I added a list of recent customers of ours here.

I also created a page for the course I am teaching: Introductory algebraic topology. The outline is there already and some of the articles have been written. The second one is Homology as a equivalence relation. Consider the question, What is a tunnel? It’s not as simple as it seems. It takes some work to find a good answer, the main part of which is: A tunnel is an equivalence class of closed surves.

Over the following months (it’s a two semester course) I’ll keep adding material as the course progresses.

I am also teaching Advanced calculus and some of this stuff will also find its way into the wiki.

A new paper that uses Pixcavator:

Down-regulation of CXCR4 and CD62L in Chronic Lymphocytic Leukemia Cells Is Triggered by B-Cell Receptor Ligation and Associated with Progressive Disease [1] by Amalia Vlad, Pierre-Antoine Deglesne, Re´mi Letestu, Ste´phane Saint-Georges, Nathalie Chevallier, Fanny Baran-Marszak, Nadine Varin-Blank, Florence Ajchenbaum-Cymbalista, and Dominique Ledoux (Cancer Research 69, 6387, August 15, 2009).

From the paper:

“Progressive cases of B-cell chronic lymphocytic leukemia (CLL) are frequently associated with lymphadenopathy, highlighting a critical role for signals emanating from the tumor environment in the accumulation of malignant B cells.”

“BCR-stimulated and unstimulated fluorescent cells were mixed in RPMI 1640/10% FCS and added together onto the endothelial cell layer. After incubation for 2 h at 37jC, the nonadherent CLL cells were washed off. Remaining adherent cells were fixed, and 10 fields from duplicate chamber slides (average of 500 cells/field) were photographed under fluorescent microscope. Red and green fluorescence were separately quantified using the Pixcavator IA 3.3 software (Intelligence Perception Co.).”

Take a look at other papers that use Pixcavator.

Topology is usually defined as the science of the spacial properties that preserved under continuous transformations. So that you can bend, stretch and shrink etc but not tear or glue. This might be the only way to capture the essence in a single sentence but this sentence is meaningless to a person who knows nothing about topology. And the question “Why do we need to study topology?” still remains.

So, we start with examples instead. They come from three seemingly unrelated areas: computer vision, cosmology, and data analysis.

Computer vision

In industrial settings one might need to consider the integrity of objects being manufactured.

The first question may be: this is a bolt holding two things together, does it still or is there a crack in it?

Could be a bone too…

The second question may be: this material is supposed to hold liquid, is it water tight or is there leakage?

In other words: does it hold water?

The third question may be: to be strong this alloy is supposed to be solid, is it or are there air bubbles?

The opposite question is: does it hold air?

Observe that we consider here three different kinds of integrity as there may be a crack but no hole or vice versa etc.

We can describe these situations informally as:

• there are cuts in the object,
• there are tunnels,
• there are voids.

These three types of “damage” correspond to cycles of dimensions 0, 1, and 2 respectively. This is why:

• The simplest object with a “cut” is two points and points are 0-dimensional.
• The simplest object with a “tunnel” is a circle and curves are 1-dimensional.
• The simplest object with a “void” is a sphere and surfaces are 2-dimensional.

Cosmology

What is the shape of the universe? What is the topology? Does it have “cuts”, “tunnels”, or “voids”?

Read the whole article.