Noncommutative Geometry

Speaker: Farzad Fathizadeh (Western)

"pseudodifferential operators and index theory 3"

Time: 15:00

Room: MC 107

Using heat equation methods, the index of an elliptic operator can be computed by a local formula. In this series of lectures, we will review the necessary analysis for defining the index of an elliptic operator, and derive a local formula for the index."

Analysis Seminar

Speaker: Rasul Shafikov (Western)

"Introductions to CR functions II"

Time: 15:30

Room: MC 108

Colloquium

Speaker: Joel Kamnitzer (University of Toronto)

"Categorical actions of sl(2) and equivalences of categories"

Time: 14:30

Room: MC 108

Actions of the Lie algebra sl(2) on vector spaces arise naturally in combinatorics, geometry, and algebra. Such an action consists of a sequence of vector spaces with linear maps between them satisfying certain relations.

From this perspective, one can define an action of sl(2) on a category to be a sequence of categories with functors between them satisfying certain relations. Such actions were studied by Chuang-Rouquier in the context of representations of the symmetric group in positive characteristic. More recently, Cautis, Licata, and the speaker studied an action of sl(2) where the categories involved were derived categories of coherent sheaves on cotangent bundles to Grassmannians. Following the ideas of Chuang-Rouquier, we used this sl(2) action to construct an equivalence of derived categories between different cotangent bundles of Grassmannians.

Hodge Theory

Speaker: Richard Gonzales (Western)

"Chern classes of differentiable vector bundles."

Time: 10:00

Room: MC 106

Geometry and Topology

Speaker: Matthias Franz (Western)

"Equivariant cohomology and structures up to homotopy"

Time: 11:30

Room: MC 108

I will discuss how A-infinity algebras and other structures 'up to homotopy' can be used to compute equivariant cohomology and, more generally, the cohomology of fibre bundles. The resulting constructions lead to an 'up to homotopy' version of Koszul duality as described by Goresky-Kottwitz-MacPherson. As application, I will express the integral cohomology of smooth, non-compact toric varieties purely in terms of fan data.

Noncommutative Geometry

Speaker: Farzad Fathizadeh (Western)

"Pseudodifferential operators and index theory 4"

Time: 14:30

Room: MC 107

Using heat equation methods, the index of an elliptic operator can be computed by a local formula. In this series of lectures, we will review the necessary analysis for defining the index of an elliptic operator, and derive a local formula for the index.

Colloquium

Speaker: Askold Khovanskii (University of Toronto)

"Hilbert theorem on degree of projective variety and Kushnirenko theorem"

Time: 15:30

Room: MC108

According to the Kushnirenko theorem the number of solutions in (C^*)ⁿ of a generic system of equations P_1=...=P_n=0 with given Newton polyhedra Delta(P_1)=...=Delta(P_n)=Delta equals to n! V(Delta), where V(Delta) --- n-dimensional volume of Delta. I will present an elementary proof of this theorem using the famous Hilbert theorem on degree of projective variety. If time permits I will present a simple proof of the Hilbert theorem.

Analysis Seminar

Speaker: Rasul Shafikov (Western)

"Introductions to CR functions III"

Time: 15:30

Room: MC 108

(sum_{k=1}ⁿ a_k b_k )² ≤ (sum_{k=1}ⁿ a_k²) (sum_{k=1}ⁿ b_k²)

Noncommutative Geometry

Speaker: Mohammad Hassanzadeh (Western)

"Eilenberg -Zilber and Kunneth formulas for (co)cyclic modules"

Time: 16:30

Room: MC 108

ABSTRACT: I shall talk about Eilenberg-Zilber and Kunneth formulas for cocyclic and cyclic modules. Then we will apply it to the special case of Connes-Moscovici cocylic module for Hopf algebras. This will be an introduction for my main goal which is cup product and coproduct for Hopf cyllic cohomology of Hopf algebras.

Noncommutative Geometry

Speaker: Sheldon Joyner (Western)

"Integration in free groups"

Time: 15:00

Room: MC 107

Abstract: This talk is a survey of K.-T. Chen's dissertation work (as expounded in a 1951 paper of the same name) in which he demonstrated the utility of his elementary theory of integration in free groups, in informing about subgroups of these groups. This integration provides the classical framework for the non-commutative Fourier transform developed by Kapranov, which will be the subject of upcoming seminar talks.

Hodge Theory

Speaker: Richard Gonzales (Western)

"Geometric construction of Chern classes."

Time: 10:00

Room: MC 106

Algebra Seminar

Speaker: Richard Gonzales (Western)

"Equivariant Cohomology. Part I"

Time: 14:30

Room: MC 107

Our plan is to give an overview of equivariant cohomology for torus actions as it is understood after Goresky, Kottwitz and MacPherson.

Geometry and Topology

Speaker: Martin Pinsonnault (Western)

"Rigidity phenomena in symplectic geometry I"

Time: 11:30

Room: MC 108

Symplectic geometry provides the mathematical framework of classical mechanics in its Hamiltonian formulation, and also underlies modern approaches to quantum theories. About twenty years ago, the introduction of new ideas and techniques has led to the discovery of "symplectic rigidity" phenomena that have no counterparts in classical differential geometry. Since then, symplectic geometry evolved into an independent field of research at the crossroads of geometric topology, algebraic geometry, dynamical systems, and gauge theory. The study of symplectic diffeomorphisms is at the heart of understanding symplectic rigidity phenomena. In this introductory talk, I will give an (somewhat biased) overview of symplectic rigidity, focussing mainly on homotopy-theoretic, geometric, and algebraic properties of symplectomorphism groups, as well as on properties of embedded symplectic balls and Lagrangian submanifolds.

Noncommutative Geometry

Speaker: (Western)

"NCG learning seminar"

Time: 15:00

Room: MC 107

Noncommutative Geometry

Speaker: Farzad Fathizadeh (Western)

"Pseudodifferential operators and index theory 5"

Time: 16:00

Room: MC 107

Analysis Seminar

Speaker: Janusz Adamus (Western)

"On the holomorphic closure dimension of real analytic sets I"

Time: 15:30

Room: MC 108

I will discuss some recent results of a joint work with professor Rasul Shafikov, concerning a CR geometry problem, with a complex (yet simple) local analytic solution.

Pizza Seminar

Speaker: Graham Denham (Western)

"lattice points, magic squares, and counting"

Time: 16:30

Room: MC 107

Noncommutative Geometry

Speaker: Ali Moatadelro (Western)

"The CKM invariant in noncommutative geometry 1"

Time: 15:00

Room: MC 107

Noncommutative Geometry

Speaker: Mohammad Hassanzadeh (Western)

"Eilenberg -Zilber and Kunneth formulas for (co)cyclic modules 2"

Time: 16:00

Room: MC 107

Colloquium

Speaker: John Bell (Western)

"The cohesiveness of the continuum and other mathematical objects"

Time: 15:30

Room: MC 108

Hodge Theory

Speaker: Richard Gonzales (Western)

"Functoriality of Chern classes. Examples."

Time: 10:00

Room: MC 106

Algebra Seminar

Speaker: Richard Gonzales (Western)

"Equivariant Cohomology. Part II"

Time: 14:30

Room: MC 107

Our plan is to give an overview of equivariant cohomology for torus actions as it is understood after Goresky, Kottwitz and MacPherson.

Geometry and Topology

Speaker: Martin Pinsonnault (Western)

"Rigidity phenomena in symplectic geometry II"

Time: 11:30

Room: MC 108

In this second talk, I will explain how we can probe the space of symplectic embeddings of balls in some four-manifolds through the analysis of symplectomorphism groups. After explaining the general framework, I will focus on some rational symplectic four-manifolds for which that analysis can be carried through effectively using pseudo-holomorphic curves techniques and the theory of deformations of complex structures.

Noncommutative Geometry

Speaker: (Western)

"NCG Learning Seminar"

Time: 15:00

Room: MC 107

Noncommutative Geometry

Speaker: Farzad Fathizadeh (Western)

"Pseudodifferential operators and index theory 6"

Time: 14:00

Room: MC 104

Analysis Seminar

Speaker: Janusz Adamus (Western)

"On the holomorphic closure dimension of real analytic sets II"

Time: 15:30

Room: MC 108

I will discuss some recent results of a joint work with professor Rasul Shafikov, concerning a CR geometry problem, with a complex (yet simple) local analytic solution.

Noncommutative Geometry

Speaker: Ali Moatadelro (Western)

"The CKM invariant in noncommutative geometry 2"

Time: 15:00

Room: MC 107

Noncommutative Geometry

Speaker: Mohammad Hassanzadeh (Western)

"Eilenberg -Zilber and Kunneth formulas for (co)cyclic modules 3"

Time: 16:00

Room: MC 107

Colloquium

Speaker: Rick Jardine (Western)

"Path categories and concurrency"

Time: 14:30

Room: MC 108

The path category P(K) of a simplicial complex K is a category which is built from vertices (objects) and 1-simplices (morphisms), subject to commutativity conditions associated to the 2-simplices of K. This construction extends to a functor from simplicial sets to categories which is left adjoint to the nerve.

Here is why one cares: path category morphisms specialize to execution paths in higher dimensional automata. These objects are geometric models for behaviour of parallel processing systems, and techniques are required to distinguish execution paths between states in such a system. This calculational problem is non-trivial, since the path category functor is not a standard homotopy invariant and produces categories with little extra structure. The known viable lines of attack arise from higher category theory and homotopy coherence theory.

Algebra Seminar

Speaker: Manfred Kolster (McMaster University)

"The Coates-Sinnott Conjecture"

Time: 14:30

Room: MC 107

35 years ago Coates and Sinnott formulated a K-theoretic analog of an even more classical result of Stickelberger's about annihilation of class groups of abelian fields. The talk will report on the current standing of this conjecture, the "correct" formulation in terms of motivic cohomology, and its relation to Iwasawa theory.

Geometry and Topology

Speaker:

"No lecture"

Time: 11:30

Room: MC 108

Noncommutative Geometry

Speaker: (Western)

"NCG Learning Seminar"

Time: 15:00

Room: MC 107

Noncommutative Geometry

Speaker: Farzad Fathizadeh (Western)

"Pseudodifferential operators and index theory 7"

Time: 16:00

Room: MC 107

Using heat equation methods, the index of an elliptic operator can be computed by a local formula. In this series of lectures, we will review the necessary analysis for defining the index of an elliptic operator, and derive a local formula for the index.

Analysis Seminar

Speaker: Frédéric Rochon (Toronto)

"A local families index formula for d-bar operators on punctured Riemann surfaces"

Time: 15:30

Room: MC 108

Using heat kernel methods developed by Vaillant, we will show how to obtain a local index formula for families of d-bar operators parametrized by the Teichmuller space of Riemann surfaces of genus g with n punctures. The formula also holds on the corresponding moduli space in the sense of orbifolds where it can be written in terms of Mumford-Morita-Miller classes. As we will indicate, the degree two part of the formula gives the curvature of the corresponding determinant line bundle equipped with the Quillen connection, a result originally obtained by Takhtajan and Zograf.

Noncommutative Geometry

Speaker: Ali Moatadelro (Western)

"The CKM invariant in noncommutative geometry 3"

Time: 15:00

Room: MC 107

Noncommutative Geometry

Speaker: Mohammad Hassanzadeh (Western)

"Eilenberg -Zilber and Kunneth formulas for (co)cyclic modules 4"

Time: 16:00

Room: MC 107

Algebra Seminar

Speaker: Lex Renner (Western)

"The H-polynomial of a group embedding"

Time: 14:30

Room: MC 107

The Poincaré polynomial of a Weyl group calculates the Betti numbers of the projective homogeneous space G/B, while the h-vector of a simple polytope calculates the Betti numbers of the corresponding rationally smooth toric variety. There is a common generalization of these two extremes called the H-polynomial. It applies to projective, homogeneous spaces, toric varieties and, much more generally, to any algebraic variety X where there is a connected, solvable, algebraic group acting with a finite number of orbits. We illustrate this situation by describing the H-polynomials of certain projective (G x G)-varieties X, where G is a semisimple group and B is a Borel subgroup of G. This description is made possible by finding an appropriate cellular decomposition for X and then describing the cells combinatorially in terms of the underlying monoid of (B x B)-orbits. The most familiar example here is the wonderful compactification of a semisimple group of adjoint type.

Geometry and Topology

Speaker: Tatyana Foth (Western)

"TBA"

Time: 11:30

Room: MC 108

Geometry and Topology

Speaker: Tatyana Foth (Western)

"Varieties of complex Lie algebras"

Time: 11:30

Room: MC 108

I will give a brief historical introduction and then will report on joint work with M. Tvalavadze, where we prove, in particular, that the first integral homology group of an irreducible component of the variety of graded n-dimensional complex Lie algebras, under certain assumptions, is trivial.

Analysis Seminar

Speaker: Kiumars Kaveh (Toronto)

"Convex bodies in algebraic geometry"

Time: 15:30

Room: MC 107

I will show how to associate a convex body to a finite dimensional subspace L of rational functions on an n-dimensional variety X. This rather simple construction generalizes the well-known construction of a Newton polytope (in toric geometry). We will see how volume of this convex body is responsible for the number of solutions of a generic system of equations f_1= ...= f_n = 0 from L, which can be regarded as a far generalization of Kushnirenko theorem. This then enables one to apply inequalities in convex geometry to algebraic geometry. As an example, using classical isoperimetric inequality, we get simple and elementary proofs of Hodge index theorem as well as Alexander-Fenchel inequality (for mixed volume of convex bodies). If time permits I'll discuss the relation with the concept of integral closure of a subspace of rational functions.

Noncommutative Geometry

Speaker: (Western)

"NCG Learning Seminar"

Time: 16:30

Room: MC 104

Noncommutative Geometry

Speaker: Farzad Fathi Zadeh (Western)

"Weyl's law and noncommutative geometry"

Time: 11:00

Room: MC 104

Analysis Seminar

Speaker: Marc Laforest (Polytechnique, Montréal.)

"Conservation laws and kinetic relations for nonconvex systems"

Time: 15:30

Room: MC 108

Nonconvex conservation laws appear naturally as models of the dynamics of materials with phase boundaries, like thin film flows. More generaly, they occur in systems for which the total energy of the molecular system is nonconvex. In such systems, the second law of thermodynamics, which imposes that entropy production be positive, is not sufficient to obtain uniqueness of solutions. In fact, a kinetic relation is necessary to relate the driving force and the speed of the propagating phase boundary. This kinetic relation supplements the usual entropy condition (an inequality) by specifiying the rate of entropy production.

In this talk, we give a general introduction to the theory of convex and nonconvex conservation laws. We conclude with a discussion of more recent results obtained in collaboration with Philippe G. LeFloch of Paris VI.

Noncommutative Geometry

Speaker: Ali Moatadelro (Western)

"The CKM invariant in noncommutative geometry 4"

Time: 15:00

Room: MC 107

Noncommutative Geometry

Speaker: Mohammad Hassanzadeh (Western)

"Eilenberg -Zilber and Kunneth formulas for (co)cyclic modules 5"

Time: 16:00

Room: MC 107

Colloquium

Speaker: Mike Roth (Queen's University)

"Cup product of line bundles on homogeneous varieties"

Time: 14:30

Room: MC 108

One of the most beautiful and important theorems in representation theory is the Borel-Weil-Bott theorem, which produces all of the irreducible representations of a semi-simple Lie group G (for instance GL_n) in the cohomology groups of a specific algebraic variety X constructed from G.

For the purposes of representation theory these cohomology groups are usually just treated as vector spaces, but because they come from geometry, they have a richer internal structure. In particular, there is a cup product map defined on any two such groups which maps to a third.

It was not previously known how to compute the effects of this map.

This talk will discuss the complete solution to this problem for all semi-simple groups G, as well as the related representation-theoretic problem of which components of a tensor product can be realized through such a cup product map.

Most of the talk will be a discussion of the representation theory of G and the Borel-Weil-Bott theorem.

This is a joint project with Ivan Dimitrov.

Algebra Seminar

Speaker: Sheldon Joyner (Western)

"Hopf algebras of polylogarithms"

Time: 14:30

Room: MC 107

The appearance of multiple zeta values in Theoretical Physics as well as in computations of Drinfel'd and Deligne stimulated great interest. One strategy leading to a better understanding of these numbers has been to study the polylogarithm functions (which go back to Euler). Since these functions admit expressions as both sums and iterated integrals, they give rise to two distinct Hopf algebra structures. It is believed that all polynomial relations satisfied by such numbers are known, and arise from these structures. In this talk, we outline this theory, and introduce the use of complex iterated integrals to define a Hopf algebra structure on polylogarithm functions at non-integer values.