Noncommutative Geometry

Speaker: Asghar Ghorbanpour (Western)

"Quantum dynamical systems I: Geodesic flow in NCG"

Time: 14:30

Room: MC 107

This talk is an introduction to the classical dynamical systems and the way it can be generalized to the quantum setting. We shall start with a Hamiltonian on the phase space and show that how the flow can be lifted on the observables after quantization. The Heisenberg equation can be obtained from its classical counterpart, i.e. Hamilton's equations. It is a well known fact that the geodesic flow of a Riemannian manifold $M$ is the flow of the Hamiltonian given by $H(q,p)=g_q(p,p)/2$. This can be generalized to spectral triples. We also show that for a finitely summable (even) spectral triple $(A,H,D)$ the analogue of the geodesic flow on the bounded operators of H, is given by $$F_t(T)=e^{it|D|}T e^{-it|D|}.$$ Finally I shall recall Egorov's theorem.

This is the first talk in a series of talks on "quantum dynamical systems and their properties" which will be jointly delivered by Ali Fathi and Asghar Ghorbanpour.

Geometry and Topology

Speaker: Marcy Robertson (UWO)

"Schematic Homotopy Types of Operads"

Time: 15:30

Room: MC 108

The rational homotopy type $X_{\mathbb{Q}}$ of an arbitrary space $X$ has pro-nilpotent homotopy type. As a consequence, pro-algebraic homotopy invariants of the space $X$ are not accessible through the space $X_{\mathbb{Q}}$. In order to develop a substitute of rational homotopy theory for non-nilpotent spaces Toen introduced the notion of a pointed schematic homotopy type over a field $\mathbb{k}$, $(X\times k)^{sch}.$

In his recent study of the pro-nilpotent Grothendieck-Teichmuller group via operads, Fresse makes use of the rational homotopy type of the little $2$-disks operad $E_2$. As a first step in the extension of Fresse's program to the pro-algebraic case we discuss the existence of a schematization of the little $2$-disks operad.

Homotopy Theory

Speaker: Martin Frankland (Western)

"Relation between completion and localization"

Time: 14:30

Room: MC 108

Noncommutative Geometry

Speaker: Mitsuru Wilson (Western)

"Formal Deformation Theory of Poisson Manifolds"

Time: 14:30

Room: MC 107

Pioneer works in formal deformation theory goes back to the late 60's, where a noncommutative space from a classical algebra of smooth functions was constructed. In his '98 paper Kontsevich proved the existence of a deformation called star products, on $C^\infty(M)$ where the product is understood to be a formal power series in one variable, on a Poisson manifold. In this talk I will define and derive an explicit formula on $\mathbb{R}^n$ of the star product. Furthermore I will explain Kontsevich's proof inasmuch elementary words as possible.

Algebra Seminar

Speaker: Tom Baird (Memorial University)

"Moduli spaces of vector bundles over a real curve"

Time: 14:30

Room: MC 108

In a seminal paper in 1983, Atiyah and Bott calculated the Betti numbers of the moduli space of holomorphic bundles over a complex curve using Morse theory of the Yang-Mills functional. In this talk, I will explain how to adapt the Atiyah-Bott method to calculate Z/2-Betti numbers of the moduli space of real/quaternionic vector bundles over a real curve. I will also report on work in progress on calculating rational Betti numbers.

Geometry and Topology

Speaker: Martin Frankland (Western)

"Completed power operations for Morava $E$-theory"

Time: 15:30

Room: MC 108

Morava $E$-theory is an important cohomology theory in chromatic homotopy theory. Using work of Ando, Hopkins, and Strickland, Rezk described the algebraic structure found in the homotopy of $K(n)$-local commutative $E$-algebras via a monad on $E_*$-modules that encodes all power operations. However, the construction does not see that the homotopy of a $K(n)$-local spectrum is $L$-complete (in the sense of Greenlees-May and Hovey-Strickland). We improve the construction to a monad on $L$-complete $E_*$-modules, and discuss some applications. Joint with Tobias Barthel.

Dept Oral Exam

Speaker: Fatemeh Bagherzadeh (Western)

"Galois groups and order spaces"

Time: 13:00

Room: MC 108

In this talk it is considered the Galois point of view on determining the structure of space of ordering of fields via considering small Galois quotients of absolute Galois groups of Pythagorean fields. We use mainly Galois theoretic, group theoretic and combinatorial arguments. When a simple invariants of order space determine this order space completely ? Some interesting cases when this happen will be described.

Homotopy Theory

Speaker: Omar Ortiz (Western)

"Rational homotopy theory via commutative dg algebras"

Time: 14:30

Room: MC 108

Colloquium

Speaker: Matt Kahle (Ohio State)

"Spectral methods in random topology"

Time: 15:30

Room: MC 108

Random topology is the study of topological invariants of random topological spaces. In this talk I will briefly survey work on topology of random simplicial complexes, starting with 1-dimensional models, i.e. random graphs. The Erdos-Renyi theorem characterizes the threshold edge probability where the random graph becomes connected, and we now know several different generalizations of this theorem to higher dimensions. In this talk, I'll discuss recent progress in proving such theorems by understanding eigenvalues of random matrices. I will not assume any particular topology or probability prerequisites, and the talk will aim to be self contained. Part of this is joint work with Chris Hoffman and Elliot Paquette.

Analysis Seminar

Speaker: Eduardo Zeron (CINVESTAV Instituto Politecnico Nacional Mexico)

"Lagrangian, totally real, and rationally convex manifolds. Three of the kind?"

Time: 11:30

Room: MC 107

Lagrangian and totally real submanifolds are two objects deeply related because of their definitions. The tangent space of a totally real manifold meets its complex rotation in only one point, the origin; while the tangent space of a Lagrangian submanifold is orthogonal to its complex rotation. One should notice that there are totally real 3-spheres in $\mathbb C^3$, but these spheres cannot be Lagrangian.

Around 1995 Duval and Sibony introduced a new relation between totally real, Lagrangian, and rationally convex manifolds. They proved that, at least for compact totally real submanifolds, rational convexity is equivalent to be Lagrangian for some appropriate Kaehler form.

Moreover, in a recent paper Cieliebak and Eliashberg have proved that, for $n>2$, the closure of a bounded domain in $\mathbb C^n$ is isotopic to a rationally convex set if and only if it admits a defining Morse function with no critical points of index strictly larger than n. This result implies in particular that there are smooth 3-spheres in $\mathbb C^3$ with a compact and rationally convex tubular neighbourhood. These smooth 3-spheres cannot be Lagrangian.

Algebra Seminar

Speaker: Cameron L. Stewart (Waterloo)

"Arithmetic and transcendence"

Time: 14:30

Room: MC 108

Techniques developed for transcendental number theory have had many surprising applications in the study of purely arithmetical questions. The aim of the talk will be to discuss this phenomenon.

Noncommutative Geometry

Speaker: Ali Fathi Baghbadorani (Western)

"Quantum Ergodicity"

Time: 15:30

Room: MC 108

I will first explain the notion of ergodicity for classical dynamical systems and will go over some of the well known examples of such systems. I will then introduce the notion of quantum ergodicity for quantized Hamiltonian systems and discuss some open problems.

Noncommutative Geometry

Speaker: Ali Fathi Baghbadorani (Western)

"Quantum Ergodicity II"

Time: 14:30

Room: MC 107

After introducing the notion of a quantum dynamical system and ergodicity of such systems, I will explain the example of the quantum Kronecker map on noncommutative 2-torus. If time allows I will also explain the notion of semi-classical ergodicity (quantum ergodicity) that arises in quantization of chaotic Hamiltonian systems.

Geometry and Topology

Speaker: Mike Misamore (UWO)

"An Etale van Kampen Theorem for Simplicial Sheaves"

Time: 15:30

Room: MC 108

A comparison of the etale homotopy type of a representable geometrically pointed simplicial sheaf X and that of its standard and twisted fibred sites is demonstrated. This result is applied to yield an etale van Kampen theorem for representable geometrically pointed connected simplicial sheaves under suitable hypotheses on the ambient site.

Homotopy Theory

Speaker: Enxin Wu (Western)

"Rational homotopy theory via dg Lie algebras"

Time: 14:30

Room: MC 108

Noncommutative Geometry

Speaker: Alim Eshmatov (Western)

"An Introduction to Homological Mirror Symmetry"

Time: 14:30

Room: MC 107

Index Theory Seminar

Speaker: Sean Fitzpatrick (Western)

"Axioms for the topological index"

Time: 14:00

Room: MC 107

I will continue last week's talk by listing the axioms for the topological index, and then sketching how these axioms can be used to prove the index theorem (with emphasis on the sketching).

Colloquium

Speaker: Masoud Khalkhali (Western)

"A noncommutative view of zeta regularized determinants and analytic torsion"

Time: 15:30

Room: MC 108

I shall first recall the classical theory of Ray-Singer analytic torsion, and conformal anomaly, for families of elliptic operators. I will mostly focus on families of elliptic operators on Riemann surfaces. The methods used here are based on ideas of spectral geometry and hence stand a chance of extension to a noncommutative setting. The extensions, when possible, are however quite nontrivial and involve many new elements and difficult computations. I shall then look at some known examples of noncommutative Riemann surfaces, the noncommutative elliptic curves equipped with curved metrics, and sketch the progress made in the last few years in understanding their conformal and spectral geometry. Scalar curvature can be defined by study of special values of spectral zeta functions. In particular I shall explain a formula for scalar curvature obtained in my joint work with F. Fathizadeh (and independently by Connes and Moscovici). This formula plays an important role for further study of noncommutative spectral geometry of noncommutative tori.

Analysis Seminar

Speaker: Sean Fitzpatrick (Western)

"Almost CR quantization"

Time: 11:30

Room: MC 107

Given a $G$-invariant almost CR structure on a manifold $M$ one can construct a first-order differential operator whose restriction to the fibres of the CR structure resembles the Dolbeault-Dirac operator on an almost Hermitian manifold. This operator defines a virtual $G$-representation that is infinite-dimensional; however, given an additional assumption on the group action we can compute the character of this representation as a generalized function on $G$. To justify the use of the word "quantization" I'll sketch some parallels with geometric quantization that occur when some additional structure is imposed on the almost CR structure. When the almost CR structure is integrable, we will see the appearance of the tangential Cauchy-Riemann complex in this approach.

Algebra Seminar

Speaker: Omar Ortiz (Western)

"Schubert calculus meet p-compact groups"

Time: 14:30

Room: MC 108

The theory of p-compact groups deals with the homotopy analogues of compact Lie groups, and has been traditionally studied from the homotopy theory point of view. In this talk I will present some connections between this theory and the Schubert calculus, in a more algebro-combinatorial style. In particular I will focus on the different descriptions of the torus-equivariant cohomology of p-compact flag varieties, generalizing the theory of Bruhat graphs and results of Goresky-Kottwitz-MacPherson.

Algebra Seminar

Speaker: Patrick D. F. Ion (Michigan)

"Geometry and the Discrete Fourier Transform"

Time: 15:30

Room: MC 108

We'll see a relationship between some elementary geometry and the discrete Fourier transform, which offers a starting point for excursions into polynomials, complex analysis, interpolation and circulant matrices. It has turned up in practical statistics, fluid mechanics, sculpture, and elsewhere, as well as providing intriguing pictures for the motions of the $N$-body problem. It's behind what has been popularized by Kalman as the most marvelous theorem in mathematics.

Dept Oral Exam

Speaker: Martin Van Hoof (Western)

"Symplectomorphism Groups of Weighted Projective Spaces and Related Embedding Spaces"

Time: 17:00

Room: MC 108

We consider weighted projective spaces and homotopy properties of their symplectomorphism groups. In the case of one singularity, the symplectomorphism group is weakly homotopy equivalent to the Kahler isometry group of a certain Hirzebruch surface that corresponds to the resolution of the singularity. In the case of multiple singularities, the symplectomorphism groups are weakly equivalent to tori. These computations allow us to investigate some properties of related embedding spaces.

Geometry and Topology

Speaker: Rick Jardine (Western)

"Simplicial sheaves, cocycles, and torsors"

Time: 15:30

Room: MC 108

This talk will be a partial introduction to the general subject area. We will discuss the homotopy theory of simplicial sheaves and presheaves, along with its interpretation in non-abelian cohomology and the theory of stacks.

This lecture was first given at a conference at the Fields Institute during the summer of 2013.

Analysis Seminar

Speaker: Yan Xu (Dongbei University of Finance and Economics)

"The ubiquitous B-splines from approximation to Combinatorics: Eulerian numbers, polytopes and Askey scheme"

Time: 14:30

Room: MC 108

Spline functions are original derived from approximation theory then were applied to established the theory of wavelets which is the most important tool in CAGD founded by engineers at the first beginning. They are also closely related with some branches in pure mathematics, such as Combinatorics and Asymptotic analysis. Due to the somewhat high distance between these fields, people working in Enumerative Combinatorics, Partition functions and Asymptotic analysis of biorthogonal systems do not seem to be fully aware of the results on the B-splines. This presentation may shed some lights on the connection between the related problems arising in different fields such as B-splines and Eulerian polynomials, Box-spline and the volume of polytopes, the asymptotic properties of splines and Askey scheme. It may useful to bring together some areas of research which have developed independently without much knowledge of each other.

Homotopy Theory

Speaker: Hugo Bacard (Western)

"p-adic homotopy theory"

Time: 14:30

Room: MC 108

Noncommutative Geometry

Speaker: Alim Eshmatov (Western)

"An Introduction to Homological Mirror Symmetry II"

Time: 14:30

Room: MC 107

Index Theory Seminar

Speaker: Matthias Franz (Western)

"The Lefschetz fixed-point formula"

Time: 14:00

Room: MC 107

Colloquium

Speaker: Richard Hind (Notre Dame University)

"Quantitative results in symplectic geometry"

Time: 15:30

Room: MC 108

Symplectic geometry sometimes seems much like Riemannian geometry, that is, rigid, and other times resembles differential topology, that is, there is much more flexibility in constructions. We will discuss examples like embedding and isotopy problems which are rigid only within a certain range of parameters. Quantitative symplectic geometry aims to determine the critical transition parameters.

PhD Thesis Defence

Speaker: Martin VanHoof (Western)

""Symplectomorphism Groups of Weighted Projective Spaces and Related Embedding Spaces""

Time: 14:00

Room: MC 108

We consider weighted projective spaces and homotopy properties of their symplectomorphism groups. In the case of one singularity, the symplectomorphism group is weakly homotopy equivalent to the Kahler isometry group of a certain Hirzebruch surface that corresponds to the resolution of the singularity. In the case of multiple singularities, the symplectomorphism groups are weakly equivalent to tori. These computations allow us to investigate some properties of related embedding spaces.

Algebra Seminar

Speaker: Marcy Robertson (Western)

"Schematic homotopy types of operads"

Time: 14:30

Room: MC 107

The rational homotopy type $X_{\mathbb{Q}}$ of an arbitrary space $X$ has pro-nilpotent homotopy type. As a consequence, pro-algebraic homotopy invariants of the space $X$ are not accessible through the space $X_{\mathbb{Q}}$. In order to develop a substitute of rational homotopy theory for non-nilpotent spaces Toen introduced the notion of a pointed schematic homotopy type over a field $\mathbb{k}$, $(X\times k)^{sch}.$

In his recent study of the pro-nilpotent Grothendieck - Teichm$\mathrm{\ddot{u}}$ller group via operads, Fresse makes use of the rational homotopy type of the little $2$-disks operad $E_2$. As a first step in the extension of Fresse's program to the pro-algebraic case we discuss the existence of a schematization of the little $2$-disks operad.

Geometry and Topology

Speaker: Sanjeevi Krishnan (Univ. of Pennsylvania)

"Directed Poincare Duality"

Time: 15:30

Room: MC 107

Sheaves of semimodules on locally ordered spaces model real-world local constraints on dynamical systems indescribable in the language of modules. This talk generalizes a homotopical construction of singular sheaf (co)homology for semimodule-valued sheaves over locally ordered spaces; terminal compactifications of ordered Euclidean space play the role of spheres. Examples of singular (co)homology semimodules include flows on a network subject to capacity constraints and causal singularities on spacetimes. Various calculational tools, incorporating previous cubical approximation theorems, will be presented. The main result presented is a Poincare Duality for sheaves over spacetimes and generalizations admitting top-dimensional topological singularities. Calculations on spacetime surfaces and directed graphs follow.

This talk assumes no familiarity with directed topology or semimodules.

Analysis Seminar

Speaker: Grigoris Fournodavlos (University of Toronto)

"On a characterization of Arakelian sets"

Time: 14:30

Room: MC 108

We are going to discuss some standard problems regarding uniform approximation in the complex domain. In particular, we are going to focus our attention on Arakelian's theorem, i.e., uniform holomorphic approximation on closed sets. These topics have been extensively studied in the past since the celebrated theorem by Mergelyan appeared in the early 50's. What we are going to address in this talk is perhaps a not so noted topological viewpoint of the subject. More precisely, we will motivate the following characterization of Arakelian sets and examine applications: "A closed set $F\subset \mathbb{C}$ is an Arakelian set in the plane, if and only if it has a base of simply connected open neighborhoods."

Homotopy Theory

Speaker: Mike Misamore (Western)

"Morava K-theories and the chromatic tower"

Time: 14:30

Room: MC 108

Comprehensive Exam Presentation

Speaker: Josue Rosario-Ortega (Western)

"Complex Monge-Ampère Equations, the Space of Kähler metrics and complexification of Hamiltonian flows"

Time: 13:00

Room: MC 108

In this talk I will explain some geometric aspects of Complex Monge-Ampère Equations and its relation with the space of Kähler metrics in a fixed cohomology class. I will also explain some recent applications to the complexification of Hamiltonian flows on a compact Kähler manifold.

Algebra Seminar

Speaker: Daniel Schaeppi (Western)

"Which tensor categories come from algebraic geometry?"

Time: 14:30

Room: MC 107

Tannakian duality as developed by Grothendieck, Saavedra, Deligne and Milne is a duality between geometric objects (affine group schemes, gerbes) and tensor categories (Tannakian categories). A Tannakian category is a tensor category equipped with additional structure which ensures that it is equivalent to the category of representations of an affine group scheme or gerbe. In characteristic zero Deligne has found a particularly simple description of Tannakian categories. This description is convenient since it only involves talking about properties of the tensor category. The properties in question are enough to construct the required additional structure.

Tannakian duality can be extended to a broader class of geometric objects including schemes and certain algebraic stacks. The corresponding tensor categories (that is, the categories of coherent sheaves on these objects) are the weakly Tannakian categories. I will review the notion of weakly Tannakian category, and I will talk about work in progress to generalize Deligne's description of Tannakian categories in characteristic zero to an intrinsic description of weakly Tannakian categories in characteristic zero.