Noncommutative Geometry

Speaker: Jason Haradyn (Western)

"NCG Learning Seminar: Isospectral and Nonisometric Plane Domains (4)"

Time: 14:30

Room: MC 107

We will construct Buser's example of two Schreier graphs that are isospectral but not isomprphic. The proof of this has a wonderful connection to representation theory, and some useful pre-trace formulae will be reviewed. We will then start the construction of isospectral, non-isometric planar domains that will be concluded in part 5 of this series of talks.

Geometry and Topology

Speaker: Derek Krepski (Western)

"Geometric quantization and group-valued moment maps"

Time: 15:30

Room: MC 108

Originally aimed at understanding the relationship between classical and quantum mechanics, geometric quantization is a construction whose ideas originated in representation theory, in the work of Kirillov, Kostant and Souriau during the late 1960's. From the symplectic geometry perspective, geometric quantization produces representations of compact Lie groups starting from `geometric' (i.e. Hamiltonian) group actions of such Lie groups. Some ideas surrounding this construction will be discussed, including their adaptation to the theory of 'group-valued' moment maps, which is a finite-dimensional model of the theory of Hamiltonian loop group actions. Applications in this context (re)produce so-called Verlinde formulas of conformal field theory for simply connected compact Lie groups. For non-simply connected Lie groups, Verlinde formulas have been conjectured, and this approach verifies the conjectured formulas for $G=SO(3)$.

Analysis Seminar

Speaker: Dusty Grundmeier (University of Michigan)

"Rigidity of CR Mappings for Hyperquadrics"

Time: 15:30

Room: MC 108

This is joint work with Jiri Lebl and Liz Vivas. We prove that the rank of a Hermitian form on the space of holomorphic polynomials can be bounded by a constant depending only on the maximum rank of the form restricted to affine manifolds. As an application we prove a result along the lines of the Baouendi-Huang and Baouendi-Ebenfelt-Huang rigidity theorems for CR mappings between hyperquadrics. If we have a real-analytic CR mapping of a hyperquadric not equivalent to a sphere to another hyperquadric Q(A,B), then either the image of the mapping is contained in a complex affine subspace or A is bounded by a constant depending only on B.

Noncommutative Geometry

Speaker: Asghar Ghorbanppour (Western)

"NCG Learning Seminar: Spin^c Structure and Dirac Operators"

Time: 14:30

Room: MC 107

The irreducible (real) Clifford modules play a very important role in the theory of Dirac operators. The obstruction for the existence of such a module in real and complex case is different. For the real one, vector bundle should have vanishing second Stiefel-Whitney class, however, in the complex case the second Stiefel-Whitney class only needs to be mod 2 reduction of an integral class, equivalently, the vector bundle admits a $spin^c$ structure. In this talk we will examine the obstruction for the existence of $spin^c$ structure, then the construction of complex spinor bundle and the $spin^c$ connection on it and finally Dirac operator on the spinor fields will be discussed.

Colloquium

Speaker: Oliver Roendigs (Osnabrueck)

"The Grothedieck ring of varieties"

Time: 15:30

Room: MC 108

The Grothendieck ring of varieties over a field is a bookkeeping device for invariants of varieties which preserve the relation [X] = [Z]+[X-Z] whenever Z is a closed subvariety of X. Examples of such invariants include counting points if the field in question is finite, or the topological Euler characteristic if the field is the complex numbers. After introducing the Grothendieck ring and some invariants, I will discuss a certain invariant which involves the A^1-homotopy type of Morel and Voevodsky.

Algebra Seminar

Speaker: Jochen G$\mathrm{\ddot{a}}$rtner (Heidelberg)

"Higher Massey products in the cohomology of pro-$p$-extensions"

Time: 10:30

Room: MC 108

What do the 'picture hanging problem' and 'Borromean rings' have in common? Their solution can be described by Milnor invariants in link theory, or equivalently by higher cohomological Massey products.

As noticed by B. Mazur, M. Morishita et al, there is a remarkable analogy between the theory of links and pro-$p$-extensions of number fields with ramification restricted to a finite set of primes. We discuss this analogy and give an arithmetic interpretation of Massey products in low degrees. It turns out that certain symmetry relations in the topological world carry over to number theory in special cases only.

We report on the work on applications of higher Massey products in order to construct so-called mild pro-$p$-groups and investigate recent progress in the theory of tamely ramified pro-$p$-extensions by J. Labute and A. Schmidt.

Algebra Seminar

Speaker: Christian Maire (Universit$\mathrm{\acute{e}}$ de Franche-Comt$\mathrm{\acute{e}}$)

"Example of arithmetic mild pro-$p$-groups"

Time: 14:30

Room: MC 108

In this talk, we will show how to obtain mild pro-$p$-groups in the arithmetic context.

Noncommutative Geometry

Speaker: Mingcong Zeng (Western)

"NCG Learning Seminar: A proof of Bott periodicity theorem (2)"

Time: 14:30

Room: MC 107

This talk is dedicated to the proof of Bott periodicity. First we generalize the clutching function to vector bundles over $X \times S^2$, then we can simplify the clutching function, first to a Laurent polynomial, then to a polynomial, finally to a linear function. And by the discussion on the linear clutching function, we can finally decompose it into a vector bundle with trivial clutching function and another one with clutching function $z$. Finally, we can construct a inverse by the simplified clutching function for the external product to prove that it is an isomorphism.

Noncommutative Geometry

Speaker: Alimjon Eshmatov (Western)

"Noncommutative Symplectic Geometry (1)"

Time: 14:30

Room: MC 107

In the first of a series of talks, I will try to give a basic idea of Noncommutative Geometry (due to M. Artin, Y. Manin, M.Kontsevich ...) which is sort of parallel to Connes' NCG. I will recall some basic facts and explain Kontsevich's idea of studying NCG through Representation varieties (Rep- functor).

Geometry and Topology

Speaker: Kyle Ormsby (MIT)

"Cancelled"

Time: 15:30

Room: MC 108

Cancelled

Analysis Seminar

Speaker: Wayne Grey (Western)

"Amalgam spaces"

Time: 15:30

Room: MC 108

TBA

Noncommutative Geometry

Speaker: Alim Eshmatov (Western)

"Noncommutative Symplectic Geometry (2)"

Time: 14:30

Room: MC 107

In this talk, we will discuss a notion of noncommutative symplectic structure and Calabi-Yau algebras. I will give some examples and some results related to these structures.

Noncommutative Geometry

Speaker: Josue Rosario-Ortega (Western)

"NCG Learning Seminar: Geometric Quantization"

Time: 11:30

Room: MC 108

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.

Noncommutative Geometry

Speaker: Dave Penneys (University of Toronto)

"Constructing subfactors with the jellyfish algorithm"

Time: 14:30

Room: MC 106

I will start with a basic introduction to planar algebras. I will then discuss recent joint work with Scott Morrison (arXiv:1208.3637) and recent joint work with Stephen Bigelow (arXiv:1208:1564). With Morrison, we construct a new exotic subfactor planar algebra using Bigelow's jellyfish algorithm. With Bigelow, we determine exactly when a planar algebra has a presentation by generators and jellyfish relations.

Geometry and Topology

Speaker: Anatoly Preygel (University of California-Berkeley)

"Thom-Sebastiani & duality for matrix factorizations"

Time: 15:30

Room: MC 108

The Z/2-graded dg-category MF of matrix factorizations is a categorical singularity invariant appearing on the algebro-geometric side of mirror symmetry. Though it lives in commutative algebra, and the first definitions were very explicit, it turns out to admit a few descriptions very much in the spirit of homotopy theory. After recalling a bit about sheaf theories in derived algebraic geometry, we'll sketch at least one of these definitions and show how it can be used to give conceptual proofs of some fundamental properties of MF.