Colloquium

Speaker: Greg Smith (Queen's University)

"Syzygies and Polytopes"

Time: 15:30 - 16:30

Room: MC 108

How does one see geometry in a system of polynomial equations? In this talk, we will examine the relationship between the geometry of a projective variety and the homogeneous equations that define it. After surveying some of the classic results, we will discuss recent developments for toric varieties.

Analysis Seminar

Speaker: Ilya Kossovskiy (Western)

"Systems of PDE's associated to CR-manifolds and applications"

Time: 15:30 - 16:30

Room: MC 108

In this series of 3 talks I will first state the general concept of a system of PDE's associated to a non-degenerate CR-manifold. The idea goes back to Lie, Cartan and Segre, and it was undeservedly forgotten. The PDE-approach was recently reviewed by A.Sukhov and J.Merker and enabled the latter one to obtain some interesting results in CR-geometry. The classical results of S.Lie and the recent results of J.Merker will be stated on the second lecture. Finally, on the last lecture I will tell about a recent result with R.Shafikov concerning extension of holomorphic mappings where the PDE-approach was successfully applied as well.

Algebra Seminar

Speaker: Detlev Hoffmann (Dortmund)

"Sums of squares"

Time: 10:30 - 12:00

Room: MC 107

Sums of squares have been a research topic for as long as people have studied algebra and number theory. In modern language, some of the central questions are as follows. Let R be a ring with 1. Which elements in R can be written as sums of squares of elements in R? If an element is a sum of squares, how many squares are needed to write it as such? We give a survey of a few (of the many) classical and more recent results and open problems, focusing on fields, simple (or division) algebras and commutative rings.

Colloquium

Speaker: Masoud Khalkhali (Western)

"Weyl Law at 101"

Time: 15:30 - 16:30

Room: MC 108

In 1911 Hermann Weyl proved his famous law on the asymptotic distribution of eigenvalues of Laplacians on a bounded domain. Answering a question posed by physicists, Lorentz and Sommerfeld among others, he showed that the statistics of large eigenvalues determines the  volume and dimension of such a domain. This result, which nowadays is paraphrased as``one can hear the volume and dimension of a bounded domain", is the foundation stone of a remarkable edifice  of modern  mathematics known as spectral geometry. The  ultimate goal is to know how much of the geometry and topology of a Riemannian manifold is encoded in its spectrum. Forexample, is it true that isospectral manifolds are isometric? That is,  Can one hear the shape of a drum? While the answer is in general negative, we  know that one can  hear the total scalar curvature, Betti numbers, and signature   of a closed Riemannian manifold. In fact an  infinite sequence of spectral invariants, known as DeWitt--Gilkey--Seeley coefficients can be defined and  computed from the short time asymptotic of the heat kernel. Another set of ideas in spectral geometry concerns with different types of trace formulae and applications to number theory, quantum physics, and quantum chaos. In some sense this even goes back to the very origins of the Weyl  law in quantum mechanics and in deriving Planck's radiation formula from it.

In this talk I shall outline some of these connections and then focus on our current joint work with Farzad Fathizadeh and show how some of these ideas can be imported to the world of noncommutative geometry of Alain Connes. In fact without spectral geometry there could be no noncommutative geometry!  In particular I shall highlight our recent proof of a Weyl law for noncommutative tori equipped with a general metric.

Algebra Seminar

Speaker: Detlev Hoffmann (Dortmund)

"Differential forms, Milnor K-theory and bilinear forms under field extensions"

Time: 14:30 - 15:30

Room: MC 108

Let $F$ be a field of characteristic $p>0$, and let $X$ be an integral affine hypersurface over $F$ with function field $K=F(X)$. We determine the kernels of the restriction maps given by extending scalars from $F$ to $K$ for the following algebraic objects: the space of absolute Kahler differentials, Milnor $K$-theory modulo $p$, and the Witt ring of symmetric bilinear forms in the case $p=2$. All these kernels have a surprisingly explicit description. This is joint work by Andrew Dolphin and myself, the case of Milnor $K$-theory being due to Stephen Scully who uses our results on differential forms.

Noncommutative Geometry

Speaker: Sajad Sadeghi (Western)

"NCG Learning Seminar: Introduction to Operator Algebra I: Gelfand-Naimark Theorems"

Time: 14:30 - 15:30

Room: MC 107

In this lecture, we are going to present two Gelfand-Naimark theorems: Firstly, it is shown that every commutative C*-algebra is isomorphic to the algebra of continuous functions on its spectrum, vanishing at infinity. Finally, a fundamental theorem, known as GNS construction will be discussed. Actually it is shown that every C*-algebra can be seen as a C*-subalgebra of the algebra of bounded linear operators on a Hilbert space.

Geometry and Topology

Speaker: Milena Pabiniak (University of Toronto)

"Lower bounds on Gromov width of coadjoint orbits through the Gelfand-Tsetlin pattern."

Time: 15:30 - 16:30

Room: MC 108

Gromov width of a symplectic manifold M is a supremum of capacities of balls that can be symplectically embedded into M. The definition was motivated by the Gromov's Non-Squeezing Theorem which says that maps preserving symplectic structure form a proper subset of volume preserving maps.

Let G be a compact connected Lie group G, T its maximal torus, and $\lambda$ be a point in the chosen positive Weyl chamber.

The group G acts on the dual of its Lie algebra by coadjoint action. The coadjoint orbit, M, through $\lambda$ is canonically a symplectic manifold. Therefore we can ask the question of its Gromov width.

In many known cases the width is exactly the minimum over the set of positive results of pairing $\lambda$ with coroots of G:

$$\min \{ \langle \alpha_j^{\vee},\lambda \rangle; \alpha_j \textrm{ a coroot, }\langle \alpha_j^{\vee},\lambda \rangle>0\}.$$

For example, this result holds if G is the unitary group and M is a complex Grassmannian or a complete flag manifold satisfying some additional integrality conditions.

We use the torus action coming from the Gelfand-Tsetlin system to construct symplectic embeddings of balls. In this way we prove that the above formula gives the lower bound for Gromov width of U(n) and SO(n) coadjoint orbits.

In the talk I will describe the Gelfand-Tsetlin system and concentrate mostly on the case of regular U(n) orbits.

Analysis Seminar

Speaker: Ilya Kossovskiy (Western)

"Systems of PDE's associated to CR-manifolds and applications"

Time: 15:30 - 16:30

Room: MC 108

In this series of 3 talks I will first state the general concept of a system of PDE's associated to a non-degenerate CR-manifold. The idea goes back to Lie, Cartan and Segre, and it was undeservedly forgotten. The PDE-approach was recently reviewed by A.Sukhov and J.Merker and enabled the latter one to obtain some interesting results in CR-geometry. The classical results of S.Lie and the recent results of J.Merker will be stated on the second lecture. Finally, on the last lecture I will tell about a recent result with R.Shafikov concerning extension of holomorphic mappings where the PDE-approach was successfully applied as well.

Noncommutative Geometry

Speaker: Ali Fathi (Western)

"Noncommutative Chern-Simons Gauge Theory I"

Time: 14:30 - 16:30

Room: MC 107

Noncommutative Geometry

Speaker: Sajad Sadeghi (Western)

"NCG Learning Seminar: Introduction to Operator Algebra II: Gelfand-Naimark Theorems"

Time: 14:30 - 15:30

Room: MC 107

In this lecture we are going to talk about an equivalence between the category of compact Hausdorff topological spaces and the category of commutative unital C*-algebras. We also introduce spectrum of an element in a unital C*-algebra and give some examples. Then we present GNS construction.

Geometry and Topology

Speaker: Hugo Bacard (Western)

"co-Segal categories"

Time: 15:30 - 16:30

Room: MC 108

For a symmetric monoidal model category $\mathscr{M} = (\underline{M};\otimes; I)$, we develop a theory of weakly enriched categories over $\mathscr{M}$, called co-Segal $\mathscr{M}$-categories. By ‘weakly enriched’ we mean a sort of enriched category where there is no prescription of a composition; but rather we allow many possible compositions, where each of them is associative up-to homotopies. Their definition derives from the philosophy of classical Segal categories; and just like for Segal categories they give rise to higher categorical structures. In this talk I will present the theory of co-Segal categories and will give the first results on their homotopy theory.

Analysis Seminar

Speaker: Ilya Kossovskiy (Western)

"Systems of PDE's associated to CR-manifolds and applications"

Time: 15:30 - 16:30

Room: MC 108

In this series of 3 talks I will first state the general concept of a system of PDE's associated to a non-degenerate CR-manifold. The idea goes back to Lie, Cartan and Segre, and it was undeservedly forgotten. The PDE-approach was recently reviewed by A.Sukhov and J.Merker and enabled the latter one to obtain some interesting results in CR-geometry. The classical results of S.Lie and the recent results of J.Merker will be stated on the second lecture. Finally, on the last lecture I will tell about a recent result with R.Shafikov concerning extension of holomorphic mappings where the PDE-approach was successfully applied as well.

Noncommutative Geometry

Speaker: Ali Fathi (Western)

"Noncommutative Chern-Simons Gauge Theory II: The Algebraic Setting"

Time: 14:30 - 16:30

Room: MC 107

Algebra Seminar

Speaker: Lex Renner (Western)

"Quasi-invariant theory"

Time: 14:30 - 15:30

Room: MC 108

One of the main themes of invariant theory is to relate the $G$-invariant regular functions, of a regular action $G\times X\to X$, to some suitable quotient morphism $\pi : X\to Y$. However, there are examples to show that the naive attempt $X\mapsto k[X]^G$ does not lead directly to any appealing conclusion. Indeed, $k[X]^G$ may not be finitely generated, or it may not be "large" enough to separate the $G$-orbits of $G\times X\to X$, even generically.

The purpose of this talk is to discuss some basic results of "quasi-invariant theory". The main ideas here have their roots in the work of Hilbert, Zariski, Nagata, and Rosenlicht. Our major purpose is to assess the influence of quasi-invariant rational functions and $G$-invariant divisors on the problem of constructing a useful quotient object of a regular action $G\times X\to X$.