Geometry and Topology

Speaker:

"No seminar"

Time: 15:30 - 16:30

Room: MC 108

Analysis Seminar

Speaker: Patrick Speissegger (McMaster)

"O-minimal transition maps and Roussarie's finite cyclicity conjecture"

Time: 15:30 - 16:30

Room: MC 108

Let F be the family of all polynomial vector fields of degree d in the plane. Hilbert's 16th problem conjectures that there is a finite bound on the number of limit cycles of the vector fields belonging to F. This as yet open problem (if d is at least 2) has a tantalizingly model-theoretic flavor, but no model-theoretic framework has been discovered so far to capture it. On the other hand, Roussarie's finite cyclicity conjecture reduces the problem to a localized (in the parameter space) one. In recent joint work with Kaiser and Rolin, we used o-minimality (a branch of model theory) to establish Roussarie's conjecture in a very special case. I will survey our approach, with an emphasis on the role o-minimality plays in obtaining a finite upper bound on the number of limit cycles.

Colloquium

Speaker: Stephen M. Watt (Western)

"Algorithms for symbolic polynomials"

Time: 15:30 - 16:30

Room: MC 108

We wish to compute with polynomials where the exponents are not known in advance. Expressions of this sort arise frequently in practice, for example in the analysis of algorithms, and it is difficult to work with them effectively in current computer algebra systems. There are various simple operations we must be able to perform, such as squaring $x^{2n}-1$ to get $x^{4n}-2x^{2n}+1$, or differentiating to get $2nx^{2n-1}$. Other operations are less obvious.

We consider the case where multivariate polynomials can have exponents that are themselves integer-valued multivariate polynomials. We call these objects "symbolic polynomials" and show they form a unique factorization domain, naturally related to the polynomial ring. We present algorithms to compute their GCD, factorization and functional decomposition.

Algebra Seminar

Speaker: Dan Christensen (Western)

"The dual of the Steenrod algebra"

Time: 14:30 - 15:30

Room: MC 108

This is the second talk in the Steenrod algebra learning seminar.

Geometry and Topology

Speaker: Joe Neisendorfer (Rochester)

"Homotopy groups with coefficients"

Time: 15:30 - 16:30

Room: MC 108

How should homotopy groups with coefficients in an abelian group be defined? There are three criteria. They should be functors on the homotopy category of pointed spaces. They should satisfy a universal coefficient theorem. They should have long exact sequences related to fibrations.

For coefficients in finitely generated abelian groups, such functors exist and are corepresentable. For rational coefficients, such functors exist but it is a theorem of Kan and Whitehead that they are not corepresentable.

In the case of finite coefficients, one would like that the homotopy groups have a global exponent which is the same as that of the coefficient group. The question reduces to the so-called co H-space exponents of Moore spaces. In dimensions 4 and higher, these exponent questions are easy but the answer can be surprising. For example, groups with mod 2 coefficients can have exponent 4.

The case of the exponent of the 3 dimensional homotopy group has some subteties which are addressed by application of a variation of the classical Hopf invariants introduced by Hopf.

Stable Homotopy

Speaker: Dan Christensen (Western)

"The dual of the Steenrod algebra: part 2"

Time: 11:30 - 12:30

Room: MC 107

Colloquium

Speaker: David Riley (Western)

"On Köthe's Conjecture and its kissing cousin, the Kurosh Problem"

Time: 15:30 - 16:30

Room: MC 108

The most famous open problem in the area of nil algebras is the Köthe Conjecture, first posed in 1930, which asserts that if a ring has no nonzero nil ideals then it has no nonzero nil one-sided ideals. This is a fundamental question about the general structure of rings, and a thorough understanding of nil and nilpotent rings is necessary for any serious attempt to understand general rings. The most famous problem about algebraic algebras is the Kurosh Problem, which is of a similar vintage and asks whether the knowledge that a finitely generated algebra is algebraic over a base field is sufficient to ensure that the algebra is finite dimensional. This is untrue in general, as demonstated by Golod and Shafarevich in 1964. However, many partial positive results are known, and the borderline between positive and negative solutions of the Kurosh Problem is still being investigated. There are close connections between these two general themes; for example, the Golod-Shafarevich algebras are infinite dimensional finitely generated nil algebras that are not nilpotent. My talk will be a short survey of the current state of these two themes, including more on their relationship.

Algebra Seminar

Speaker: Christopher Brav (Toronto)

"Stability conditions and Kleinian singularities"

Time: 14:30 - 15:30

Room: MC 108

We review slope stability for coherent sheaves on algebraic curves and then discuss Tom Bridgeland's generalization of stability to triangulated categories. For a certain triangulated category associated to a Kleinian singularity, Bridgeland conjectured that a connected component of the space of stability conditions should be the universal cover of a $K(G,1)$ for $G$ a generalized braid group and showed this is the case for $G$ the classical braid group. We generalize this to braid groups of types ADE. This is joint work with Hugh Thomas.

Geometry and Topology

Speaker: Alexander Nenashev (York)

"Symplectically oriented cohomology theories in algebraic geometry"

Time: 13:30 - 14:30

Room: MC 107

It is about symmetric and skew-symmetric bilinear forms on vector spaces and vector bundles, Witt theory for algebraic varieties, Pontrjagin classes, Thom operators, and orientations on cohomology theories.

Analysis Seminar

Speaker: Damir Kinzebulatov (Toronto)

"Almost periodic holomorphic functions on coverings of complex manifolds"

Time: 15:30 - 16:30

Room: MC 108

H.Bohr's theory of almost periodic functions has numerous applications to various areas of mathematics. Two branches of this theory, both extending the classical setting of almost periodic functions on reals, were particularly rich on interesting and deep results: holomorphic almost periodic functions on tube domains and almost periodic functions on topological groups. This talk is devoted to a natural link between these two concepts - holomorphic almost periodic functions on coverings of complex manifolds, their function-theoretic properties and the `sprouts' of the theory of analytic sheaves on the corresponding Bohr compactifications of the coverings.

This is joint work with Alexander Brudnyi.

Stable Homotopy

Speaker: Enxin Wu (Western)

"Freeness of modules over the Steenrod algebra"

Time: 11:30 - 12:30

Room: MC 108

Algebra Seminar

Speaker: Lex Renner (Western)

"Observable actions of algebraic groups"

Time: 14:30 - 15:30

Room: MC 108

Let $G$ be an affine algebraic group and let $X$ be an irreducible, affine variety. Assume that $G$ acts on $X$ via $G \times X \to X$. The action is called stable if there exists a nonempty, open subset $U\subseteq X$ consisting entirely of closed $G$-orbits. The action is called observable if for any proper, $G$-invariant, closed subset $Y\subseteq X$ there is a nonzero invariant function $f\in k[X]^G$ such that $f|_Y = 0$. It is easy to prove that "observable implies stable" but the two notions are not the same for general groups. We discuss a useful geometric characterization of observability. We then discuss some of the following questions and illustrate them with the appropriate examples.

(1) When is the action $H \times G\to G$, by left translation, observable?
(2) Does the characterization simplify if G is unipotent? solvable? reductive?
(3) What happens if $X$ is factorial? reducible?
(4) Is $int : G\times G\to G$, $(g,x)\mapsto gxg^{-1}$, always observable?
(5) Can we generalize to the case where $X$ is projective and $G \times X \to X$ is linearizable?

Geometry and Topology

Speaker: Jeffrey Morton (Western)

"Extended TQFT by 2-Linearization"

Time: 15:30 - 16:30

Room: MC 108

In this talk, I will describe a 2-functor, called "2-linearization", from spans of groupoids into 2-vector spaces (C-linear abelian categories). Using groupoids representing moduli stacks of flat connections, this gives rise, for every finite group G, to an "extended topological quantum field theory", a 2-functorial invariant for manifolds with corners. I will also discuss how to extend this to compact Lie groups, including measures on stacks, and a generalization of the category of distintegrations (a nice category of measure spaces) to stacks.