Noncommutative Geometry

Speaker: Josue Rosario-Ortega (Western)

"NCG Learning Seminar: Geometric Quantization (2)"

Time: 14:30 - 16:00

Room: MC 107

To quantize a classical system we have to consider the kinematic relation between the classical and quantum case: In the quantum case the states of a system are represented by the rays in a Hilbert space H and the observables by a collection of symmetric operators on H. In the classical case the state space is a symplectic manifold M and the observables are the algebra of smooth functions on M. The kinematic problem is: given M and its symplectic form is it possible to reconstruct the Hilbert space H and the symmetric operators? Geometric quantization gives a well defined procedure to construct the Hilbert space H and the operators corresponding to the classical observables. This procedure also satisfies the Dirac's quantum conditions. In this talk I will discuss these constructions in detail and the three stages of geometric quantization: pre-quantization, polarization and metaplectic correction.

Geometry and Topology

Speaker: Kirsten Wickelgren (Harvard)

"Investigating the section conjecture"

Time: 15:30 - 16:30

Room: MC 108

Grothendieck's anabelian conjectures predict that the etale fundamental group is a fully faithful functor from certain anabelian schemes to profinite groups with Galois action, or equivalently that the maps between anabelian schemes are the same as maps between their etale homotopy types. This is analogous to an equivalence between fixed points and homotopy fixed points for Galois actions. We will discuss the anabelian conjectures and their topological analogues, and in the case of maps from the spectrum of a field, relate certain nilpotent obstructions to Massey products. As a corollary one has that the order n Massey product vanishes, where x denotes the image of x in k* under the Kummer map k* -> H^1(Gal(kbar/k), Z_l(1)).

Analysis Seminar

Speaker: Sasha Kocic (University of Mississippi)

"Renormalization and rigidity theory for circle diffeomorphisms with breaks"

Time: 15:30 - 16:30

Room: MC 108

Rigidity theory of circle diffeomorphisms, which concerns smooth conjugacy to a rigid rotation, is a classic problem in dynamical systems initiated by Arnol'd and settled by Herman and Yoccoz. We present complete renormalization and rigidity theory for circle maps with breaks, i.e., circle diffeomorphisms with a single singular point where the derivative has a jump discontinuity. We prove that renormalizations of any two C^{2+alpha}-smooth (alpha>0) circle maps with breaks, with the same irrational rotation number and the same size of the break, approach each other exponentially fast. As a corollary, we obtain a strong rigidity statement for such maps: for almost all irrational rotation numbers, any two circle maps with breaks, with the same rotation number and the same size of the break, are C^1-smoothly conjugate to each other. As we proved earlier, the latter result cannot be extended to all irrational rotation numbers. (This joint work with Kostya Khanin)

Analysis Seminar

Speaker: Rasul Shafikov (Western)

"On local polynomial convexity of the union of two Lagrangian submanifolds in C^n"

Time: 15:30 - 16:30

Room: MC 108

I will discuss the property of polynomial convexity of compacts in C^n, and the outline the proof of local convexity of two Lagrangian submanifolds in C^n at a point of transversal intersection.

Geometry and Topology

Speaker: Jeffrey Morton (SUNY Buffalo State)

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

Time: 15:30 - 16:30

Room: MC 107

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.

Noncommutative Geometry

Speaker: Masoud Khalkhali (Western)

"The heat equation proof of the Atiyah-Singer index theorem (1)"

Time: 14:30 - 16:00

Room: MC 107

I shall survey the main steps in the heat equation proof of the Atiyah-Singer index theorem for Dirac operators on spin manifolds.

Noncommutative Geometry

Speaker: Travis Ens (Western)

"NCG Learning Seminar: Feynman's diagrams and Feynman's theorem (1)"

Time: 10:30 - 12:00

Room: MC 107

Algebra Seminar

Speaker: Mehdi Garrousian (Western and U. Windsor)

"On the Koszulity of the algebra of reciprocal forms"

Time: 14:30 - 15:30

Room: MC 108

The Orlik-Terao algebra of a hyperplane arrangement is the algebra of the rational functions with poles on an arrangement of hyperplanes. This is a commutative analogue of the cohomology algebra of the complement and defines the so-called reciprocal plane which is a an irreducible projective variety. In a recent joint work with G. Denham and S. Tohaneanu, we give a decomposition result about the OT algebra whenever the underlying combinatorics has a certain genericity structure. This result implies that the OT algebra is Koszul if the arrangement is nicely filtered. As consequences, we give several other results such as characterizations for the OT algebra to be a quadratic complete intersection.

Analysis Seminar

Speaker: Kaushal Verma (Indian Institute of Science, Bangalore)

"Fatou Bieberbach domains"

Time: 15:30 - 16:30

Room: MC 108

This will be a survey talk, accessible to all, on Fatou Bieberbach domains.