Limited Term Assistant Professor Research Research Areas: Additive Combinatorics Analysis Research Interests: 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. 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 set in R^n has Hausdorff dimension n. 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!). Furthermore, this conjecture is deeply connected to Euclidean harmonic analysis which adds to its importance (Fefferman's ball multiplier counter example, Stein's restriction conjecture, Schrodinger's equation, Bochner Reisz conjecture, Local smoothing for wave equation etc.). 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). 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 (things were much more technically and theoretically challenging than the finite field case however). 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 pesky defiant little volleyball 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 new for me. Additive/arithmetic combinatorics concerns results about additive structure in integers/other additive groups. Important classical results include: Green-Tao theorem: primes have arbitrarily long APs Szemeredi's theorem: if you have "enough" density on the integers you have arbitrarily long APs. This result has 3 proofs: Szemerdi's original combinatorial proof, Furstenberg's ergodic theory proof, Gower's Fourier analysis/functional analysis proof. Terry Tao calls this is a Rosetta stone for connecting different branches of mathematics. Also shoutout to Roth's density increment proof for 3 APs (this philosophy of achieving non trivial goals by incremental progress is some ubiquitous mathematical thing like induction on scales in Kakeya). Vinogradov's theorem: any odd number can be written as a sum of three primes. This is taking us closer to (analytic) number theory and names like Ramanujan, Hardy, Littlewood, 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 or continuous, there are intersting results in both. It get 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 amongst other things. For preprints/papers so far, here is my arxiv page: https://arxiv.org/search/math?searchtype=author&query=Choudhuri,+M+R Here is my PhD thesis from UBC: https://open.library.ubc.ca/soa/cIRcle/collections/ubctheses/24/items/1.0449434 I'm also curious about algebraic combinatorics, especially when there is some geometry involved. My background in these topics are very poor, but you only live once
