Noncommutative Geometry

Speaker: Ali Moatadelro (Western)

"Representation theory of compact quantum groups with examples, lecture 5. Irreducible representations of SU(3), continued."

Time: 09:30 - 11:30

Room: MC 107

In this series of lectures, we will discuss basic examples of compact quantum groups and their (finite dimensional) representations. We will start with reviewing the classical theory. We shall classify all finite dimensional irreducible representations of compact Lie groups SU(2) and SU(3). Then we will proceed to the general theory of representation of compact Lie groups and will discuss several important results including the highest weight theory, the Peter-Weyl decomposition theorem, and also the Borel-Weil-Bott construction of representations. Finally, we will see how much of the theory holds in the quantum case.

Geometry and Topology

Speaker: Bert Guillou (University of Illinois)

"G-spectra are spectral Mackey functors"

Time: 15:30 - 16:30

Room: MC 107

Equivariant spectra have received a good deal of attention lately due to their central role in the Hill-Hopklins-Ravenel proof of the Kervaire invariant one problem. I will describe joint work with Peter May that provides an alternative model for equivariant spectra (indexed on a complete G-universe).

Analysis Seminar

Speaker: Ilya Kossovskiy (Western)

"Envelopes of holomorphy for real submanifolds in a complex space"

Time: 15:30 - 16:30

Room: MC 107

One of the most impressive phenomena in several complex variables is the phenomenon of forced analytic continuation for holomorphic functions. The biggest domain, to which the family of all holomorphic functions extends, is called the envelope of holomorphy of a domain or of a real submanifold in a complex space. Envelopes of holomorphy have some nice geometric description, making them in a sense similar to convex hulls of domains and submanifolds in a Euclidian space. In the present talk we discuss some classical theorems for domains of holomorphy as well as some new results for real submanifolds in a complex space.

Noncommutative Geometry

Speaker: Ali Moatadelro (Western)

"Holomorphic structures on the quantum projective space"

Time: 14:30 - 16:30

Room: MC 107

We review the notion of holomorphic vector bundles in noncommutative geometry and then define holomorphic structures on canonical line bundles on the quantum projective space. The space of holomorphic sections of these line bundles will determine the quantum homogeneous coordinate ring of the quantum projective space. We also show that the holomorphic structure is naturally represented by a twisted positive Hochschild cocycle.

Symplectic Learning Seminar

Speaker: M. VanHoof (Western)

"Connectedness of Symp groups, part II"

Time: 13:00 - 14:30

Room: MC 105C

This is a continuation of our discussion of the connectedness of $Symp(CP^2)$ an other symplectomorphism groups.

Colloquium

Speaker: Rahim Moosa (Waterloo)

"A Hasse-Schmidt approach to expansions of algebraic geometry"

Time: 15:30 - 16:30

Room: MC 107

Differential-algebraic geometry is the expansion of algebraic geometry where in addition to polynomial equations one considers (partial) algebraic differential equations. Similarly, using automorphisms instead of derivations, we have difference-algebraic geometry. In this talk I will give an introduction to these subjects and to describe recent work with Tom Scanlon in which we develop a new foundation that unifies and generalises them. Our approach is informed by the theory of iterative Hasse-Schmidt derivations and is based upon an abstract notion of "prolongation" for an algebraic variety.

Algebra Seminar

Speaker: Andrey Minchenko (Western)

"Differential representations of $SL(2)$"

Time: 14:30 - 15:30

Room: MC 107

In order to describe linear representations of a group, it is sufficient to find all its indecomposable representations. It is known that indecomposable algebraic representations of G=SL(2) correspond to irreducible subrepresentations of G in the ring R of polynomials in two variables x and y. Given a derivation ' on the ground field, R extends to a G-representation R' by adding variables x', y', x", y", etc. We will investigate indecomposable subrepresentations of R' and discuss their relation to description of all differential representations of G. If time permits, I will describe the category of differential representations of tori.

Noncommutative Geometry

Speaker: Ali Moatadelro (Western)

"Representation theory of compact quantum groups with examples, lecture 6, Irreducible representations of semisimple Lie agbras."

Time: 09:30 - 11:30

Room: MC 107

In this series of lectures, we will discuss basic examples of compact quantum groups and their (finite dimensional) representations. We will start with reviewing the classical theory. We shall classify all finite dimensional irreducible representations of compact Lie groups SU(2) and SU(3). Then we will proceed to the general theory of representation of compact Lie groups and will discuss several important results including the highest weight theory, the Peter-Weyl decomposition theorem, and also the Borel-Weil-Bott construction of representations. Finally, we will see how much of the theory holds in the quantum case.

Geometry and Topology

Speaker: Clark Barwick (MIT)

"Algebraic K-theory of $\infty$-categories"

Time: 15:30 - 16:30

Room: MC 107

In joint work with John Rognes, we show how to transfer the technologies and results of Quillen and Waldhausen in higher algebraic $K$-theory to the context of $\infty$-categories. Analogues of the $S_{\bullet}$ and $Q$-constructions — as well as versions of the additivity, localization, and d evissage theorems — are among the results we find in this new context. As a motivation for this work, we discuss a conjecture of Hopkins, Waldhausen, and Rognes on the algebraic $K$-theory of $BP\langle n\rangle$.

Analysis Seminar

Speaker: Ilya Kossovskiy (Western)

"Envelopes of holomorphy for real submanifolds in a complex space II"

Time: 15:30 - 16:30

Room: MC 107

One of the most impressive phenomena in several complex variables is the phenomenon of forced analytic continuation for holomorphic functions. The biggest domain, to which the family of all holomorphic functions extends, is called the envelope of holomorphy of a domain or of a real submanifold in a complex space. Envelopes of holomorphy have some nice geometric description, making them in a sense similar to convex hulls of domains and submanifolds in a Euclidian space. In the present talk we discuss some classical theorems for domains of holomorphy as well as some new results for real submanifolds in a complex space.

Noncommutative Geometry

Speaker: Ali Fathi (Western)

"Cyclic cohomology 9, K-theory for C^*-algebras (II): Basic K-groups, continued."

Time: 14:30 - 16:30

Room: MC 107

Cyclic (co)homology is the noncommutative analogue of de Rham (co)homology and as such plays an important role in noncommutative geometry and its applications (in operator algebras, index theory, ...) A variant of it, topological Hochschild and cyclic homology, plays an important role in algebraic K-theory as well.

We will give a series of lectures on the subject (2 hours per week), starting from basic material and gradually building towards more advanced stuff. Outline: 1. Basic homological algebra in abelian categories 2. Hochschild (co)homology; computations (Hochschild-Kostant-Rosenberg, group algebras) 3. Cyclic (co)ohomology, Connes' spectral sequence; computations (relation with de Rham, group algebras); cyclic category. 4. K-theory and K-homology, 5. Connes-Chern character 6. An index formula 7. Applications to idempotent conjectures.

The basic texts to follow are: 1. Cyclic Homology, J. L. Loday 2. Noncommutative Geometry, A. Connes 3. Noncommutative Differential Geometry, Publication math. IHES, 1985, A. Connes. 4. Basic noncommutative geometry, Masoud Khalkhali

Colloquium

Speaker: Heydar Radjavi (Waterloo)

"When small parts imply small wholes"

Time: 15:30 - 16:30

Room: MC 107

Roughly speaking, this talk is about questions of the following general type on collections of matrices or linear operators: if we know something about the collection to be "small" in some sense, when can we say the collection is itself small? A classical example of the kind of results we are interested in is the old theorem that if a group G of complex matrices is irreducible (i.e., the members of G have no nontrivial invariant subspace in common), and if the traces of members of G form a finite set, then G is itself finite. We'll consider (a) measures of smallness other than finiteness, e.g., boundedness; and (b) linear functionals other than just trace.

Algebra Seminar

Speaker: Ekaterina Shemyakova (Western)

"Darboux transformation methods for integrable equations: some classical and own results"

Time: 14:30 - 15:30

Room: MC 107

Darboux transformation methods for integrable equations can be classified as differential equations, algebra, analysis, geometry of surfaces, mathematical physics or theoretic computer algebra. In my talk I shall give some introdution into the area intertwining some classical results with some of my own.

Noncommutative Geometry

Speaker: Asghar Ghorbanpour (Western)

"K-homology (I. Very beginning of the theory)"

Time: 14:30 - 16:30

Room: MC 107

Having well-known isomorphism between groups K^0(X)\otimes R and H^{ev}(X;R), for compact space X, one can think of (generalized) homological theory related to K-theory same as homology theory of spaces for cohomology theory. This theory can be constructed in an abstract way by using topological N-dual space of X. M. F. Atiyah in his paper "Global Theory of Elliptic operators" showed how we can find representatives for elements of the theory, so called K-homology, by abstract elliptic operators on X. In this talk we will review atiyah's paper, it will include following parts:

- K-theory as a generalized cohomology theory and its appropriate way to define homology theory related to it.

- Elliptic operators on a compact manifolds and their abstract analogue for general compact spaces.

- K-index of elliptic operators and k-homology

Symplectic Learning Seminar

Speaker: M. VanHoof (Western)

"Resolving singularities of weighted projective spaces"

Time: 13:10 - 15:10

Room: MC 104

We will show how to resolve the singularities of the symplectic orbifold $(CP^2(1,2,q), \omega_{FS})$, where $q > 0$ is an odd positive integer. The approach we take involves symplectic cutting, and this is the first step towards trying to understand the symplectomorphism group of the orbifold in question. A guiding philosophy is to try to keep the number of blow ups to a minimum (in this case, 3) since the symplectomorphism group becomes much harder to understand after a certain number of blow ups."

Geometry and Topology

Speaker: Pablo Pelaez (Duisburg-Essen)

"Rigid Motivic Homotopy Groups"

Time: 15:30 - 16:30

Room: MC 108

We will recall Voevodsky's definition of the presheaves of rigid motivic homotopy groups and show that they admit a canonical structure of presheaves with transfers when we consider rational coefficients and quasi-excellent base schemes.

Noncommutative Geometry

Speaker: Farzad Fathizadeh (York University)

"A Noncommutative Residue for Pseudodifferential Operators on the Noncommutative Two Torus"

Time: 12:30 - 13:30

Room: MC 107

I will first explain a pseudodifferential calculus for the canonical dynamical system associated to the noncommutative two torus which is a special case of Connes' pseudodifferential calculus for C^* dynamical systems. Then I will report on a recent joint work with M. W. Wong where we define a noncommutative residue for classical pseudodifferential operators on the noncommutative two torus, and prove that up to a constant multiple it is the unique trace on the algebra of classical pseudodifferential operators modulo infinitely smoothing operators.

Pizza Seminar

Speaker: Mehdi Garrousian (Western)

"Sudoku Enumeration and Hyperplane Arrangements"

Time: 16:30 - 17:30

Room: MC 107

There are various mathematical models and solution strategies for the sudoku puzzle. In this expository talk, I will use the language of graph theory and hyperplane arrangements to approach the sudoku enumeration problem. In particular, I will show how sudokus suggest a finite field interpretation of the characteristic polynomial of hyperplane arrangements. If time permits I will also discuss a Groebner basis algorithm for determining whether a given sudoku puzzle has a solution.

Noncommutative Geometry

Speaker: Sheldon Joyner (Western)

"The Grothendieck-Teichmuller group"

Time: 14:30 - 16:30

Room: MC 107

This talk will handle Drinfel'd's construction of the object of the title by means of deformations of braided monoidal categories.

Symplectic Learning Seminar

Speaker: M. VanHoof (Western)

"Resolving singularities of weighted projective spaces (2)"

Time: 10:30 - 12:00

Room: MC 104

This is the second part of our discussion of singularities of weighted projective spaces and their resolutions.

Symplectic Learning Seminar

Speaker: M. Mousavi (Western)

"Schur-Horn-Kostant theorem for Symplectomorphisms of toric manifolds"

Time: 13:30 - 15:00

Room: MC 104

We continue our exposition of the Schur-Horn-Kostant theorem for Symplectomorphisms groups of toric manifolds

Colloquium

Speaker: Victor Snaith (Sheffield)

"A history of the Arf-Kervaire invariant problem"

Time: 15:30 - 16:30

Room: MC 107

More than 50 years ago Michel Kervaire constructed a manifold with no differentiable structure. Constructing manifolds is the easy part - the trick is to construct an invariant, in this case the Arf-Kervaire invariant, which guarantees the existence or otherwise of the property one is after. The connection between framed manifolds and stable homotopy groups led (circa 1960) to the problem: ``Do there exist framed manifolds of odd Arf-Kervaire invariant?''. Recently Hill-Hopkins-Ravenel proved that the outcome is: ``mostly framed manifolds of Arf-Kervaire invariant one do not exist''.

Here is what Lewis Carroll has to say about non-existent things!

``I know what you're thinking about,'' said Tweedledum; ``but it isn't so, nohow.'' ``Contrariwise,'' continued Tweedledee,``if it was so, it might be; and if it were so, it would be; but as it isn't; it ain't. That's logic.''