research
The general aim of my research was to understand the fundamental structure of spacetime and, yes, this necessarily includes all the matter one can put there. Yikes! This is very, very ambitious and requires more insight than not only one human but an entire generation of researchers can possibly have. Yet science has always been a civilizational project, and that's what all problems thus far have been like. It's not unlike ants digging a long tunnel, but this one does seem to go on a bit longer than most.
I started out very interested in formulations of quantum field theory, like AQFT and (pre)factorization algebras. All of this tends to involve a very simple notion of putting one thing on top of another. And often, we want to describe something locally and then glue it together into one big thing... this leads one to descent theory, a mathematical subject due to the great(est?) 20th century mathematician Grothendieck. But this is all very abstract. Besides, what if perturbation theory isn't needed at all?
Later study, indeed, led me deep into geometry in the concrete, and my work led me to Representation theory, very broadly, and in particular Supersymmetric QFTs (with 8 supercharges) and their link to Integrable systems and Isomonodromic deformations. Supersymmetric theories engineered by String theory provide us with theories which are not only consistent, but also exactly computable. This is truly amazing - usually we might only think of the harmonic oscillator or the Kepler problem as "computable" - but in this case the perturbative series terminate and the nonperturbative sectors contain all the content of the theory. In turn, they are computable by equivariant localization, once we equip our space with a toric action. All in all, we get some wild links.
For instance, my research was focused on the so-called Painlevé transcendents. In a sense, these functions are to elementary functions what irrationals are to the rational numbers, and in general they arise from the very sensible requirement of a) them solving a differential equation and b) not having movable critical points. That is, a moveable (ie, dependent on the initial conditions) pole is a-ok, but a branch cut is not. Think about it. If you're solving a differential equation by using analytical continuation, you go along some path and reconstruct the solution. If there are branch cuts which you know beforehand, you take care about crossing them. But if a branch cut develops due to your starting point, you can potentially cross it without ever knowing it, and end up somewhere else on the universal cover.
These functions are fundamental to physics, and they're already starting to pop up more and more. This trend is clearly unstoppable.
In my work, I found how certain "twists" on these exactly computable susy partition functions solve these equations! There are M2 brane defects in a 6d space with splits and exhibits a 4d-2d duality involved. The deep reason is quite complicated, and there's no need to go into it. Besides this, I've been looking into ways to use an exact version of the Geometric transition of Ooguri-Vafa fame to compute the same functions, which uses 3d supersymmetric theories which describe systems of M2 branes.
And these are all very simple problems, at heart. As I said, putting one thing on top of another (matter on top of space). What exactly is this matter, what is space, how it interacts and how you can do all of the above coherently - this is the main issue. Exact calculations enable us to go far, but these are still not realistic models. For one, there is no adjoint matter in the "real world". There are two ways out: one is to deform these peculiar theories away from their special, highly symmetric formulation. Another is to say: we've seen some beautiful connections in geometry, and all the effort was worth it. I'm not sure if the last one is a "way out". But science is a human affair - finding Beauty, Truth and Goodness makes us all better off.
Besides, I have this deep conviction that all of Nature can be described in a few simple ideas. And history teaches us that things get unified once we learn how to speak its language. Most of this language can, at least this far, be described using (higher) category theory, and all of that can be formulated in various String theories. Perhaps that's not it, at all, perhaps the way are automata? Perhaps the two are linked?
A while ago, my own intellectual mandibles (forget not the ants!) were happily chipping away at more specialized subjects: noncommutative geometry and quantum groups. I was also looking into why some classical, nontrivially constrained systems describing a relativistic spinning top (which is a picturesque model for electron spin) yield spacetime noncommutativity when quantized. Honestly, it might even be expected from basic, introductory NRQM that it might not be that simple to treat spin as an "arrow". The matter is of course more subtle, since a gauge symmetry eliminates both the spin vector and its conjugate momenta as observables and leaves their antisymmetrized product as an observable, which to first order also corresponds to the generator of Lorentz symmetries. It really makes my head spin, at least.
On the other hand, there is the problem of putting spacetime noncommutativity in by hand, which becomes relevant at high energy scales. Well, by itself it's no problem, but the consequences need to be worked out. One immediate result is a Hopf algebroid structure on the algebra of functions, which carries a lot of information about spacetime. Well, the phase space is actually a twisted Hopf algebroid, the twist itself being a special operator which determines everything from multipication to addition of momenta to particle statistics. To keep myself from rambling, I will just say that there are a lot of interesting physical and mathematical problems - the two are really "forced" to hold hands in this particular image of spacetime!
Some of my other interests are jazz (I play sax, guitar and a bit of piano), philosophy, weightlifting, and late night conversations.
The general aim of my research was to understand the fundamental structure of spacetime and, yes, this necessarily includes all the matter one can put there. Yikes! This is very, very ambitious and requires more insight than not only one human but an entire generation of researchers can possibly have. Yet science has always been a civilizational project, and that's what all problems thus far have been like. It's not unlike ants digging a long tunnel, but this one does seem to go on a bit longer than most.
I started out very interested in formulations of quantum field theory, like AQFT and (pre)factorization algebras. All of this tends to involve a very simple notion of putting one thing on top of another. And often, we want to describe something locally and then glue it together into one big thing... this leads one to descent theory, a mathematical subject due to the great(est?) 20th century mathematician Grothendieck. But this is all very abstract. Besides, what if perturbation theory isn't needed at all?
Later study, indeed, led me deep into geometry in the concrete, and my work led me to Representation theory, very broadly, and in particular Supersymmetric QFTs (with 8 supercharges) and their link to Integrable systems and Isomonodromic deformations. Supersymmetric theories engineered by String theory provide us with theories which are not only consistent, but also exactly computable. This is truly amazing - usually we might only think of the harmonic oscillator or the Kepler problem as "computable" - but in this case the perturbative series terminate and the nonperturbative sectors contain all the content of the theory. In turn, they are computable by equivariant localization, once we equip our space with a toric action. All in all, we get some wild links.
For instance, my research was focused on the so-called Painlevé transcendents. In a sense, these functions are to elementary functions what irrationals are to the rational numbers, and in general they arise from the very sensible requirement of a) them solving a differential equation and b) not having movable critical points. That is, a moveable (ie, dependent on the initial conditions) pole is a-ok, but a branch cut is not. Think about it. If you're solving a differential equation by using analytical continuation, you go along some path and reconstruct the solution. If there are branch cuts which you know beforehand, you take care about crossing them. But if a branch cut develops due to your starting point, you can potentially cross it without ever knowing it, and end up somewhere else on the universal cover.
These functions are fundamental to physics, and they're already starting to pop up more and more. This trend is clearly unstoppable.
In my work, I found how certain "twists" on these exactly computable susy partition functions solve these equations! There are M2 brane defects in a 6d space with splits and exhibits a 4d-2d duality involved. The deep reason is quite complicated, and there's no need to go into it. Besides this, I've been looking into ways to use an exact version of the Geometric transition of Ooguri-Vafa fame to compute the same functions, which uses 3d supersymmetric theories which describe systems of M2 branes.
And these are all very simple problems, at heart. As I said, putting one thing on top of another (matter on top of space). What exactly is this matter, what is space, how it interacts and how you can do all of the above coherently - this is the main issue. Exact calculations enable us to go far, but these are still not realistic models. For one, there is no adjoint matter in the "real world". There are two ways out: one is to deform these peculiar theories away from their special, highly symmetric formulation. Another is to say: we've seen some beautiful connections in geometry, and all the effort was worth it. I'm not sure if the last one is a "way out". But science is a human affair - finding Beauty, Truth and Goodness makes us all better off.
Besides, I have this deep conviction that all of Nature can be described in a few simple ideas. And history teaches us that things get unified once we learn how to speak its language. Most of this language can, at least this far, be described using (higher) category theory, and all of that can be formulated in various String theories. Perhaps that's not it, at all, perhaps the way are automata? Perhaps the two are linked?
A while ago, my own intellectual mandibles (forget not the ants!) were happily chipping away at more specialized subjects: noncommutative geometry and quantum groups. I was also looking into why some classical, nontrivially constrained systems describing a relativistic spinning top (which is a picturesque model for electron spin) yield spacetime noncommutativity when quantized. Honestly, it might even be expected from basic, introductory NRQM that it might not be that simple to treat spin as an "arrow". The matter is of course more subtle, since a gauge symmetry eliminates both the spin vector and its conjugate momenta as observables and leaves their antisymmetrized product as an observable, which to first order also corresponds to the generator of Lorentz symmetries. It really makes my head spin, at least.
On the other hand, there is the problem of putting spacetime noncommutativity in by hand, which becomes relevant at high energy scales. Well, by itself it's no problem, but the consequences need to be worked out. One immediate result is a Hopf algebroid structure on the algebra of functions, which carries a lot of information about spacetime. Well, the phase space is actually a twisted Hopf algebroid, the twist itself being a special operator which determines everything from multipication to addition of momenta to particle statistics. To keep myself from rambling, I will just say that there are a lot of interesting physical and mathematical problems - the two are really "forced" to hold hands in this particular image of spacetime!
Some of my other interests are jazz (I play sax, guitar and a bit of piano), philosophy, weightlifting, and late night conversations.