Geometry and Topology

Speaker: Ivan Kobyzev (Western)

"$K$-theory of root stacks and its application to equivariant $K$-theory"

Time: 15:30

Room: MC 107

There is a result from the 1980's that allows us to describe the equivariant $K$-theory of curves: if $X$ and $Y$ are curves, G is a finite reducible group and $Y = X/G$, then we can write $K_G(X)$ in terms of $K(Y)$ and some representation rings associated to the group. Prof. Dhillon and I have generalized this result to any dimension using the description of the category of coherent sheaves on a root stack given by Borne and Vistolli.

Analysis Seminar

Speaker: Tatyana Barron (Western)

"Toeplitz operators on hyperkahler and multisymplectic manifolds"

Time: 14:30

Room: MC 107

I will report on results obtained in a recent paper with Baran Serajelahi. I will describe quantization constructions, obtained from Berezin-Toeplitz quantization, for an n-dimensional compact Kahler manifold regarded as a (2n-1)-plectic manifold, and for a hyperkahler manifold.

Comprehensive Exam Presentation

Speaker: Mohsen Mollahajiaghaei (Western)

"Resonance Varieties of Graphical Arrangements"

Time: 14:30

Room: MC 108

To each differential-graded algebra and element a\in A^1, we associate a cochain complex, where the map is defined by the multiplication by a. The degree l resonance variety is the set of elements a in A^1 such that the l-th cohomology is not zero. It is shown that The degree l resonance variety, up to ambient linear isomorphism, is an invariant of A. The characteristic varieties of a space are the jump loci for homology of rank 1 local systems. The main motivation for the study of resonance varieties comes from the tangent cone, which there is a close relation between the degree-one resonance varieties to the characteristic varieties, where the tangent cone of W at 1 is the algebraic subset TC_1(W) of C^n defined by the initial ideal in(J) \subset S. In this talk we describe the degree-one resonance variety. We will be particularly interested in the resonance varieties of graphical arrangements.

Noncommutative Geometry

Speaker: Ali Fathi (Western University (PhD Candidate))

"Regularized traces of elliptic operators II"

Time: 15:00

Room: MC 107

I will explain the construction of Kontsevich-Vishik canonical trace on non-integer order classical pseudodifferential operators. This construction has it roots in the old methods of extracting a finite part from a divergent sum or integral (infra-red and ultra-violet divergence), used by mathematicians and physicists. If time permits I will explain some of the results on generalizations of this construction to noncommutative setting.

Graduate Seminar

Speaker: Sina Hazratpour (Western)

"Logical structure(s) of quantum theory"

Time: 13:00

Room: MC 106

In this talk, we will explore few logical foundations for quantum theory. I will briefly introduce logical syntax and semantic, and the important notion of Lindebaum Algebra. After acquiring essential logical equipment, I will discuss Partial Boolean Algebras as an algebraic model to formulate Kochen- Specker theorem in pure logical setting. I will as well mention the toposical foundation for quantum theory and the formulation of Kochen- Specker theorem in this setting. During the talk, some historical and philosophical issues will also be addressed.

Homotopy Theory

Speaker: Martin Frankland (Western)

"The generation theorem for stable homotopy"

Time: 14:00

Room: MC 107

We will present a theorem due to J. Cohen that the stable homotopy groups of spheres are generated under higher Toda brackets by the classes in Adams filtration one: the Hopf classes as well as the first alpha element (for odd primes).

Algebra Seminar

Speaker: Graham Denham (Western)

"Combinatorial covers and cohomological vanishing"

Time: 15:30

Room: MC 107

We construct a combinatorial framework for proving cohomological vanishing results on certain classes of spaces, by means of a Mayer-Vietoris-type spectral sequence and certain Cohen-Macaulayness hypotheses. The spaces include complex hyperplane complements, their De Concini-Procesi compactifications, and configuration spaces of points in tori. In particular, we generalize classical vanishing results due to Kohno, Esnault, Schechtman and Vieweg, and recent work of Davis, Januszkiewicz, Leary and Okun.

This is joint work with Alex Suciu and Sergey Yuzvinsky.

Noncommutative Geometry

Speaker: Asghar Ghorbanpour (Western University)

"Dirac operators by example"

Time: 11:00

Room: MC 106

After a quick review of the general theory of the Dirac operators and fixing some notations, I will construct the Dirac operator for some of Riemannian manifolds. This will include torus with conformally perturbed flat metric, 2-sphere and 3-spheres with the round metric and if time allows me, I will construct the Dirac operator for the Robertson-Walker metrics.

Analysis Seminar

Speaker: Janusz Adamus (Western)

"On finite determinacy of the local geometry of analytic maps (Part I)"

Time: 14:30

Room: MC 107

Given a (real or complex) analytic map $f=(f_1,\dots,f_n):X\to\mathbb{K}^n$, one can consider its approximations by maps $T_d(f)$ whose coordinates are Taylor polynomials of the $f_i$ of degree $d$. We will show that the continuity of the family of fibres of $f$ is finitely determined. That is, it is already determined by the polynomial maps $T_d(f)$ for $d$ sufficiently large. As a consequence, we also obtain finite determinacy of analytic complete intersections. This is joint work with H. Seyedinejad.

Noncommutative Geometry

Speaker: Alimjon Eshmatov (Western University (Assistant Professor and Postdoctoral Fellow))

"Noncommutative Poisson Structures"

Time: 15:00

Room: MC 107

In this talk, we will discuss Noncommutative Poisson Structure which was introduced by Crawley-Boevey and how it fits nicely with Kontsevich-Rosenberg principle. We will also give some examples. If time allows, we will also discuss its relation to Van den Bergh's Double Poisson Algebras.

Graduate Seminar

Speaker: Asghar Ghorbanpour (Western)

"An introduction to the spectral action principle"

Time: 13:00

Room: MC 106

The spectral action principle was introduced by Connes and Chamseddine to formulate the notion of action for spectral triples. This talk will be an introduction to this principle and to the so called spectral action functional. After a quick review of the theory of Dirac operators and spectral triples, we will show how one can produce classical actions, such as Einstein-Hilbert or Yang-Mills action, by considering the spectral action functional for a suitable Dirac operator.

Homotopy Theory

Speaker: Cihan Okay (Western)

"Hopf invariant one problem"

Time: 14:00

Room: MC 107

I will sketch Adams' original proof of the Hopf invariant one problem.

Algebra Seminar

Speaker: Nicole Lemire (Western)

"Four-dimensional algebraic tori"

Time: 15:30

Room: MC 107

We investigate the rationality of four-dimensional algebraic tori and the associated equivariant birational linearisation problem. We connect these problems to the question of determining when an algebraic group is (stably) Cayley - that is (stably) equivariantly birationally isomorphic to its Lie algebra and earlier joint work on the (stably) Cayley problem with Popov, Reichstein, Borovoi and Kunyavskii.

Noncommutative Geometry

Speaker: Asghar Ghorbanpour (Western University)

"Bochner-type formulas in spin geometry"

Time: 11:00

Room: MC 106

A formula that expresses a Laplace-type operator, i.e an operator with the metric tensor as their leading symbol, as sum of the Laplacian of a connection and an endomorphism, is called Bochner-type formula. The goal of this talk is to review the general form of Bochner type formulas and focus on some specific examples from the spin geometry, such as Lichnerowicz formula for Dirac and twisted Dirac operators.

Analysis Seminar

Speaker: Janusz Adamus (Western)

"On finite determinacy of the local geometry of analytic maps (Part II)"

Time: 14:30

Room: MC 107

Given a (real or complex) analytic map $f=(f_1,\dots,f_n):X\to\mathbb{K}^n$, one can consider its approximations by maps $T_d(f)$ whose coordinates are Taylor polynomials of the $f_i$ of degree $d$. We will show that the continuity of the family of fibres of $f$ is finitely determined. That is, it is already determined by the polynomial maps $T_d(f)$ for $d$ sufficiently large. As a consequence, we also obtain finite determinacy of analytic complete intersections. This is joint work with H. Seyedinejad.

Homotopy Theory

Speaker: (Western)

"No meeting today"

Time: 14:00

Room: MC 107

We resume next week.

Algebra Seminar

Speaker: (Western)

"No seminar today"

Time: 14:30

Room: MC 107

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)