What Is Zeta Function Regularization?

1 mainThere is a popular mathematical result which says that the sum of all natural numbers is -1/12. I have discussed it in detail here. This looks very unintuitive to a first time observer. In fact, most people would say that this is some kind of mathematical trickery. How can a bunch of positive numbers sum up to a negative fraction, right? Actually, there is a very real purpose to this whole thing of adding up all the natural numbers to get a negative fraction as the result. However, our general sense tells us that this shouldn’t be possible. The discussion in one of my previous blog posts was about the mathematics involved in this result. This discussion is more about the underlying fundamentals and where these results come from. So how do we explain this situation? Where is it used in real life?   Continue reading

What Is A Singularity?

1 mainYou must have heard the term “singularity” being used in various contexts. We see it in mathematics, physics, circuit design, mechanics, fluid dynamics, etc. The word “singular” means something that is extraordinary, unique, and strange. When we talk about singularity in mathematics, we usually refer to the uniqueness of mathematical objects. In particular, singularities refer to the points where the mathematical objects are not well-behaved i.e. we can’t define them for those points. In higher mathematics, we usually define functions to explain the behavior of any system. These functions are the mathematical objects under consideration, and we will see how to understand their singularities. But why do we care about singularities? Is it just a mathematical concept or do we ever see it in real life?   Continue reading

1 + 2 + 3 + 4 + 5 + …. = -1/12

sum of all natural numbersWait a minute, what? That looks very very wrong. You must be wondering if the title is some horrendous typo, right? Well, let me assure you that it’s absolutely not! The sum of all natural numbers is equal to -1/12. This blog post is not just about mathematical trickery either. The equation in the title is actually a very important result used in theoretical physics, particularly in string theory. Now how can that be possible? Are physicists really that bad at mathematics? That can’t be it! What is the proof behind this? Do we ever encounter it in real life?   Continue reading

Quantum Encryption And Black Holes – Part 2/2

1 mainIn the previous post, we discussed about the concepts of quantum encryption and black holes. We also talked about how we do cryptography in the subatomic world. This blog post is a continuation of that discussion. As the title suggests, the overarching theme is the relationship between quantum encryption and black holes. Let’s continue talking about it then. Although quantum encryption looks extremely robust in theory, how practical is it? What do we know about its security and how is it related to black holes? We know that nothing can escape from black holes, so we need a way to understand more about the black holes.   Continue reading

Quantum Encryption And Black Holes – Part 1/2

1 mainIs that really the title? It looks like two random things mashed up together. Doesn’t make much sense, right? Well, recent research suggests that quantum encryption and black holes may be related. A proposed mathematical proof outlines the way in which information behaves in coded messages, and this may have implications for black holes. The proof basically suggests that the radiation spit out by black holes may retain information about them. The research not only focuses on encoding communications in quantum mechanical systems, but also addresses a long-standing question for theoretical physicists: What exactly happens to all the stuff that falls into a black hole? Is it possible to retrieve any information about the black hole?   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