Geometry and Topology

Speaker: Tatyana Foth (Western)

"TBA"

Time: 11:30 - 12:30

Room: MC 108

Geometry and Topology

Speaker: Tatyana Foth (Western)

"Varieties of complex Lie algebras"

Time: 11:30 - 12:30

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 - 16:30

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 - 18:00

Room: MC 104

Noncommutative Geometry

Speaker: Farzad Fathi Zadeh (Western)

"Weyl's law and noncommutative geometry"

Time: 11:00 - 12:00

Room: MC 104

Analysis Seminar

Speaker: Marc Laforest (Polytechnique, Montréal.)

"Conservation laws and kinetic relations for nonconvex systems"

Time: 15:30 - 16:30

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 - 16:00

Room: MC 107

Noncommutative Geometry

Speaker: Mohammad Hassanzadeh (Western)

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

Time: 16:00 - 17:00

Room: MC 107

Colloquium

Speaker: Mike Roth (Queen's University)

"Cup product of line bundles on homogeneous varieties"

Time: 14:30 - 15:30

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 - 15:30

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.