Unfortunately things have gotten more busy so I haven’t been writing on the blog. However, I found a new way to prepare talks that I really like: I type out my notes in a way which I hope is readable to someone who has not been to the talk. It is halfway between an expository paper and lecture notes and the advantage is that I have a product which I can share on my webpage (and I find it more useful than slides — I am starting to dislike slides presentations for seminar talks). Anyway, I’ll advertise some of these here (I’ve never actually posted about my own research on this blog, so it’s a first!):

1. I’ve been working with Laurent Gruson and Jerzy Weyman on finding geometric interpretations for orbits in “Vinberg -representations”. I gave a talk on this at Princeton (notes) and Michigan (notes). The Princeton talk is more introductory in nature, and even though there is overlap, the two sets of notes should complement one another (for the record, both talks were approximately 1 hour, 45 minutes).

2. A separate project with Weyman involves trying to understand Koszul homology for certain classes of determinantal-like ideals. The motivation comes from trying to classify minimal free resolutions over quadric hypersurface rings and in trying to understand a certain result of Koike and Terada in combinatorial representation theory. I’m giving a talk on this tomorrow at Michigan (notes).

It’s a bit time-consuming, but I think preparing notes like this can be very useful, especially for projects which haven’t been finished yet (it helps me gain direction). I hope more people try it!

-Steven

Yan and I recently put a paper on the arXiv that enumerates the graded (3+1)-avoiding posets.  In this post, I kill the adjective “graded” and talk a bit about what (3+1)-avoiding posets are and why they’re interesting.  If you don’t know what a poset is, I’ve included the definition in Note 0 at the bottom of the post.

As with any object as general as posets, we are mostly interested not in results about all posets, but rather in finding particular families of posets with interesting or unexpected properties.  One such family of posets are the (3+1)-avoiding posets.  These are the posets that do not contain four elements, say a, b, c, and d, such that a < b < c and d is incomparable to the other three.  A short digression to explain the name “(3+1)-avoiding”: one natural class of posets are the chains, finite total orders like the first example in the previous paragraph.  A natural name for the chain with vertices is n, so the chain with three vertices is 3 and the chain with one vertex (the only poset with exactly one vertex) is 1.  There are several natural operations that we can use to combine posets, including the disjoint union, denoted “+”.  Thus, 3+1 is the poset that you get if you add a single isolated vertex to a three-element chain, and a poset is (3+1)-avoiding if it has no four elements that induce the poset 3+1.

I gave a talk at the MIT applied math seminar recently. Some people wanted a blog post, so here it is.

Some of my mathematical hobbies include probability and machine learning. I have recently realized that many simple but effective ideas of these fields really all come from one thing: an independence assumption. It was only until I saw the same example in several different guises, however, before I really caught on. As something Occam would surely approve of, the extreme naiveté this approach embraces can actually go a long way. Our key player is simple: we say that and are conditionally independent given if . This can be written in the more symmetric form . Now I will tell a few stories. Most of these should be old for a specialist, but I hope I’ve included some remarks that even they may appreciate.

Some of my research last year had to do with notions of “positivity” in Heegaard Floer homology and knot theory (a phrase which I’m borrowing from a paper of Matt Hedden). As a simple example of positivity, a braid (and more generally, a knot) is called positive if it has a diagram containing only positive crossings:

I’ve been learning some physics from Allan Adams and some physics students here at MIT, and I’ve suddenly realized that there are a lot of contextual tricks I take for granted when I’m doing physics, which make the physicists’ lives easier and sometimes really irritates the mathematicians. The first two examples I can think of is differentiating under the integral and never checking convergence, though both of these really fall under the bigger umbrella of assuming everything is well-behaved (which probably accounts for 80% of the mathematical gripes I’ve seen against physicists). Now, I’m a happy supporter of this “wishful thinking” practice: to use a programming analogy, I think of this habit as the lazy evaluation version of having good definitions, and as a lover of Python generators I totally appreciate the idea of saying “we’ll figure out the right definitions later since they actually exist.”

One of the most common physical tricks, however, is not of this category. It is the curiously natural framework: “we have a consistent idea of units.” Here’s a perfectly sound argument to get something that is not entirely obvious:

One of the seven Millennium Prize Problems stated by the Clay Mathematics Institute is the P vs. NP problem, which asks: is P = NP? What are P and NP exactly? In this post, I’ll go over some basic computational complexity classes: P, NP, co-NP, PSPACE, and known relationships between them, as well as intuition behind each class, without getting too technical. After reading, you should understand what the P vs. NP problem and you should get an idea of how the basic complexity classes are interrelated.

This is one of those things that convinced me that commutative algebra was cool and actually useful for solving combinatorial problems. The Frobenius problem (commonly known as the coin problem, of which a special case is the McNugget problem) asks: given positive integers with , what is the largest integer that cannot be written in the form  for nonnegative integers ? For demonstrative purposes, I’ll show you how you can use commutative algebra to solve the case , but the principle generalizes to larger . At the end, I’ll speculate on how one might use this to solve higher-dimensional versions of the Frobenius problem, whatever that might mean.

Probably every mathematician is familiar with Cantor’s diagonal argument for proving that there are uncountably many real numbers, but less well-known is the proof of the existence of an undecidable problem in computer science, which also uses Cantor’s diagonal argument. I thought it was really cool when I first learned it last year.

Hi everyone! I’m Alan, and I hope to contribute posts on connections between math and computer science. I majored in math and minored in CS as an undergrad, and will be a first year grad student in the computer science department at MIT in the fall. My interests on the math side include combinatorics, combinatorial games, and commutative algebra. On the computer science side, I’m interested in algorithms, complexity theory, quantum computing, and artificial intelligence. The underlying theme behind my interests is the application of mathematical knowledge to do things more efficiently or prove that they’re hard to do, and as such my posts will have particular emphasis on this theme.

My underdeveloped academic homepage can be found here.

In my spare time, I devoutly follow hedonism by enjoying food, music, movies and video games.

This post is just a public service announcement: the function RandomTableau[p] in Mathematica is meant to give a random standard Young tableau of shape p. Though the documentation isn’t actually so helpful as to tell you explicitly what probability distribution it is allegedly generating, the only natural option is uniform over all tableaux of that shape. However, the function RandomTableau[p] does not generate tableaux uniformly. The rest of this post quickly outlines the evidence for this claim.