Geometry and Topology

Speaker: Graham Denham (Western)

"The tropical construction of de Concini and Procesi's wonderful models"

Time: 15:30 - 16:30

Room: MC 107

In 1995, de Concini and Procesi investigated certain iterative blowups of affine space along intersections of linear subspaces, their wonderful models, a fundamental example being the Fulton-Macpherson configuration space compactification. In doing so, they developed suitable combinatorics to describe, among other things, the cohomology of the wonderful models.

In 2006, Feichtner and Yuzvinsky constructed smooth toric varieties from de Concini and Procesi's combinatorial data, and found that, for any arrangement of hyperplanes, the cohomology ring of the de Concini-Procesi wonderful model is isomorphic to the Chow ring of their toric variety. Their argument is indirect, via the combinatorics defining the rings in question.

I will outline a toric construction of de Concini and Procesi's wonderful models for hyperplane arrangements. This is an example of Tevelev's notion of a tropical compactification. One advantage is that it provides a geometric explanation of Feichtner and Yuzvinsky's isomorphism.

Analysis Seminar

Speaker: Seyed Mehdi Mousavi (Western)

"An Infinite-Dimensional Maximal Torus and Shur-Horn-Kostant Convexity"

Time: 14:40 - 15:30

Room: MC 107

One of the main notion introduced in the study of finite dimensional compact Lie groups is the so-called maximal torus. In 1997, Bao and Ratiu discovered an infinite dimensional subgroup in the group of the volume-preserving diffeomorphisms of the 2-dimensional annulus that can potentially play the role of a maximal torus. They showed this subgroup is a path-connected submanifold which is flat and totally geodesic with respect to the hydrodynamic metric. Moreover it is a maximal abelian subgroup (with a finite Weyl group). This suggested that part of finite dimensional Lie group theory may be extended to the volume-preserving diffeomorphisms of the annulus. Indeed, in a later work, Bloch, Flaschka and Ratiu showed that after an appropriate completion of the spaces considered, a version of Schur-Horn-Kostant convexity theorem holds. El-Hadrami extended these results to the case of the unit sphere and CP^{2}, found a candidate for the maximal torus in the symplectomorphism group of symplectic toric manifolds, and then conjectured that some results in previous works can be extended to those groups. However, a gap in El-Hadrami’s arguments was later discovered.

In two talks we discuss some possible extensions and corrections to El-Hadrami´s work. We also mention the Schur-Horn-Kostant convexity theorem for the symplectomorphism groups of toric manifolds.

Noncommutative Geometry

Speaker: Masoud Khalkhali (Western)

"Theorema Egregium and Gauss-Bonnet Theorem for Surfaces"

Time: 14:30 - 16:30

Room: MC 108

Since this year we shall be busy with curavture in noncommutative geometry, I thought I should start with the most fundamental classical incarnation of this notion: Gauss' theory of curvature for surfaces, and what it can teach us. All are welcome!

When Gauss, in his celebrated paper of 1827, {\it Disquisitiones generales circa superficies curvas} {(\it General investigations of curved surfaces)} after a long series of calculations

eventually showed that the extrinsically defined curvature of a

surface can be expressed entirely in terms of its intrinsic metric (= the

first fundamental form), he got so excited that he called the obvious corollary of this result Theorema Egregium (The Remarkable Theorem).

Gauss's formidable curvature formula, and the closely related {\it Gauss-Bonnet theorem} is the foundation stone for

all of differential geometry, as it was later shown by Riemann in 1859

that the curvature of higher dimensional manifolds can be understood

purely in terms of curvatures of its two dimensional submanifolds.

Theorema Egregium can also be regarded as the infinitesimal form of,

and in fact is equivalent to, the celebrated Gauss-Bonnet Theorem. This paper of Gauss is the single most important work in the entire history of differential geometry.

Colloquium

Speaker: Rasul Shafikov (Western)

"Lagrangian immersions, polynomial convexity, and Whitney umbrellas"

Time: 15:30 - 16:30

Room: MC 107

An embedding $\phi: S \to \mathbb R^4$ from a real surface is called Lagrangian if $\phi^* \omega =0$, where $\omega$ is the standard symplectic form on $\mathbb R^4$. Gromov's theorem (1985) on the existence of a holomorphic disc attached to a compact Lagrangian submanifold of $\mathbb C^n$ provides topological obstructions for Lagrangian embeddings (or immersions) of compact surfaces. However, Givental (1986) showed that such maps always exist if we allow singularities that are either self-intersections or open Whitney umbrellas.

Existence of holomorphic discs attached to a submanifold $X$ of $\mathbb C^n$ is related to the question of polynomial convexity of $X$. I will discuss the joint work with Alexandre Sukhov, where we show that if a Lagrangian surface $X \subset \mathbb C^2$ has an isolated singularity which is a Whitney umbrella, then near the singularity the surface $X$ is locally polynomially convex.

Algebra Seminar

Speaker: Matthias Franz (Western)

"Equivariant cohomology and syzygies"

Time: 14:30 - 15:30

Room: MC 107

The GKM method is a powerful way to compute the equivariant (and ordinary) cohomology of many spaces with torus actions. So far it has been applied to so-called equivariantly formal $T$-spaces, which include compact Hamiltonian $T$-manifolds.

In this talk I will explain that the GKM method is valid for a much larger class of $T$-spaces. The explanation is based on a new interpretation of a sequence originally due to Atiyah and Bredon, and involves the notion of syzygies as used in commutative algebra. I will also exhibit a surprising relation between the equivariant Poincaré pairing and the GKM description.

This is joint work with Chris Allday and Volker Puppe.

Geometry and Topology

Speaker: (Western)

"No Seminar--Thanksgiving"

Time: 15:30 - 16:30

Room: MC 108

Analysis Seminar

Speaker: Rasul Shafikov (Western)

"Lagrangian immersions, polynomial convexity, and Whitney umbrellas, II"

Time: 14:40 - 15:30

Room: MC 107

This is a continuation of the talk from October 6. Details will be given of the proof that a Lagrangian surface $X\subset \mathbb C^2$ near an isolated singularity which is a Whitney umbrella is locally polynomially convex. In this talk I will discuss the connection between polynomial convexity of surfaces and their characteristic foliation.

Pizza Seminar

Speaker: Martin Pinsonnault (Western)

"The unsolvability by radicals of the quintic"

Time: 16:00 - 17:00

Room: MC 107

The aim of the talk is to prove the unsolvability by radicals of the quintic (in fact of the general \(n^{\text{th}}\) degree equation for \(n\geq 5\)). This famous theorem was first proved by N. Abel and P. Ruffini around 1821. However, a complete understanding of \(\textit{solvability}\) had to wait for Evariste Galois and his introduction of group theory in a 1831 manuscript that was miraculously found by Liouville in 1843. We will present a proof of the Abel-Ruffini theorem, very close to Galois's own exposition that uses only elementary properties of groups, rings, and fields as they are taught in a first course in abstract algebra.

Noncommutative Geometry

Speaker: Asghar Ghorbanpour (Western Phd Student)

"Spectral Triples (I. Definition and Examples)"

Time: 14:30 - 16:00

Room: MC 108

Geometric operators defined on a compact Riemannian manifold, e.g. Laplacian, Dirac, provide a framework in which we can investigate some geometric properties while we are completely working with algebra of operators on Hilbert spaces and commutators and spectral analysis of operators. In this setting we will have objects called spectral triples introduced by Alain Connes, which will play role of differential calculus on our (noncommutative) spaces.

A spectral triple is a triple (A,H,D) in which A is an involutive algebra (plays role of $C^\infty (M)) and H is Hilbert space on which A acts continuously (it is analogous of the space of the sections of vector bundle which D acts on) and D is an operator (it is our first order elliptic differential operator) which has some properties.

This talk is the first session of a series of talks in which we will investigate different properties and examples and objects related to spectral triples. The talk will start with definition of spectral triples and we shall go through classical examples to show where the ideas come from and at the end a spectral triple defined on NC-torus will be discussed.

Colloquium

Speaker: Rosona Eldred (UIUC/ Hamburg)

"Goodwillie's Calculus of Functors"

Time: 15:30 - 16:30

Room: MC 107

A reasonable first approximation of a space X is its homology. Similarly, a first approximation to a functor F may be given by a homology theory, which is a particularly nice linear functor. This is the first stage of a Taylor tower of functors approximating F, developed by Goodwillie. One nice property of this tower is that within the ``radius of convergence'' of F, its value on a space X may be determined by the value of the limit of its Taylor tower on X. Functors of special interest are the identity functor of spaces and Waldhausen's Algebraic K-theory of a space, A(X). We will give an introduction to the calculus of functors, incorporating known results about these two functors.

Algebra Seminar

Speaker: Vladimir Miransky (Western)

"Theory of Quantum Hall Effect in monolayer and bilayer graphene"

Time: 14:30 - 15:30

Room: MC 107

I describe the theory of the quantum Hall effect in monolayer and bilayer graphene based on the magnetic catalysis effect. The role of the symmetry and its breakdown in this phenomenon is discussed.

Analysis Seminar

Speaker: Rasul Shafikov (Western)

"Lagrangian immersions, polynomial convexity, and Whitney umbrellas, III"

Time: 14:40 - 15:30

Room: MC 107

This is a continuation of the talk from October 11. Details will be given of the proof that a Lagrangian surface $X\subset \mathbb C^2$ near an isolated singularity which is a Whitney umbrella is locally polynomially convex. In this talk I will discuss Bruno's construction of normal forms for the dynamical system that determines the phase portrait of the characteristic foliation.

Colloquium

Speaker: David McKinnon (Waterloo)

"Rational points repel each other, so they can't be densely packed"

Time: 15:30 - 16:30

Room: MC 107

Well, actually, we only think that rational points repel each other, so the title is merely conjectural in general. In particular, there is a conjecture of Batyrev and Manin that says that on certain kinds of algebraic varieties, points with rational coordinates are not very densely packed. There is another conjecture, due to Vojta, that says that subvarieties of algebraic varieties cannot be approximated too well by rational points. The purpose of my talk will be to make all these notions precise, to explain what they have to do with one another, and to explain why Vojta's conjecture sometimes implies that of Batyrev and Manin.

Noncommutative Geometry

Speaker: Masoud Khalkhali (Western)

"Theorema Egregium and Gauss-Bonnet Theorem for Surfaces (2)"

Time: 10:30 - 12:30

Room: MC 108

Since this year we shall be busy with curavture in noncommutative geometry, I thought I should start with the most fundamental classical incarnation of this notion: Gauss' theory of curvature for surfaces, and what it can teach us. All are welcome!

When Gauss, in his celebrated paper of 1827, {\it Disquisitiones generales circa superficies curvas} {(\it General investigations of curved surfaces)} after a long series of calculations

eventually showed that the extrinsically defined curvature of a

surface can be expressed entirely in terms of its intrinsic metric (= the

first fundamental form), he got so excited that he called the obvious corollary of this result Theorema Egregium (The Remarkable Theorem).

Gauss's formidable curvature formula, and the closely related {\it Gauss-Bonnet theorem} is the foundation stone for

all of differential geometry, as it was later shown by Riemann in 1859

that the curvature of higher dimensional manifolds can be understood

purely in terms of curvatures of its two dimensional submanifolds.

Theorema Egregium can also be regarded as the infinitesimal form of,

and in fact is equivalent to, the celebrated Gauss-Bonnet Theorem. This paper of Gauss is the single most important work in the entire history of differential geometry.

Algebra Seminar

Speaker: Masoud Khalkhali & Farzad Fathizadeh (Western and York)

"Curvature in noncommutative geometry II"

Time: 15:40 - 17:00

Room: MC 107

In this talk, I will continue the lecture given by Masoud Khalkhali on our recent joint work on the Gauss-Bonnet theorem and scalar curvature for the noncommutative two torus, in the context of Alain Connes' noncommutative differential geometry. I will first construct the Connes-Tretkoff spectral triple encoding the metric information on this $C^*$-algebra so that we view it as a noncommutative Riemannian manifold equipped with a general metric. Then I will recall a spectral definition for its scalar curvature, and will illustrate the process of finding a local expression for the curvature by employing a special case of Connes' pseudodifferential calculus for $C^*$-dynamical systems by means of which one can pursue the heat kernel scheme of elliptic differential operators and index theory. I should mention that recently Connes and Moscovici also found precisely the same formula independently. At the end I will explain how this formula fits into our earlier work which extends the Gauss-Bonnet theorem of Connes and Tretkoff to general conformal structures on noncommutative two tori.

Analysis Seminar

Speaker: Sudeshna Basu (George Washington University)

"Non compact Trace class operators and Schatten-class operators in p-adic Hilbert spaces"

Time: 14:00 - 15:00

Room: MC 107

We introduce Schatten-class operators in p-adic Hilbert spaces and study their properties and give several examples. We also show that the Trace class operators in p-adic Hilbert spaces strictly contains the class of completely continuous operators.This gives a totally new perspective to the space of compact operators in p-adic Hilbert spaces. Apart from its intrinsic interest as discussed in this paper, p-adic functional analysis is a fast growing research area and has numerous applications in differential equations, statistics, quantum physics, dynamical systems, cognitive sciences, psychology and sociology, to name a few.

Geometry and Topology

Speaker: Ajneet Dhillon (Western)

"The Fundamental Group Scheme-An Overview"

Time: 15:30 - 16:30

Room: MC 107

Analysis Seminar

Speaker: Seyed Mohammad Hadi Seyedinejad (Western)

"Testing local regularity of complex analytic mappings by fibred powers"

Time: 14:40 - 15:30

Room: MC 107

This two-session talk will be concerned with holomorphic mappings between complex analytic sets (or more generally, analytic spaces). Local regularity of such a mapping can be measured by uniformity (or lack thereof) of the family of its fibres. In the first part of the talk, we will discuss the general idea of testing local regularity (like openness or flatness) by passing to fibred powers of a given map. The second session will be devoted to a recent joint work with Janusz Adamus: We establish an analytic version of flatness descent to prove a criterion for flatness of a holomorphic mapping with singular target. Previously, the best analogous result had been known only for the case of smooth targets.

Pizza Seminar

Speaker: Masoud Khalkhali (Western)

"A Topological proof of Abel-Ruffini theorem"

Time: 16:00 - 17:00

Room: MC 107

Galois referred to his theory as the ``Theory of Ambiguities". In his last letter he writes: “Mes principales m'editations depuis quelque temps etaient dirigees sur l’application a l’analyse transcendante de la theorie de l’ambiguite. Il s’agissait de voir a priori dans une relation entre quantites ou fonctions transcendantes quels echanges on pouvait faire, quelles quantites on pouvait substituer aux quantites donnees sans que la relation put cesser d’avoir lieu. Cela fait reconnaitre tout de suite l’impossibilite de beaucoup d’expressions que l’on pourrait chercher. Mais je n’ai pas le temps et mes id´ees ne sont pas encore bien d´eveloppees sur ce terrain qui est immense...”

This idea of Galois is indeed so rich and the territory is so vast that even after 200 years of mathematics we are still not sure it has fully delivered all its potential. I shall introduce the idea of analytic continuation and use it to define the monodromy of an algebraic function, as an instance of the application of ambiguities in analysis and algebra. I shall then indicate how this quickly leads to a proof of impossibility of solving general quintics by radicals, the Abel-Ruffini theorem.

Noncommutative Geometry

Speaker: Ali Fathi (Western)

"Gauss-Bonnet Formula for Hypersurfaces"

Time: 10:30 - 12:00

Room: MC 108

Gauss–Bonnet theorem or Gauss–Bonnet formula is one of the star attractions of modern differential geometry. It states that for a compact oriented manifold M, the "curvatura integra" over M is equal to a multiple of Euler Characteristic of M.

We shall give an "extrinsic" proof for M as an embedded submanifold (actually an even dimensional hyper surface)of a Euclidean space. The proof is heavily based on the Poincare-Hopf index theorem which states that the sum of indexes of a smooth vector field over M is equal to the Euler characteristic of M.

Algebra Seminar

Speaker: Mohab Safey El Din (Universite Pierre et Marie Curie (Paris 6), INRIA Paris-Rocquencourt Research Centre)

"Polynomial System Solving over the Reals: Algorithms, Complexity, Implementations and Applications"

Time: 14:30 - 15:30

Room: MC 320

Solving non-linear algebraic problems is one of the major challenges in scientific computing. In several areas of engineering sciences, algebraic problems encode geometric conditions on variables taking their values over the reals. Thus, most of the time, one aims to obtain some informations on the real solution set of polynomial systems. The resolution of these problems often has a complexity which is exponential in the number of variables.

In this talk, I will review some geometric and algebraic techniques which enable to obtain fast practical algorithms meeting the best known complexity bounds. These algorithms are implemented in the maple package (RAGlib: The Real Algebraic Geometry library) which has the feature to provide algorithms of asymptotically optimal algorithms in real geometry. Its practical performances will be discussed and some applications will be presented.

This talk is based on joint work with J.C. Faugere, A. Greuet, E. Schost, PJ Spaenlehauer and L. Zhi.

Geometry and Topology

Speaker: Matthias Franz (Western)

"Equivariant cohomology and syzygies"

Time: 15:30 - 16:30

Room: MC 107

The GKM method is a powerful way to compute the equivariant (and ordinary) cohomology of many spaces with torus actions. So far it has been applied to so-called equivariantly formal $T$-spaces, which include compact Hamiltonian $T$-manifolds.

In this talk I will explain that the GKM method is valid for a much larger class of $T$-spaces. The explanation is based on a new interpretation of a sequence originally due to Atiyah and Bredon, and involves the notion of syzygies as used in commutative algebra. I will also exhibit a surprising relation between the equivariant Poincaré pairing and the GKM description.

This is joint work with Chris Allday and Volker Puppe.