Algebra Seminar

Speaker: Lila Kari (Western)

"The many facets of natural computing"

Time: 14:30

Room: MC 108

Natural computing is the field of research that investigates models and computational techniques inspired by nature and, dually, attempts to understand the world around us in terms of information processing. It is a highly interdisciplinary field that connects the natural sciences with mathematical and computational science, both at the level of information technology and at the level of fundamental research. As a matter of fact, natural computing areas and topics come in many flavours, including pure theoretical research, algorithms and software applications, as well as biology, chemistry and physics experimental laboratory research.

In this talk, we describe models and computing paradigms abstracted from natural phenomena as diverse as self-reproduction, the functioning of the brain, Darwinian evolution, group behaviour, the immune system, the characteristics of life, cell membranes, and morphogenesis. These paradigms can be implemented either on traditional electronic hardware or on alternative physical media such as biomolecular (DNA, RNA) computing, or trapped-ion quantum computing devices. Dually, we briefly describe several natural processes that can be viewed as information processing, such as gene regulatory networks, protein-protein interaction networks, biological transport networks, and gene assembly in unicellular organisms. In the same vein, we list efforts to understand biological systems by engineering semisynthetic organisms, and to understand the universe from the point of view of information processing.

The talk is based on the review article "The Many Facets of Natural Computing", L. Kari, G. Rozenberg, "Communication of the ACM", October 2008.

Geometry and Topology

Speaker: Jose Malagon Lopez (Western)

"Some computations in equivariant algebraic cobordism"

Time: 15:30

Room: MC 108

Noncommutative Geometry

Speaker: Ali Motadelro (Western)

"Metric aspects of noncommutative geometry 1"

Time: 14:00

Room: MC 106

Metric noncommutative geometry: In this series of talks, I am going to review some metric aspects of noncommutative geometry due to Alain Connes. To be more specific, I will discuss four formulas in Riemannian geometry and formulate them in algebraic forms, so that they can be considered in “noncommutative spaces” as well. These four formulas are concerned about geodesic distance, volume form, space of gauge potentials and Yang-Mills functional action. In the first talk last week, we looked at the spectral triple of a Riemannian manifold which in a sense captures our algebraic data. We also saw a formula for geodesic distance using just this piece of information. For the next talk, I'm planning to discuss volume forms and space of gauge potentials.

Noncommutative Geometry

Speaker: Mehdi Mousavi (Western)

"Equivariant de Rham Cohomology 1"

Time: 14:00

Room: MC 106

I will try to cover the following topics: 1. Equivariant cohomology: Definition, motivation, 2. Two constructions of E, 3. Cartan Model and some examples.

Colloquium

Speaker: Tony Geramita (Queen's)

"Sums of squares: evolution of an idea"

Time: 15:30

Room: MC 108

Beginning with Fermat's characterization of primes which are the sum of two squares, I would like to show how this naturally leads to Waring's Problem for Integers and then to Waring's Problem for Homogeneous Polynomials.

One half of this latter problem has been solved in recent years and I will explain the nature of the approach to that solution through the study of the higher secant varieties of Veronese varieties and the study of non-reduced subschemes of projective n-space.

Despite the technical sounding terms above, the talk is aimed at a general audience and I will keep the technicalities to a minimum.

Stable Homotopy

Speaker: Sam Isaacson (Western)

"Vanishing Lines in Ext"

Time: 10:30

Room: MC 108

Algebra Seminar

Speaker: Tony Geramita (Queen's)

"Generalizations of the notion of "Rank of a Matrix""

Time: 14:30

Room: MC 108

The notion of rank for matrices (or $2$-tensors) is well understood but the generalizations to higher order tensors is much less well understood and is an active area of research with applications in Biology, Statistics, Computing and Signal Processing.

I will explain the nature of the problem, the notions of "tensor rank" and "border rank" and how problems in this area have been studied through the lens of Higher Secant Varieties of Segre Varieties and the cohomology of non-reduced varieties whose support is a union of linear spaces. I will mention some recent progress on the problems that have been done by me and my collaborators M. V. Catalisano and A. Gimigliano and also some conjectures and open problems. If time permits I will also explain some recent work of Landsberg, Weyman and ourselves on the defining ideals of these higher order secant varieties.

Geometry and Topology

Speaker: Priyavrat Deshpande (Western)

"Arrangements of Submanifolds"

Time: 15:30

Room: MC 108

A real arrangement of hyperplanes is a collection of finitely many hyperplanes in a real vector space. It is known that the combinatorics of the intersections of these hyperplanes contains substantial information about the topology of the complement of the hyperplanes in the real as well as complexified space. For example, the cohomology of the complexified complement can be expressed in terms of the intersection lattice associated with the arrangement. The face poset of an arrangement defines a simplicial complex (the Salvetti complex) which has the homotopy type of this complement.

Noncommutative Geometry

Speaker: Ali Motadelro (Western)

"Metric aspects of noncommutative geometry II"

Time: 14:00

Room: MC 106

Metric noncommutative geometry: In this series of talks, I am going to review some metric aspects of noncommutative geometry due to Alain Connes. To be more specific, I will discuss four formulas in Riemannian geometry and formulate them in algebraic forms, so that they can be considered in “noncommutative spaces” as well. These four formulas are concerned about geodesic distance, volume form, space of gauge potentials and Yang-Mills functional action. In the first talk last week, we looked at the spectral triple of a Riemannian manifold which in a sense captures our algebraic data. We also saw a formula for geodesic distance using just this piece of information. For the next talk, I'm planning to discuss volume forms and space of gauge potentials.

Pizza Seminar

Speaker: Farzad Fathizadeh (Western)

"What does the spectral theorem say?"

Time: 17:00

Room: MC 107

The Spectral Theorem, and the closely related Spectral Multiplicity Theory is a gem of modern mathematics. It is about the structure, and complete classification, up to unitary equivalence, of normal operators on a Hilbert space. This theorem is the generalization of the theorem in linear algebra which says that every normal, in particular selfadjoint, matrix is unitarily equivalent to a diagonal matrix; or, in simple terms, is diagonalizable in an orthonormal basis. The extension of this result to infinite dimensions is by no means obvious and involves many new subtle phenomena that have no analogue in finite dimensions. The final result has many applications to pure and applied mathematics, mathematical physics, and quantum mechanics. In this talk, a proof of the spectral theorem for Hermitian operators on a Hilbert space will be outlined and some applications will be discussed. This talk should be accessible to undergraduate students.

Noncommutative Geometry

Speaker: Mehdi Mousavi (Western)

"Equivariant de Rham cohomology II"

Time: 14:00

Room: MC 106

Stacks Seminar

Speaker: Jeffrey Morton (Western)

"Topological and Smooth Groupoids"

Time: 13:30

Room: MC 107

In this talk, I will give some basic definitions and facts about topological and Lie groupoids. I will describe some examples involving group actions and configuration spaces of geometric structures. I will outline some work analogs of standard constructions in differential geometry, such as differential forms, in this context. I will also describe how these same examples can be described in terms of other kinds of structures: stacks (specifically, topological and differentiable stacks), and $C*$-algebras associated to the groupoids. This talk will be a fairly introductory presentation of the subjects of this seminar.

Colloquium

Speaker: Edward Bierstone (Toronto)

"Resolution except for minimal singularities"

Time: 15:30

Room: MC 108

The subject of the talk is resolution of singularities in algebraic or analytic geometry. Resolution of singularities leads to a space with only normal crossings singularities (i.e., transverse self-intersections). It therefore makes sense to consider normal crossings singularities acceptable from the start, and try to resolve singularities except for normal crossings. We will discuss the following question (a variant of a problem of Janos Kollar). Can we find the smallest class of singularities S with the following properties: (1) S includes all normal-crossings singularities; (2) every variety or space X admits a proper mapping f: X' --> X such that X' has only singularities in S, and f is an isomorphism over the locus of points of X having only singularities in S? No technical background will be assumed.

Stable Homotopy

Speaker: Enxin Wu (Western)

"Self-maps and Periodicity for Modules over the Steenrod Algebra"

Time: 10:30

Room: MC 108

Algebra Seminar

Speaker: Eric Schost (Western)

"Computing roadmaps"

Time: 14:30

Room: MC 108

Consider the following questions (coming for instance from motion planning problems): given two points on a real algebraic set $S$, do they belong to the same connected component? If so, how can we connect them?

Canny introduced "roadmaps" as a way to reduce such problems to computations with curves only. Given $s$ polynomial equations with rational coefficients, of degree $d$ in $n$ variables, Canny's algorithm, and its generalizations by Basu, Pollack and Roy, have a cost polynomial in $(s D^n)^n$.

This is depressingly high; as a result, none of these algorithms is practical for realistic instances. Indeed, one would rather expect a cost polynomial in $s D^n$. I will present ongoing work with Mohab Safey El Din toward this goal.

Symplectic Learning Seminar

Speaker: M. VanHoof (Western)

"Symplectic Cutting II"

Time: 13:30

Room: MC 105c

Applications of the symplectic cutting construction will be discussed.

Geometry and Topology

Speaker: Kirill Zainoulline (Ottawa)

"Degree formula for connective K-theory"

Time: 15:30

Room: MC 108

We use the degree formula for connective K-theory to study rational contractions of algebraic varieties. As an application we obtain a condition of rational incompressibility of algebraic varieties and a version of the index reduction formula. Examples include complete intersection, rationally connected varieties, twisted forms of abelian varieties and Calabi-Yau varieties.

Noncommutative Geometry

Speaker: Ali Motadelro (Western)

"Metric aspects of noncommutative geometry III"

Time: 14:00

Room: MC 106

Metric noncommutative geometry: In this series of talks, I am going to review some metric aspects of noncommutative geometry due to Alain Connes. To be more specific, I will discuss four formulas in Riemannian geometry and formulate them in algebraic forms, so that they can be considered in “noncommutative spaces” as well. These four formulas are concerned about geodesic distance, volume form, space of gauge potentials and Yang-Mills functional action. In the first talk last week, we looked at the spectral triple of a Riemannian manifold which in a sense captures our algebraic data. We also saw a formula for geodesic distance using just this piece of information. For the next talk, I'm planning to discuss volume forms and space of gauge potentials.

Analysis Seminar

Speaker: Vladimir Chernov (Dartmouth College)

"Topological Properties of Manifolds admitting a $Y^x$-Riemannian metric"

Time: 15:30

Room: MC 108

A complete Riemannian manifold $(M, g)$ is a $Y^x_l$-manifold if every geodesic $\gamma(t)$ parametrized by arc length and originating at a point $\gamma(0)=x$ satisfies $\gamma(l)=x$ for $0\neq l\in \mathbb R$. Berard-Bergery proved that if $(M,g)$ is a $Y^x_l$-manifold of dimension $m>1$, then $M$ is a closed manifold with finite fundamental group, and the ring $H^*(M, \mathbb Q)$ is generated by one element. We say that $(M,g)$ is a $Y^x$-manifold if for every $\epsilon >0$ there exists $l$ with $|l|>\epsilon$ such that for every geodesic $\gamma(t)$ parametrized by arc length and originating at $x$, the point $\gamma(l)$ is $\epsilon$-close to $x$. We use Low's notion of refocussing Lorentzian manifolds to show that if $(M, g)$ is a $Y^x$-manifold of dimension $m>1$, then $M$ is a closed manifold with finite fundamental group. If $\dim M=2, 3$ and $(M,g)$ is a $Y^x$-manifold, then $(M, \tilde g)$ is a $Y^x_l$-manifold for some metric $\tilde g$.

Pizza Seminar

Speaker: Siyavus Acar (Western)

"Circular Billiards"

Time: 17:00

Room: MC 107

There is an old question in optics that has been called Alhazen's Problem. The name Alhazen honours an Arab scholar Ibn-al-Haytham who flourished 1000 years ago. The problem itself can be traced further back, at least to Ptolemy's Optics written around AD 150. The problem - while considered one of the 100 great problems of elementary mathematics - is very easy to state: Given two arbitrary balls on a circular billiard table, how does one aim the object ball so that it hits the target ball after one bounce off the rim. In this talk we introduce various methods of approach that has been studied, but mainly focus on the number of solutions and their distribution on the table.

Noncommutative Geometry

Speaker: Mehdi Mousavi (Western)

"Equivariant de Rham cohomology III"

Time: 14:00

Room: MC 106

I will try to cover the following topics: 1. Equivariant cohomology: Definition, motivation, 2. Two constructions of E, 3. Cartan Model and some examples.

Colloquium

Speaker: Vladimir Chernov (Dartmouth College)

"Legendrian links, causality, and the Low conjecture"

Time: 15:30

Room: MC 108

Two points x,y in a spacetime X are said to be causally related if there is a nonspacelike curve between them, i.e. if one can get from one point to the other moving not faster than the light speed. For globally hyperbolic spacetimes X the light rays through a point x form a Legendrian sphere $S_x$ in the contact manifold N of all light rays in X. We show that if the universal cover of a level set of a timelike function is an open manifold, then x and y are causally related exactly when the Legendrian link $(S_x, S_y)$ is nontrivial in N. In particular this proves the Low conejcture and the Legendrian Low conjecture formulated by Natario and Tod.

Stacks Seminar

Speaker: Ajneet Dhillon (Western)

"Stacks"

Time: 11:30

Room: MC 107

This talk will give a historical motivation for stacks from the nonexistence of moduli spaces, give a non-technical definition of a stack, and describe some examples.

Stable Homotopy

Speaker: Enxin Wu (Western)

"Margolis' killing construction"

Time: 13:30

Room: MC 108

Algebra Seminar

Speaker: Richard Gonzales (Western)

"Group embeddings and cohomology"

Time: 14:30

Room: MC 108

Let $G$ be a reductive group. A $G\times G$-variety $X$ is called an equivariant compactification of $G$ if $X$ is normal, projective, and contains $G$ as an open and dense orbit. Regular compactifications and reductive embeddings are the main source of examples. In the first case, the equivariant cohomology ring has been explicitely described by Bifet, de Concini, Procesi and Brion. Loosely speaking, it depends mostly on the torus embedding part and the structure of the $G\times G$-orbits. As for the second class, Renner has found that they have a canonical cell decomposition based on underlying monoid data. My goal in this talk is to give an overview of the theory of group embeddings, putting more emphasis on the monoid approach, and to describe the structure of the so called rational cells. Finally, I will explain how such cellular decompositions could lead to a further application of GKM theory to the study of reductive embeddings.