Ashika Jayanthy’s Blog

I have a small ritual before decisions that matter. Three coins, thrown six times. The resulting pattern is a hexagram, one of sixty-four that tells me something. Not about the future. About the present state of my own thinking.  The I Ching is, at its mechanical core, a sampling device. The traditional yarrow stalk method generates lines through a procedure weighted so that not all outcomes are equally likely. The coin method is simpler: three coins, each heads or tails, combined to produce one of four line types. Do this six times and you have drawn a point from a space of sixty-four possibilities. A 6-bit binary string. A hexagram. What strikes me about this is not the mysticism that has accumulated around it over millennia, but the structure underneath. You are not choosing. You are sampling. And the question you bring to the coins acts as a constraint on how you interpret what you draw, which means you are, without knowing it, doing something very close to what Edwin Jaynes sp...

Here’s a question that sits underneath a lot of mathematical biology, usually unasked: why this mathematical object and not another? Why differential equations for population dynamics, why information theory for neural coding, why networks for protein interactions? The answer is rarely arbitrary. The best mathematical frameworks don’t get imposed on biological systems from outside. They get recognized, pulled out of structure that was already there, waiting for the right language. Group theory and DNA is one of the cleaner examples of this recognition. I want to explain why. A group, in the mathematical sense, is a set of transformations that can be composed, reversed, and that includes doing nothing at all. The integers under addition form a group. Rotations of a sphere form a group. What groups measure, in the deepest sense, is symmetry: the structure of what stays the same when something changes. This turns out to be exactly the right question to ask about DNA. Consider what every m...

  V4 didn’t just ‘work’, it separated real DNA from randomized sequences in a way SU(2) completely failed to do. First, a bit about V4:  The Klein four-group V4 is the simplest non-trivial example of an abelian group that is not cyclic. It has four elements — the identity e, and three elements a, b, ab — each of order 2, meaning every element is its own inverse. The group table is entirely determined by this: any two distinct non-identity elements multiply to give the third. What makes V4 attractive for DNA is its natural fit with four bases, its asymmetric metric structure when you weight transitions and transversions differently, and the fact that its irreducible representations decompose cleanly into four components, one for each codon position, as it turns out. When you assign each DNA base to a V4 element and multiply bases along a sequence window as a path-ordered product, the probability that the product returns to the identity encodes the statistical structure of the s...

I came up with an idea some time ago. Let me lay it out here for your pleasure. DNA is full of symmetries. What if we used group structure to probe it? Based on this intuition I decided to map each base in DNA to a specific group generator. The choice of generator took some time. I wanted something of dimension 3 or 4 to naturally fit the DNA bases, which are 4 in number. I figured that if the underlying symmetry of DNA bases was captured in the relationships of group generators to each other, this should show it. Finally I decided that I needed a continuous Lie group to better capture the changes on a manifold. I wanted to represent the DNA sequence as a continuous path on a manifold, where each base moves the path in a direction determined by its group generator, rather than treating the sequence as discrete jumps between states. I decided to use SU(2). The generators of this group are the 3 Pauli matrices, and I added the identity. I was convinced I had something. I didn’t. Let me l...

There is a principle in probability theory that says: given what you know, assume nothing else. State your constraints: probabilities must sum to one, the average of some measurable quantity must equal some observed value, and then choose the distribution that is maximally noncommittal about everything beyond those constraints. This is the maximum entropy principle, and from it falls most of statistical mechanics: the Boltzmann distribution, the partition function, thermodynamics itself. Occam’s razor says prefer the hypothesis that assumes least. The principle of least action says nature takes the path that extremizes action, not the most dramatic path, not an arbitrary one, the one that is in some precise sense most economical. In chemistry, systems settle into minimum energy configurations. In Bayesian inference, maximum entropy priors are the ones that smuggle in the fewest assumptions about what you don’t know. These are different principles, stated in different languages, applied...

If the first phase was curiosity, and the second was learning how to think, the third has been about learning how to work. The change was not dramatic. There was no moment where everything suddenly became easy or clear. The way I approached problems stopped depending on who was guiding me and started becoming something I could reproduce on my own. I began to notice patterns in how I made progress. When something felt confusing, it was usually because I had skipped a definition or was trying to move too quickly. Slowing down helped. Writing things out helped. Returning to first principles helped. These were simple ideas, but applying them consistently made a difference. Over time, this became a method. Not a rigid system, but a way of working that I could rely on. I would start by grounding myself in the basic objects of a subject, understanding what they were and how they behaved. From there, I would work through problems, not just to get answers, but to see how the ideas moved. When s...