Geometry and Topology

Speaker: William Slofstra (Waterloo)

"Schubert varieties and inversion hyperplane arrangements"

Time: 15:30

Room: MC 107

Freeness is an interesting algebraic property of complex hyperplane arrangements. The standard examples of free arrangements are the Coxeter arrangements, which consist of the hyperplanes normal to the elements of a finite root system. It is a natural (open) question to determine when a subarrangement of a Coxeter arrangement is free. Surprisingly, for the inversion subarrangements this question seems to be closely connected to the combinatorics of Coxeter groups and Schubert varieties. I will talk about two aspects of this connection: (1) the equality between the exponents of a rationally smooth Schubert variety and the exponents of the corresponding inversion arrangement, and (2) a criterion for freeness of inversion arrangements using root-system pattern avoidance.

Noncommutative Geometry

Speaker: (Western)

"Learning Seminar"

Time: 11:30

Room: MC 107

We continue with: ---The index problem and characteristic classes via Chern-Weil theory, ---Miraculous cancellations, Getzler's supersymmetric proof of the Atiyah-Singer index theorem, special cases: Gauss-Bonnet-Chern, Hirzebruch signature theorem, and Riemann-Roch. Baris Ugurcan (UWO) will speak in the second part.

Colloquium

Speaker: Lance Littlejohn (Baylor University)

"Glazman-Krein-Naimark theory, left-definite theory and the square of the Legendre polynomials differential operator"

Time: 15:30

Room: MC 107

As an application of a general left-definite spectral theory, Everitt, Littlejohn and Wellman, in 2002, developed the left-definite theory associated with the classical Legendre self-adjoint second-order differential operator $A$ in $L^2(-1,1)$ having the Legendre polynomials $\{P_{n}\}_{n=0}^{\infty}$ as eigenfunctions. As a particular consequence, they explicitly determine the domain $\mathcal{D}(A^2)$ of the self-adjoint operator $A^{2}$. However, this domain, in their characterization, does not contain boundary conditions. In fact, this is a general feature of the left-definite approach developed by Littlejohn and Wellman. Yet, the square of the second-order Legendre expression is in the limit-4 case at each end point $x=\pm1$ in $L^2(-1,1)$ meaning that $\mathcal{D}(A^2)$ should exhibit four boundary conditions. In this talk, after a gentle `crash course' on left-definite theory and the classical Glazman-Krein-Naimark (GKN) theory, we show that $\mathcal{D}(A^2)$ can, in fact, be expressed using four (separated) boundary conditions. In addition, we determine a new characterization of $\mathcal{D}(A^2)$ that involves four non-GKN boundary conditions. These new boundary conditions are surprisingly simple - and natural - and are equivalent to the boundary conditions obtained from the GKN theory.

Algebra Seminar

Speaker: Andrei Minchenko (Weizmann Institute of Science)

"Simple Lie conformal algebras"

Time: 14:30

Room: MC 107

The notion of a Lie conformal algebra (LCA) comes from physics, and is related to the operator product expansion. An LCA is a module over a ring of differential operators with constant coefficients, and with a bracket which may be seen as a deformation of a Lie bracket. LCA are related to linearly compact differential Lie algebras via the so-called annihilation functor. Using this observation and Cartan's classification of linearly compact simple Lie algebras, Bakalov, D'Andrea and Kac classified finite simple LCA in 2000.

I will define the notion of LCA over a ring $R$ of differential operators with not necessarily constant coefficients, extending the known one for $R=K[x]$. I will explain why it is natural to study such an object and will suggest an approach for the classification of finite simple LCA over arbitrary differential fields.

Colloquium

Speaker: Ian Hambleton (McMaster)

"Manifolds and symmetry"

Time: 15:30

Room: MC 107

This will be a survey talk about connections between the topology of a manifold and its group of symmetries. I will illustrate this theme by discussing finite group actions on spheres and products of spheres, and infinite discrete groups acting properly discontinuously on products of spheres and Euclidean spaces.

Geometry and Topology

Speaker: Jennifer Vaughan (Univ. of Toronto)

"Dynamical Invariance of a New Metaplectic-c Quantization Condition"

Time: 15:30

Room: MC 107

Metaplectic-c quantization was developed by Robinson and Rawnsley as an alternative to the classical Kostant-Souriau quantization procedure with half-form correction. Given a metaplectic-c quantized symplectic manifold and a real-valued function on that manifold, we propose a condition under which a regular value of the function is a quantized energy level for the system. We discuss the properties of this condition, and we give the quantized energy levels of the harmonic oscillator and the hydrogen atom.

Noncommutative Geometry

Speaker: (Western)

"Learning Seminar"

Time: 11:30

Room: MC 107

This week we continue with:

---The index problem for elliptic PDE's, characteristic classes via Chern-Weil theory,

---Miraculous cancellations, Getzler's supersymmetric proof of the Atiyah-Singer index theorem, special cases: Gauss-Bonnet-Chern, Hirzebruch signature theorem, and Riemann-Roch.

Analysis Seminar

Speaker: David Barrett (University of Michigan, Ann Arbor)

"Sums of CR functions from competing CR structures"

Time: 15:30

Room: MC 107

This talk will consider the problem of characterizing the sum of CR functions from two competing (oppositely-oriented) CR structures sharing the same maximal complex subspace, in two specific scenarios.

In the first scenario the two structures are simply conjugate to each other and the functions in question are pluriharmonic boundary values. (This problem has an extensive history, but some new results will be presented.) In the second scenario the two structures are related by projective duality considerations.

In both cases special attention will be paid to two-dimensional circular domains. This is joint work with Dusty Grundmeier.

Colloquium

Speaker: Matthew Satriano (Waterloo)

"Stacky resolutions and applications"

Time: 15:30

Room: MC 107

We will not assume any prior knowledge of stacks for this talk. We introduce a notion of "stacky resolution" which is gives a way to study mildly singular spaces. We then discuss applications of these resolutions to group theory, Hodge theory, and toric geometry.

Graduate Seminar

Speaker: Marco Vergura (Western)

"A Giraud-type Theorem for Model Topoi"

Time: 13:30

Room: MC 108

Following the unpublished work of C. Rezk, Toposes and Homotopy Toposes, we present a formulation of the notion of model topos, intended as a model-categorical version of the classical concept of Grothendieck topos. Such a definition will be sensible enough to establish a Giraud-type theorem for model topoi. We will start by reviewing the notion of Grothendieck topos, albeit from a slightly unusual perspective which avoids the use of Grothendieck topologies. We will then state one of the possible formulation of the classical Giraud's theorem for Grothendieck topoi which characterises them axiomatically as categories satisfying suitable internal properties. An important role in this result is played by the concept of weak descent. Using our definition of Grothendieck topoi and its equivalent interpretation which involve categories admitting a left exact small presentation, it will be relatively easy to explain how to homotopify the ordinary categorical setting (substituting presheaves categories with simplicial presheaves categories and localizations with Bousfield localizations) and get the desired notion of model topoi. We will finally state and sketch the proof of a meaningful version of Giraud's theorem for such model topoi and, if time permits, we will see how it applies to provide a nice class of examples of model topoi which present the homotopy theory of homotopy sheaves on a Grothendieck site.

Geometry and Topology

Speaker: Benoit Charbonneau (Waterloo)

"Deformations of nearly Kahler instantons"

Time: 15:30

Room: MC 107

In joint work with Derek Harland, we have developed the deformations theory for instantons on nearly Kahler six-manifolds using spinors and Dirac operators. Using this framework we identify the space of deformations of an irreducible instanton with semisimple structure group with the kernel of an elliptic operator. As an application, we show that the canonical connection on three of the four homogeneous nearly Kahler six-manifolds G/H is a rigid instanton with structure group H. In contrast, these connections admit large spaces of deformations when regarded as instantons on the tangent bundle with structure group SU(3).

Noncommutative Geometry

Speaker: (Western)

"Learning Seminar"

Time: 11:30

Room: MC 107

This week we are covering: ---The index problem for elliptic PDE's, characteristic classes via Chern-Weil theory, ---Miraculous cancellations, Getzler's supersymmetric proof of the Atiyah-Singer index theorem, special cases: Gauss-Bonnet-Chern, Hirzebruch signature theorem, and Riemann-Roch, --- Approach via path integrals and quantum mechanics.

Analysis Seminar

Speaker: Fatemeh Sharifi (Western)

"Zero-free approximation"

Time: 15:30

Room: MC 107

Let $E$ be a closed subset in the complex plane with connected complement. We define $A(E)$ to be the class of all complex continuous functions on $E$ that are holomorphic in the interior $E^0$ of $E$. The remarkable theorem of Mergelyan shows that every $f\in A(E)$ is uniformly approximable by polynomials on $E$, but is it possible to realize such an approximation by polynomials that are zero-free on $E$? This question was first proposed by J.Anderson and P.Gauthier. Recently Arthur Danielyan described a class of functions for which zero-free approximation is possible on an arbitrary $E$. I am intending to talk about the generalization of his work on Riemann surfaces.

Noncommutative Geometry

Speaker: Yanli Song (U of Toronto)

"K-homological index for proper actions"

Time: 15:30

Room: MC 108

Noncommutative Geometry

Speaker: (Western)

"Learning Seminar"

Time: 11:30

Room: MC 107

This week in NCG seminar: ---Miraculous cancellations, Getzler's supersymmetric proof of the Atiyah-Singer index theorem, special cases: Gauss-Bonnet-Chern, Hirzebruch signature theorem, and Riemann-Roch, --- Approach via path integrals and quantum mechanics.

PhD Thesis Defence

Speaker: Masoud Ataei Jaliseh (Western)

"Galois 2-extensions"

Time: 14:00

Room: MC 108

The inverse Galois problem is a major question in mathematics. For a given base field and a given finite group G, one would like to list all Galois extensions L=F such that the Galois group of L=F is G. In this work we shall solve this problem for all fields F, and for group G of unipotent 4 x 4 matrices over F2. We also list all 16 U4(F2)- extensions of Q2. The importance of these results is that they answer the inverse Galois problem in some specific cases. This is joint work with Jan Minac and Nguyen Duy Tan.

Analysis Seminar

Speaker: Tatyana Barron (Western)

"Kaehler manifolds, Toeplitz operators, and automorphic forms"

Time: 15:30

Room: MC 107

Toeplitz operators are linear operators that act on spaces of holomorphic sections of powers of a hermitian holomorphic line bundle on a Kaehler manifold. When the Kaehler manifold (say, $M$) is a compact smooth quotient of an irreducible bounded symmetric domain $D$, holomorphic sections of powers of the canonical bundle on $M$ are in correspondence with holomorphic automorphic forms on $D$. This will be mostly review. I will also mention some recent results, including several results from joint work in progress with N. Alluhaibi.

Pizza Seminar

Speaker: Masoud Khalkhali (Western)

"Why E=mc^2"

Time: 16:30

Room: MC 107

Einstein's equation E=mc^2 is probably the most famous equation in history. But what it really means and why is it true? In this lecture we shall explore the roots of this equation in geometry of spacetime and how experiment informs this geometry. Basic notions of mathematics like linearity, invariance and symmetry, and the notion of a group, play a big role here.

Basic Notions Seminar

Speaker: Ajneet Dhillon (Western)

"Riemann's existence theorem"

Time: 15:30

Room: MC 107

This talk is aimed at graduate students and is more or less an advertisement for the power of algebraic geometry. Two diverse theorems in completely different subjects will be unified in this talk. First we have the Galois correspondence in field theory. Second we have a topological theorem, the correspondence between covering spaces and subgroups of the fundamental group. Riemann's existence theorem, due to Grothendieck, is a kind of meta-theorem unifying these results. It leads to the notion of the etale fundamental group.

Graduate Seminar

Speaker: Chandrasekar Rajamani (Western)

"Introduction to Orbifolds"

Time: 13:30

Room: MC 108

This talk will introduce the concept of orbifold as a generalization of that of manifold and some basic notions about groupoids. This alternate perspective will be helpful in answering some questions about orbifolds, as for example, what is a necessary condition for an orbifold to be a global quotient.

Algebra Seminar

Speaker: Masoud Khalkhali (Western)

"Monoidal categories, Hopf algebras, and cyclic cohomology"

Time: 14:30

Room: MC 107

This talk is a report on ongoing joint work with M. Hassanzadeh and I. Shapiro, where we extend the definition of Hopf cyclic cohomology with coefficients in a Yetter-Drinfeld type module to braided monoidal categories and fusion categories. There are many algebraic structures, e.g. Drinfeld's quasi-Hopf algebras, that share some of the axioms of Hopf algebras, but not all of them. In many cases these structures are objects of a monoidal category. I shall introduce a class of monoidal categories where one can in fact define a cyclic module for its objects.

Noncommutative Geometry

Speaker: (Western)

"Learning Seminar"

Time: 11:30

Room: MC 107

We continue on: ---The index problem for elliptic PDE's, characteristic classes via Chern-Weil theory, ---Miraculous cancellations, Getzler's supersymmetric proof of the Atiyah-Singer index theorem, special cases: Gauss-Bonnet-Chern, Hirzebruch signature theorem, and Riemann-Roch, --- Approach via path integrals and quantum mechanics.

Analysis Seminar

Speaker: Lyudmila Korobenko (McMaster University)

"Orlicz Sobolev inequalities for infinitely degenerate metrics and regularity of solutions to associated equations"

Time: 15:30

Room: MC 107

The talk is concerned with regularity of weak solutions to second order infinitely degenerate elliptic equations. It is known that regularity of weak solutions can be studied by studying properties of certain metric spaces associated to the operator, namely, subunit metric spaces. The problem arising in the infinitely degenerate case is that the measures of subunit balls are non doubling. As a consequence many classical tools such as Sobolev-type inequalities become unavailable. We show that in certain cases a weaker version of Sobolev inequality can be established which allows to perform Moser iterations to obtain boundedness and continuity of weak solutions.

Colloquium

Speaker: Michel Coste (Rennes)

"Singularities in robotics"

Time: 15:30

Room: MC 107

I shall briefly present robotics, with images. I shall in particular explain the notions of serial architecture and parallel architecture. Then I shall introduce the central topic of my talk: singularities of parallel robots. I intend to explain why roboticians studying the kinematics of parallel robots are specially interested in cusps. I shall also consider some examples: asymptotic singularities of planar parallel robots, rationality of the set of singular configurations of a Stewart platform.