Homomorphism vs Homeomorphism

1 mainDid you get the joke in the picture to the left? If not, you will do so in a few minutes. I was recently reading an article and I came across the terms mentioned in the title. From the looks of it, they are very close to each other, right? In many fields within mathematics, we talk about objects and the maps between them. Now you may ask why we would want to do that? Well, transformation is one of the most fundamental things in any field. For example, how do we transform a line into a circle, or fuel into mechanical energy, or words into numbers? There are infinitely many types of transformations that can exist. Obviously, we cannot account for every single type of transformation that can possibly exist. So we limit ourselves to only the interesting ones. So what exactly is it all about? How does it even relate to the title of this blog post?   Continue reading

What Is The Poincaré Conjecture?

colored shapeBefore we start, let me put something out there. Poincaré Conjecture is one of the seven millennium problems established by the Clay Mathematics Institute. Those problems are worth a million dollars each! The Poincaré conjecture depends on the mind-numbing problem of understanding the shapes of spaces. This field of study is referred to as “topology” by mathematicians. Topology is an important field within mathematics concerned with the study of shapes, spaces and surfaces. We interact with these in our everyday lives. We see surfaces breaking and shapes getting deformed all the time. Ever wondered if there is any law governing these deformations? Or are these deformations just too random to be studied? Does Poincaré Conjecture have any real world applications?   Continue reading