Noncommutative Geometry

Speaker: Travis Ens (Western)

"NCG Learning Seminar: Loop expansion of Feynman integrals, 1-particle irreducible graphs, and Cayley's tree formula"

Time: 14:30

Room: MC 107

Algebra Seminar

Speaker: Jessie Yang (McMaster)

"Initial ideals and tropical Severi varieties"

Time: 15:30

Room: MC 108

Tropical geometry is a systematic development of the fundamental concept, "degenerations". In this talk, I will make this statement precise in the algebraic view point, namely ''Initial ideals".

We apply the tropical approach to the classical objects in algebraic geometry, "Severi varieties". Severi varieties are spaces whose points correspond to the plane curves with a given number of nodal singular points. In this tropical approach, we can obtain purely combinatorial results on Severi varieties which involve subdivisions of polygons.

Noncommutative Geometry

Speaker: Masoud Khalkhali (Western)

"Applications of the Atiyah-Singer index theorem 3: Hirzebruch signature theorem (continued)"

Time: 14:30

Room: MC 107

I shall finish proof of the Hirzebruch signature theorem today, using index theorem. I shall give a few elementary applications, including divisibility results for Pontryagin numbers and a proof of the fact that CP^2 does not admit any spin structure. On Friday we shall see a more dramatic application in the talk by Mincong.

Colloquium

Speaker: Arturo Pianzola (University of Alberta)

"Why the cylinder is a straight line "

Time: 15:30

Room: MC 108

Why the cylinder is a straight line (thoughts on a modern interpretation of affine Kac-Moody Lie algebras)

Noncommutative Geometry

Speaker: Mingcong Zeng (Western)

"NCG Learning Seminar: An exotic differential structure on $S^7$"

Time: 10:30

Room: MC 107

One interesting application of Hirzebruch signature theorem is the construction of exotic differential structure on $S^7$. In this talk I will first show the construction of the exotic $S^7$, which is the sphere bundle of the 4 dimensional vector bundle over $S^4$ by using of the first Pontrjagin class and Euler class. Then for proving it has an exotic differential structure, we use Hirzebruch signature theorem to construct a invariant and compute this invariant for standard $S^7$ and our sphere bundle to see they are different.