functions for which a different Lp norm is used in each variable. In many cases, there are simple, easy-to-compute conditions that ensure one given mixed norm space is contained in another. In other cases, only complicated, hard-to-compute conditions will do.

Dept Oral Exam

Speaker: Wayne Grey (Western)

"The inclusion problem for mixed norm Lebesgue spaces"

Time: 14:30

Room: MC 107

Mixed norm Lebesque spaces are Banach spaces of multi-variable functions for which a different Lp norm is used in each variable. In many cases, there are simple, easy-to-compute conditions that ensure one given mixed norm space is contained in another. In other cases, only complicated, hard-to-compute conditions will do.

Graduate Seminar

Speaker: Mohsen Mollahajiaghaei (Western)

"APPLICATIONS OF POLYNOMIALS IN COMBINATORICS."

Time: 13:00

Room: MC 106

Polynomials are fundamental objects in many branches in Mathematics. Several problems in combinatorics and number theory have been solved in an interesting way using polynomials. For example, Cauchy-Davenport theorem and Snevily’s Conjecture can be proved easily by this method. In this talk, we mainly focus on Combinatorial Nullstellensatz and Chevalley-Warning theorems, with some applications in combinatorial number theory. The Combinatorial Nullstellensatz is a theorem about the roots of a polynomial. Chevalley-Warning theorem deals with the number of solutions of system of polynomial equations. As a consequence of Chevalley-Warning, finite fields are quasi-algebraically closed. This had been conjectured by Emil Artin in 1935.

Homotopy Theory

Speaker: Dan Christensen (Western)

"Higher Toda brackets in triangulated categories"

Time: 14:00

Room: MC 107

I will talk about the definition of higher Toda brackets in triangulated categories and possibly about how these are determined by the triple Toda brackets.

Noncommutative Geometry

Speaker: Mitsuru Wilson (Western University (PhD Candidate))

"NCG Learning Seminar: Local index formula"

Time: 11:00

Room: MC 106

The index of a bounded operator $T\in B(H)$ of a Hilbert space $H$ is defined as the difference between the dimensions of kernel and cokernel. That is, $${\rm Ind}(T):=\dim(\ker T)-\dim({\rm coker}T)$$ This index, if defined, is called the Fredholm index. The Fredholm index of an operator on a finite dimensional Hilbert space $H$ by the dimension theorem in linear algebra. However, the case of infinite dimensional Hilbert spaces requires more delicate analysis and an operator with nonzero index exists. The celebrated local index formula in noncommutative geometry (Connes and Moscovici 1995) relates the index of Dirac type operators and the residue cocycle in the cyclic cohomology. In the classical case, this formula equates topology and geometry. In my talk, I will prove two special cases of local index formula following closely the chapter 5 in Noncommutative geometry and particle physics by Walter Van Suijlekom. If the time is allotted, I will demonstrate the strength of the formula using simple classical spectral triples such as the circle $S^1$.

Algebra Seminar

Speaker: Johannes Middeke (Western)

"On the computation of $\pi$-flat outputs for linear time-varying differential-delay systems"

Time: 15:30

Room: MC 107

A flat output of a control system allows to express its state and its inputs as a function of the flat output and its derivatives. It can be used, for example, to solve motion planning problems. We propose a variation of the definition of flatness for linear differential systems to linear differential-delay systems with time-varying coefficients which utilises a prediction operator $\pi$. We characterize $\pi$-flat outputs and provide an algorithm to efficiently compute such outputs.

(Joint work with Jean Levine, Felix Antritter and Franck Cazaurang)