Quantum Geometry

Speaker: Luuk Verhoeven (Western)

"Immersions as unbounded Kasparov products: embedding spheres inside Euclidean space"

Time: 11:30 - 13:30

Room: MC 106

Abstract: We construct an unbounded representative for the shriek class associated to the embeddings of spheres into Euclidean space. We equip this unbounded Kasparov cycle with a connection and compute the unbounded Kasparov product with the Dirac operator on R^n+1 . We find that the resulting spectral triple for the algebra C(S^n) differs from the Dirac operator on the round sphere by a so-called index cycle, whose class in KK_0(C, C) represents the multi- plicative unit. At all points we check that our construction involving the unbounded Kasparov product is compatible with the bounded Kasparov product using Kucerovsky’s criterion and we thus capture the composition law for the shriek map for these immersions at the unbounded KK-theoretical level. The necessary KK-theory will be introduced as a black box.

Geometry and Combinatorics

Speaker: Graham Denham (Western)

"Singular loci of configuration hypersurfaces"

Time: 14:30 - 15:30

Room: MC 108

A finite graph determines a Kirchhoff polynomial, which is a squarefree, homogeneous polynomial in a set of variables indexed by the edges. The Kirchhoff polynomial appears in an integrand in the study of particle interactions in high-energy physics, which provides some incentive to study the motives and periods arising from the projective hypersurface cut out by such a polynomial.

From this perspective, work of Bloch, Esnault and Kreimer (2006) suggested that the more natural object of study is, in fact, a polynomial determined by a hyperplane arrangement, which is closely related to the basis generating polynomial of the associated matroid. I will describe joint work with Mathias Schulze and Uli Walther on the singular loci of such polynomials.

Geometry and Combinatorics

Speaker: Udit Mavinkurve (Western)

"Bimonoids for hyperplane arrangements I"

Time: 14:30 - 15:30

Room: MC 108

Geometry and Topology

Speaker: Jeffrey Morton (SUNY Buffalo State)

"Higher-Categorical Symmetries and Transport Functors in Higher Gauge Theory"

Time: 15:30 - 14:30

Room: MC 108

Higher Gauge Theory (HGT) studies categorical analogs of constructions in the geometry of connections on bundles. In this talk, I will describe an HGT analog of a phenomenon that occurs in gauge theory: a groupoid of such connections arises both as a category of functors and also from a group action. I will describe a 2-categorical analog of this phenomenon, showing a relation between global and local symmetry in categorical geometry.

Quantum Geometry

Speaker: Luuk Verhoeven (Western)

"Immersions as unbounded Kasparov products: embedding spheres inside Euclidean space"

Time: 11:30 - 13:30

Room: MC 106

Speaker: Luuk Verhoeven (Western)

"Immersions as unbounded Kasparov products: embedding spheres inside Euclidean space"

Time: 11:30

Room: MC 106

Abstract: We construct an unbounded representative for the shriek class associated to the embeddings of spheres into Euclidean space. We equip this unbounded Kasparov cycle with a connection and compute the unbounded Kasparov product with the Dirac operator on R^n+1 . We find that the resulting spectral triple for the algebra C(S^n) differs from the Dirac operator on the round sphere by a so-called index cycle, whose class in KK_0(C, C) represents the multiplicative unit. At all points we check that our construction involving the unbounded Kasparov product is compatible with the bounded Kasparov product using Kucerovsky's criterion and we thus capture the composition law for the shriek map for these immersions at the unbounded KK-theoretical level. The necessary KK-theory will be introduced as a black box.

Colloquium

Speaker: Christian Maire (University of Franche-Comte)

"The Theorem of Golod-Shafarevich in number theory"

Time: 15:30 - 16:30

Room: MC 107

For a finite group group of prime power order which can be defined minimally by d generators and r relations, the Golod-Shafarevich theorem asserts that $r> d ^ 2/4$. This has important consequences in number theory. After recalling the whole context, I will present a generalization of the GS theorem, and some new consequences of this generalized version. We will take the time to present the arithmetic objects, so that the lecture is intended for a general audience.

Geometry and Combinatorics

Speaker: Ahmed Umer Ashraf (Western)

"Finding 4T relations for matroids"

Time: 14:30 - 15:30

Room: MC 108

4T relations of chord diagrams are very useful in computing Vassiliev invariant of a knot. These relations can be purely put in the language of graph theory. Stanley’s chromatic symmetric function of the graph satisfies a variant of these relations. We will describe these ideas and see the relevant notions in the matroid theoretic setting.