Mukul Rai Choudhuri

My research interests can be grouped as follows: 

Kakeya conjecture, geometric measure theory/fractal geometry, incidence geometry, Euclidean harmonic analysis:

Most of my research and background so far is regarding the Kakeya conjecture. This is a problem which lies at the interface of geometric measure theory and incidence geometry, and it has deep connections to harmonic analysis. I have a broad interest in these three interconnected areas extending beyond Kakeya. However the Kakeya conjecture is my favorite problem because of its elegant statement, multi-faceted flavour and rich history. Research in Kakeya is a unique blend of analysis, combinatorics, geometry with algebra (polynomial method) in the picture as well.

A Kakeya set is a set which has a unit line segment in every direction. Besicovitch showed one could have Lebesgue measure zero Kakeya sets but perhaps they must be big in some other sense? The Kakeya conjecture states that Kakeya sets in R^n must have Hausdorff dimension n (which is the maximum possible). Hausdorff dimension is a notion of size from geometric measure theory that is used to quantitatively study fractals. Thus, the Kakeya conjecture could be considered a problem in geometric measure theory. 

On the other hand, the Kakeya conjecture has a nice, simple, combinatorial formulation regarding how long skinny tubes overlap (which sounds like a incidence geometry problem). One can get nice partial dimension bounds like (n+2)/2 using geometric combinatorial arguments like Wolff's hairbrush argument (in which you actually construct an object which looks like a hairbrush!). Thus, the Kakeya conjecture could be considered a problem in incidence geometry.

The Kakeya conjecture has deep connections to Euclidean harmonic analysis as was discovered for the first time in 1971 by Fefferman. He gave a counter example to the ball multiplier conjecture using Kakeya sets with arbitrarily small area. This was a question about Fourier convergence that was true for n=1 and everyone expected it to be true for n>1 and it shocked the harmonic analysis community that it wasn't. It was later discovered that there was a tower of important conjectures in harmonic analysis (restriction conjecture, Bochner-Riesz, local smoothing for wave equation) which rested on the Kakeya conjecture. Very roughly speaking this is because the wave packets coming from Fourier operators tend to live on Kakeya type tube configurations and if these configurations compress too much, the wave packets would stack up too much and this would cause the operator to be unbounded. Also, the Bourgain-Demeter decoupling theorem utilized Multilinear Kakeya in its proof.

Very recently, Wang and Zahl proved the Kakeya conjecture in 3 dimensions in a major breakthrough! They used the idea of stickiness, which is a property that states when you zoom in you have (roughly speaking) the set you started with in the first place. So stickiness approximates the regularity you get in neat, nice fractals. This property was introduced by Katz-Łaba-Tao in 1999 and it took 20 years for this idea to fully blossom and resolve the 3 dimensional case (with some additional tools and ingredients, especially from recent advances in geometric measure theory like the radial projection result of Orponen-Shmerkin-Wang).

Another major storyline is the connection of Kakeya sets to algebraic geometry and differential geometry. Dvir gave a postcard 2-page proof of the finite field version of the Kakeya conjecture using polynomials in 2008. People were shocked at the simplicity of the proof and the unforeseen usage of polynomials. Inspired by this, Larry Guth, with his strong background in geometry and topology, was able to make progress on the Kakeya conjecture using polynomials (in collaboration with Joshua Zahl, Nets Katz etc.). Things were much more technically challenging while applying the polynomial method in the Euclidean setting, requiring theory from real algebraic geometry etc. Guth and Katz also proved Erdos's distinct distances problem (which is actually an incidence geometry problem in disguise) using the polynomial method in a highly celebrated result.

There is a lot of energy and excitement about solving Kakeya in higher dimensions now and it will probably involve bringing together all the math the problem has collected over its lifespan in a grand synthesis and I look forward to seeing that happen in the coming years!

Additive/arithmetic combinatorics, analytic number theory, (geometric) Ramsey theory, discrete harmonic analysis:

I am excited to be learning and doing research in these areas which are well represented at UGA and a bit new for me. Additive/arithmetic combinatorics concerns results about additive structure in integers/other additive groups. Important classical results include:

• Roth's theorem on arithmetic progressions: this states that if you have a subset of the natural numbers with positive upper density, then it contains a 3-term AP. Here is how the proof goes: suppose you have a set A without any 3 term APs, then you can show the indicator function of A has a large Fourier coefficient. That is, the exponential sum supported on the set A has high value. If A were randomly distributed, this could not happen. So, A must be skewed towards some arithmetic subprogression (this is called a Fourier bias), and we can pass to a A' with a higher density which is still 3 AP free. We iterate this density increment process until we reach our goal (this philosophy of achieving non trivial goals by incremental progress is some ubiquitous mathematical thing like induction on scale in Kakeya, or the rising sea metaphor).
• Green-Tao theorem: primes have arbitrarily long APs
• Szemeredi's theorem: this is a generalization of Roth's theorem. It says if you have "enough" density on the integers you have arbitrarily long APs (Roth's theorem only said 3 term APs). This result has 3 different proofs: Szemerdi's original combinatorial proof, Furstenberg's ergodic theory proof, Gower's Fourier analysis/functional analysis proof. Thus, Terry Tao calls this a Rosetta stone for connecting different branches of mathematics. 
• Vinogradov's theorem: any odd number can be written as a sum of three primes. This involves using a powerful technique called circle method and is taking us closer to (analytic) number theory and names like Ramanujan, Hardy, Littlewood, Hua, Vaughan, Goldbach.
• Ramsey theory: these statements usually go like "if you have enough density/quantity, then you must have some pattern/structure". The setting could be discrete (like Szemeredi's theorem) or continuous (this brings in geometric measure theory), there are intersting results in both. It gets extra interesting when there is also some geometry involved like Bourgain's theorem about simplices.
• Frieman's theorem: if A+A is small, then A has additive structure/resembles an AP.
• Sum product theorem: If A+A is small then A.A is big and vice versa, both cannot be simultaneously small. APs and GPs are good examples to get a feel for these statements.

The last of these topics is connected to the Kakeya conjecture, as was discovered by Bourgain. Kakeya and Bourgain are everywhere! 

These topics have strong connections to harmonic analysis (Euclidean and discrete). Discrete harmonic analysis is a nice, more combinatorial setting to do harmonic analysis and things can be cleaner than the often more technical Euclidean setting. Combinatorics and harmonic analysis are recurrent themes in my research one could say.