Analysis Seminar

Speaker: Debraj Chakrabarti (Tata Institute, Bangalore)

"Condition R and Proper Holomorphic mapping of equidimensional product domains"

Time: 14:30

Room: MC 107

Consider two domains $D$ and $G$ in $\mathbb{C}^n$, each of which is the product of smoothly bounded domains, and assume that each factor of $D$ satisfies condition R, i.e, the Bergman projection preserves the class of functions smooth up to the boundary. We show that any proper holomorphic map from $D$ to $G$ extends smoothly to the closures, and splits as a product of equidimensional mappings of the factors. We also consider some possible generalization to a class of piecewise smooth domains. This is joint work with Kaushal Verma.

Geometry and Combinatorics

Speaker: Mehdi Garrousian (Western)

"Tropical Geometry II"

Time: 14:00

Room: MC 104

Tropical geometry is the discrete geometry version of algebraic geometry which provides a framework for reducing classical algebraic geometry questions into combinatorial ones. We start by introducing the tropical semiring and some motivational examples from computer science. The main objective of the talk is to explain the equivalent constructions of the tropical varieties.

Colloquium

Speaker: Evgeny Poletsky (Syracuse University)

"Holomorphic Homotopy Theory"

Time: 15:30

Room: MC 107

Holomorphic homotopy theory studies continuous deformations of holomorphic mappings and the major question is when one holomorphic mapping can be continuously deformed into another holomorphic mapping via holomorphic mappings. We call such mappings h-homotopic.

The serious studies of such questions was initiated by M. Gromov in 1989 who was interested in the homotopical Oka principle: when homotopic holomorphic mappings are h-homotopic? It led to the notions of Oka and elliptic manifolds and many interesting applications. Recently h-homotopical constructions appeared on non-elliptic manifolds which are much more general. For example, in the description of B. Joricke of envelopes of holomorphy and the disk formula for plurisubharmonic subextensions by F. Larusson and the speaker. These results raised an interest to h-homotopies on general complex manifolds.

In the talk we will briefly present Gromov's theory and then discuss the h-homotopy theory for general manifolds including the results of Joricke and Larusson-Poletsky. Finally, we will show how an h-analog for the fundamental group can be introduced.

The talk will be accessible to anybody with the knowledge of the first graduate course in complex variables.

Geometry and Topology

Speaker: Thomas Fiore (University of Michigan-Dearborn)

"Waldhausen Additivity: Classical and Quasicategorical"

Time: 15:30

Room: MC 107

We given an elementary proof of Waldhausen Additivity using key ideas from earlier proofs. Then we discuss how to prove the quasicategorical version. Model category arguments do not play a role, nor do any technical results about quasicategories. This is joint work with David Gepner and Wolfgang Lueck.

Graduate Seminar

Speaker: Ali Al-Khairy (Western)

"More Properties in Category Theory"

Time: 16:30

Room: MC 107

This talk will discuss further properties of categories, such as opposite functors and changing variance, products and bifunctors, adjoint functors, and exactness.

Geometry and Combinatorics

Speaker: Mehdi Garrousian (Western)

"Tropical Geometry III"

Time: 14:00

Room: MC 104

Last time, we gave a precise definition for a tropical variety as the closure of the image of a classical variety under an evaluation map. We'll continue the analysis by giving an equivalent description in terms of initial ideals and show that a tropical variety is a subcomplex of the Groebner complex. Next interesting topics in the line are the zero tension condition and Bezout's theorem as an intro to tropical intersection theory.

Algebra Seminar

Speaker: Claudio Quadrelli (University of Milano-Bicocca)

"Bloch-Kato groups and Galois groups?"

Time: 14:40

Room: MC 107

Every profinite group is a Galois group, but which one is also an ${\textit{absolute}}$ Galois group? The cohomological implications of the Bloch-Kato conjecture -- positively solved by M.~Rost and V.~Voevodsky -- allows us to define ${\bf{Bloch-Kato}}$ ${\bf{pro-}}$$p$ ${\bf{groups}}$, which play a crucial role, since they arise naturally as maximal pro-$p$ quotients and Sylow pro-$p$ subgroups of absolute Galois groups. In this seminar I will present the state of the art of the research on Bloch-Kato groups, with a particular mention of the 'Elementary Type Conjecture' of maximal pro-$p$ Galois groups. Yet, there's still a lot of work to do: indeed every maximal pro-$p$ Galois group is equipped with an ${\textit{orientation}}$ $G_F(p)\rightarrow\mathbb{Z}_p^\times$, arising from the action on the group of the roots of unity of $p$-power order. The study of such orientation for Bloch-Kato groups will provide hopefully new results.

Ph.D. Presentation

Speaker: Tyson Davis (Western)

"Gerbes, twisted sheaves and essential dimension"

Time: 13:00

Room: MC 107

Geometry and Combinatorics

Speaker: Bruce Fontaine (University of Toronto)

"Invariant spaces and the geometric Satake correspondence"

Time: 14:30

Room: MC 108

Given a simple, simply connected Lie group G, a common object of study is the space of G invariant tensors in tensor product of G representations. In the case of a tensor product of minuscule representations, the graphical calculus of webs developed by Greg Kuperberg is a natural way to specify invariant tensors. On the other hand, the geometric Satake correspondence can be used to realize the invariant space as the top homology of a variety we call the Satake fibre. We will show that there is a connection between these the two different ways of constructing the invariant space.

Geometry and Topology

Speaker: Niles Johnson (University of Georgia)

"Modeling stable 1-types"

Time: 15:30

Room: MC 107

It is a classical result that groupoids model homotopy 1-types, in the sense that there is an equivalence between the homotopy categories, via the classifying space and fundamental groupoid functors. We extend this to stable homotopy 1-types and Picard groupoids. Using an algebraic description of Picard groupoids, we give a model for the Postnikov invariant of a stable 1-type and describe the action of the truncated sphere spectrum in these terms. We relate this data to exact sequences of Picard groupoids developed by Vitale, constructing a model for the homotopy cofiber of a map of stable 1-types. Joint with Angélica Osorno.

Analysis Seminar

Speaker: Feride Tiglay (Western)

"Integrable evolution equations on spaces of tensor densities"

Time: 14:30

Room: MC 107

In a pioneering paper V. Arnold presented a general framework within which it is possible to employ geometric and Lie theoretic techniques to study the equations of motion of a rigid body in R^3 and the equations of ideal hydrodynamics. I will describe how to extend his formalism and introduce two integrable PDE. One of the equations turns out to be closely related to the inviscid Burgers equation while the other has not been identified in any form before. These two PDE possess all the hallmarks of integrability: the existence of a Lax pair formulation, a bi-Hamiltonian structure, the presence of an infinite family of conserved quantities and the ability to write down explicitly some of its solutions. I will also talk about local well-posedness of the corresponding Cauchy problem and global existence of solutions.

Colloquium

Speaker: Remus Floricel (University of Regina)

"Structure and classification of $E_0$-semigroups"

Time: 15:30

Room: MC 108

Introduced by R.T. Powers, $E_0$-semigroups are one-parameter semigroups $\rho=\{\rho_t\}_{t\geq 0}$ of unital normal *-endomorphisms acting on von Neumann algebras, usually the von Neumann algebra $B(H)$ of all bounded linear operators on a separable Hilbert space $H$. $E_0$-semigroups can be regarded as quantum generalizations of the classical time-irreversible dynamical systems, and their study takes into account at a non-commutative level various dissipation mechanisms and state-time evolution phenomena.

It is our purpose, in this presentation, to survey the current state of knowledge of the subject, and to discuss several classification problems.

Ph.D. Presentation

Speaker: Asghar Ghorbanpour (Western)

"Spectral Zeta Function and Spectral Invariants of Noncommutative 4-Dimensional Tori"

Time: 10:00

Room: MC 108

One knows from classical dierential geometry and geometric analysis that several invariants of compact manifolds can be encoded in terms of the spectrum of geometric operators like Laplacian or Dirac operator on a manifold. By a general theorem of Seeley-DeWitt-Gilkey [5], terms in the asymptotic expansion of the heat flow associated to the Laplacian on compact manifolds, contains information like volume (Weyl's law), scalar curvature and more subtle invariants. Some of these techniques can be extended to the setting of noncommutative geometry thanks to the formalism of spectral triples and spectral zeta functions. A recent breakthrough in this area was the work of Connes-Tretko [2] and Fathizadeh-Khalkhali [3] where they proved the Gauss-Bonnet theorem for NC 2-torus. A more recent development is the work of Connes-Moscovici [1] and Fathizadeh-Khalkhali [4] where they have managed to compute the scalar curvature of the NC 2-torus by explicitly computing the value of the zeta functional \zeta_{\Delta}(s) = tr (a\Delta^{-s}). In my presentation I will go through the recent works and their main ideas and also discuss some of those ideas for NC 4-torus and some results which we have recently found for NC 4-tori. References [1] A. Connes, H. Moscovici, Modular curvature for noncommutative two-tori, arXiv:1110.3500. [2] A. Connes, P. Tretko, The Gauss-Bonnet theorem for the noncommutative two torus, arXiv:0910.0188. [3] F. Fathizadeh, M. Khalkhali, The Gauss-Bonnet theorem for noncommuta- tive two tori with a general conformal structure, arXiv:1005.4947. [4] F. Fathizadeh, M. Khalkhali, Scalar curvature for the noncommutative two torus, arXiv:1110.3511. [5] P. Gilkey, Invariance theory, the heat equation, and the Atiyah-Singer index theorem, Mathematics Lecture Series, 11. Publish or Perish, Inc., Wilming- ton, DE, 1984. 1