Geometry and Topology

Speaker: Pal Zsamboki (Western)

"QC(X)"

Time: 15:30

Room: MC 107

I will explain how to construct the symmetric monoidal infinity-category of complexes of quasi-coherent sheaves on a stack using higher algebra. Afterwards, I will talk about how this might be used to compactify moduli stacks of torsors.

Noncommutative Geometry

Speaker: (Western)

"Learning Seminar"

Time: 11:30

Room: MC 107

We continue the lectures with : ---Clifford algebras, Clifford modules, spin structures, Dirac operators, Weitzenbock formula. ---Heat kernel and its asymptotic expansion, Gilkey's formula, McKean-Singer formula.

Analysis Seminar

Speaker: Lubos Pick (Charles University, Prague)

"Traces of Sobolev functions --- old and new"

Time: 15:30

Room: MC 107

The talk will focus on the classical problem of traces of functions from Sobolev spaces, which had originated in connection with some specific problems in PDEs and then mushroomed into a separate field of research in functional analysis and the function spaces theory. One important property enjoyed by functions from the Sobolev space $W^ {m,p}(\mathbb{R}^ n )$, where $m\in \mathbb{N}$ and $p\in[1,\infty]$, is that their restrictions, called traces, to lower dimensional spaces $\mathbb{R}^ d$ can be properly defined, provided that the dimension $d$ of the relevant subspaces is not too small, depending on the values of $n$, $m$ and $p$. In such case one can ask whether some properties such as a certain degree of integrability of a trace can be expected, and, naturally, which of these properties are the best possible. We shall survey both classical and recent results concerning traces of Sobolev functions. We shall consider basic questions concerning the very existence of trace as well as deeper problems such as optimal trace embeddings involving specific function spaces.

Algebra Seminar

Speaker: Detlev Hoffmann (Technische Universität Dortmund)

"Equivalence relations for quadratic forms"

Time: 14:30

Room: MC 107

We investigate equivalence relations for quadratic forms that can be expressed in terms of algebro-geometric properties of their associated quadrics, more precisely, birational, stably birational and motivic equivalence, and isomorphism of quadrics. We provide some examples and counterexamples and highlight some important open problems.

Geometry and Topology

Speaker: Cihan Okay (Western)

"Towards a refinement of the Bloch-Kato conjecture"

Time: 15:30

Room: MC 107

The Bloch-Kato conjecture is the statement that the Galois cohomology of the absolute Galois group of a field which contains a primitive pth root of unity in mod p coefficients is isomorphic to Milnor K-theory reduced modulo p. This statement is now a theorem proved by Rost and Voevodsky. In particular it says that the cohomology ring of the absolute Galois group is generated by one dimensional classes. It is a natural question to find intermediate Galois extensions of the base field where every element in the cohomology ring decomposes into a sum of products of one dimensional classes. In degree two we answer this question by providing a tower of Galois extensions where indecomposable elements decompose in the next level of the tower. We also illustrate this refinement by directly computing the cohomology rings of superpythagorean fields and p-rigid fields. This is a joint work with J. Minac and S. Chebolu.

Noncommutative Geometry

Speaker: (Western)

"Learning Seminar"

Time: 11:30

Room: MC 107

The topics we continue are as follows: ---Clifford algebras, Clifford modules, spin structures, Dirac operators, Weizenbok formula, ---Heat kernel and its asymptotic expansion, Gilkey's formula, Mackean-Singer formula.

Analysis Seminar

Speaker: Wayne R. Grey (Western)

"Holder's inequality and mixed-norm estimates"

Time: 15:30

Room: MC 107

Estimates involving symmetric geometric means of mixed norms have appeared since at least Littlewood's $4/3$ inequality, and remain relevant. New theorems provide a simple general framework, replacing ad-hoc methods.

More flexible generalizations of H{\"o}lder's inequality, both in one variable and for mixed norms, are crucial. These reformulate the exponent condition in terms of harmonic means, and the conclusion in terms of geometric means. I will also describe a generalization to weighted means by Albuquerque, Araujo, Pellegrino, and Seoane-Sepulveda.

The key results follow from generalized H{\"o}lder, after a combinatorial argument. The basic techniques used are just the Holder and Minkowski integral inequalities, but the final results easily produce generalizations of Littlewood's $4/3$ inequality, with applications to multilinearity, Sobolev embeddings, and other topics.

Pizza Seminar

Speaker: Chris Kapulkin (Western)

"Stable marriage problem"

Time: 17:30

Room: Biological and Geological Sciences (room 0165)

Given a group of 100 men and 100 women, can we always arrange 100 (heterosexual) marriages, which would be *stable* in that no man and no woman simultaneously prefer each other over their assigned partners (which may lead to them leaving their partners and running away)? The positive answer to this question was given in 1962 by mathematicians David Gale and Lloyd Shapley and it was one of the results for which in 2012, Shapley was awarded the Nobel Prize in economics.

I will present the mathematics behind the Gale--Shapley algorithm and some of its interesting applications. Afterwards, I will discuss a few variations on the original problem, such as: the stable roommate problem and the college admission problem.

Pizza and pops to follow!

Algebra Seminar

Speaker: Thanksgiving Friday

"(no seminar)"

Time: 14:30

Room: MC 107

Noncommutative Geometry

Speaker: (Western)

"Learning Seminar"

Time: 11:30

Room: MC 107

We are covering the topics: ---Clifford algebras, Clifford modules, spin structures, Dirac operators, Weizenbok formula, ---Heat kernel and its asymptotic expansion, Gilkey's formula, Mackean-Singer formula.

Algebra Seminar

Speaker: Stefan Gille (University of Alberta)

"Milnor-Witt K-Theory"

Time: 14:30

Room: MC 107

Milnor-Witt K-theory arises in the Morel-Voevodsky homotopy theory over a field and plays a role in the classification of vector bundles over smooth schemes. Morel in collaboration with Hopkins discovered a nice presentation of these groups, which has been recently generalized by Changlong Zhong, Stephen Scully and myself to semilocal rings which contain an infinite field. In my talk I will discuss this result and also present some applications of these groups.

Colloquium

Speaker: Niayesh Afshordi (Perimeter Institute)

"(100 years of) Alice's adventures in (Relativistic) Wonderland"

Time: 15:30

Room: MC 107

"Down, down, down. Would the fall never come to an end! I wonder how many miles I have fallen by this time? she said aloud. I must be getting somewhere near the center of the earth", wrote Charles Dodgson, the English writer and mathematician, of Alice's fantastic adventure into a bizarre world of underground creatures. I shall retell the story of our centennial adventure into the relativistic wonderland, a tall tale of non-sensical creatures that includes blunders, black holes, Schroedinger's cats, and a very vibrant vacuum!

Geometry and Topology

Speaker: Francesco Sala (Western)

"Quantum toroidal algebras and K-theoretic Hall Algebra of the stack of torsion sheaves"

Time: 15:30

Room: MC 107

Starting from the works of Nakajima and Grojnowski, moduli spaces and stacks of sheaves on surfaces represent wonderful tools for the study of vertex and quantum algebras and their representations from a geometric point of view. For example, Schiffmann and Vasserot proved that the equivariant K-theory of the stack of zero-dimensional sheaves on $\mathbb C^2$ has an associative algebra structure and is isomorphic to the positive part of quantum toroidal algebra of type $\mathfrak{gl}(1)$; moreover, it acts on the equivariant K-theory of the Hilbert scheme of points on $\mathbb{C}^2$. Their result can be seen as a K-theoretic version of Nakajima-Grojnowski cohomological result for Hilbert schemes of points.

In the present talk, I would like to describe a new conjectural approach to the study of quantum toroidal algebras of type $\mathfrak{gl}(k)$ based on the study of algebra structures on the K-theory of the stacks of torsion sheaves over other noncompact surfaces (e.g. the stack of sheaves on the minimal resolution of the Du-Val singularity $\mathbb{C}^2/\mathbb{Z}_k$​, supported at an exceptional curve)​. (This is a work in progress with Olivier Schiffmann.)

Noncommutative Geometry

Speaker: (Western)

"Learning Seminar"

Time: 11:30

Room: MC 107

The topics of this session are as follows: ---Heat kernel and its asymptotic expansion, Gilkey's formula, Mackean-Singer formula, ---The index problem for elliptic PDE's, characteristic classes via Chern-Weil theory.

Analysis Seminar

Speaker:

"Talk Postponed"

Time: 15:30

Room: MC 107

Algebra Seminar

Speaker: Christian Maire (Université de Franche-Comté)

"Splitting in analytic and non-analytic extensions of number fields"

Time: 14:30

Room: MC 107

The aim of the talk is to show in two (closed) contexts the importance of the knowledge of the set of decomposed places in an infinite extension: (i) for the mu-invariant; (ii) for the exponent of the class group along a p-tower.

This is joint work with F. Hajir (U. Mass.)

Geometry and Topology

Speaker: Graham Denham (Western)

"Milnor fibres of hyperplane arrangements"

Time: 15:30

Room: MC 107

The Milnor fibration of a complex, projective hypersurface produces a smooth manifold as a regular, cyclic cover of the hypersurface complement. When the hypersurface is a union of complex hyperplanes, the Milnor fibre is part of the study of hyperplane arrangements. In this case, the hypersurface complement is well known and studied. In particular, it is a Stein manifold, a rationally formal space, and it admits a perfect Morse function.

The cohomology and the monodromy of the Milnor fibre can be understood in terms of the cohomology jump loci of the hypersurface complement. For generic hyperplane arrangements, this cohomology and monodromy representation are known and fairly straightforward, although current technique still falls short of being able to describe even the betti numbers in the case of reflection arrangements. Some combinatorial techniques can be used to construct arrangements with Milnor fibres with interesting properties that constrast with the well-behaved nature of the arrangement complements. These include integer homology torsion, non-formality, and non-trivial monodromy representations in all cohomological degrees.

This talk is based on joint work with Alex Suciu.

Noncommutative Geometry

Speaker: (Western)

"Learning Seminar"

Time: 11:30

Room: MC 107

In the first part we continue with:

--- Heat kernel and its asymptotic expansion, Gilkey's formula, Mackean-Singer formula, ---The index problem for elliptic PDE's, characteristic classes via Chern-Weil theory.

In the second part, Baris Ugurcan (Western) will talk about IFS (iterated function systems) and boundaries.

Analysis Seminar

Speaker: Kin Kwan Leung (University of Toronto)

"The Homogeneous Complex Monge-Amp\`ere Equation and Zoll Metrics"

Time: 15:30

Room: MC 107

Let $M$ be a real analytic Riemannian manifold. An adapted complex structure on $TM$ is a complex sturcture on a neighborhood of the zero section such that the leaves of the Riemann foliations are complex submanifolds. This complex structure is called entire if it may be extended to the whole of $TM$. In this complex structure, the energy function $E = g(x,v)$ is strictly plurisubharmonic and the length function $\sqrt{E}$ satisfies the homogeneous complex Monge-Amp\`ere equation. Thus $TM$ is a Stein manifold. If the leaves of the Riemann foliations are ``nice'' enough, in our case, $M$ being a Zoll sphere, we prove that $(M,g)$ must be the round sphere. The technique in the proof can be used in a more general setting to prove an ``algebraization'' result.

Pizza Seminar

Speaker: Lex Renner (Western)

"Canonical form for linear operators over C((t))"

Time: 17:30

Room: MC 108

Many of us are familiar with the Jordan canonical form for a linear operator over C, and also the rational canonical form for a linear operator over an arbitrary field F.

In this talk we consider linear operators over the field C((t)) of formal power series, and we identify a canonical form (which we call standard canonical form) for such operators based on the theorem of Newton-Puiseux. To do this we introduce the standard matrix of an irreducible polynomial over C((t)). These results provide a departure from the companion matrix approach to producing canonical forms over C((t)).

The interesting open problem here is to identify other fields F for which there is a notion of "standard canonical form". Does this depend on some kind of generalized Newton-Puiseux Theorem for F? Or is it enough to start with any field F that comes equipped with a discrete valuation R?

Algebra Seminar

Speaker: Pierre Guillot (University of Strasbourg)

"Cayley graphs and automatic sequences"

Time: 15:30

Room: MC 107

Automatic sequences are sequences produced by automata, which can be seen as directed graphs with extra decoration. Most sequences arising in combinatorics are automatic when reduced modulo a prime power. Cayley graphs, on the other hand, are directed graphs obtained from finite groups with distinguished generators.

Following an observation by Rowland, we study those sequences which can be produced by an automaton which is a Cayley graph (with extra information). For 2-automatic sequences (for which the n-th term is a computed from the digits of n in base 2, essentially) the result is particularly satisfying: a given sequence comes from a Cayley graph if and only if it enjoys a certain symmetry, which we call self-similarity.

We give an application to the computation of certain rational fractions associated to automatic sequences.

Colloquium

Speaker: Rick Jardine (Western)

"Galois groups and groupoids, and pro homotopy types."

Time: 15:30

Room: MC 107

Pro objects, such as the absolute Galois group of a field, are pervasive in algebra. They formed the basis for the original applications of homotopy theory in geometry and number theory, via étale homotopy theory. For some time, local homotopy theory and étale homotopy theory were almost orthogonal as theories, but the relationship between the two is much better understood now, and there is a theory which engulfs both.

Perhaps there is an even more general theory that is not based on pro objects, with potential geometric applications which are not bound to the étale topology. I will describe a candidate in this talk, after some teaching moments.

Algebra Seminar

Speaker: Fall Study Break

"(no seminar)"

Time: 14:30

Room: MC 107