What follows here are the (semi-technical) chronicles of the research I've done so far, intermingled with where I would like my work to go.
I started my PhD research by investigating Relative Locality - a phenomenological model of quantum gravity induced modifications to relativistic dynamics of point particles. One of the motivations behind it that I found very enticing is that when we perform experiments, we never actually measure distances, but rather we infer them from energy and momentum of quantum particles we emit and absorb and the timing of these events. This, together with the lesson from lower-dimensional models that gravitational effects modify the standard relativistic dispersion relations, motivate us to study particle dynamics on momentum spaces with non-trivial geometry. In my first work in Relative Locality, I showed that allowing such curvature can have the consequence of breaking global momentum conservation for a single propagating particle. I followed this with a construction of a curved momentum space that preserved full Lorentz symmetry. The non-commutative structure of the result turned out to be related to the discrete Snyder spacetime.
Following my work in phenomenology, I decided to shift my focus to Spin Foam models - a proposal for path integral quantization of General Relativity. The central insight in Spin Foams is that gravity can be described by imposing so-called simplicity constraints onto a topological field theory (BF theory). Using a newly developed spinor representation of spin networks, together with my collaborators, we have discovered a simpler way of imposing these constraints. This allowed us to obtain the first analytical results for behavior of transition amplitudes in 4D quantum gravity under changes of triangulation (more technically, we evaluated the 4-dimensional Pachner moves). This work gives the hope of analytically studying the non-perturbative renormalization of 4D quantum gravity, which is one of the projects I am currently working on.
Another direction I am investigating currently concerns the nature of time in quantum gravitational scenarios. The question that I am curious about is how would one go about operationally defining time intervals in extreme gravity regions. Our current relational definitions that are used in atomic clocks are only valid due to the existence of bound states - these however are not stable in extreme situations.
One of the central questions in neuroscience that interest me is how the neurons in the cerebral cortex self-organize into complex neural circuits capable of intelligent thought. This is especially intriguing due to the vast variability of types of neurons. I would like to study this question at an interface of theoretical models supplemented with simulations and techniques from deep learning, but eventually also experimentally, through studying ensembles of neurons and building artificial networks made from neurons. My goal in this would be to understand the mechanisms behind our learning and creative thinking. This understanding will allow us to build more intelligent AI systems. In the long-term I hope that this would allow us to build brain-machine interfaces capable of enough reading and writing to enable enhancing these abilities. Having the ability to expand our cognitive abilities both through connecting to artificial larger neural networks, as well as to other people, will allow us to tackle problems and challenges that currently are unimaginable or outside of our reach.
I have now spent two summers working at Microsoft Research under Jaron Lanier on mixed reality applications to physics and mathematics research. The first time, I was part of the COMRADRE (Center of Mixed Reality Advanced Development and Research) lab in Redmond, WA. There, we worked on novel applications of mixed reality on hacked together headsets called "Reality Mashers". Apart from smaller applications, like visualizing a hypercube that was superimposed onto a physical cube that a user could rotate, I prototyped an environment for visualizing mathematical equations and performing operations on them using gestures and voice input.
My second internship took place in Mountain View, CA, where I pushed the idea from my previous work further. This time I switched my work to the HoloLens platform and created a multi-user collaborative platform for mathematical research. I explored new ways of interacting with mathematical objects. For example, transformations were represented by virtual magnifying lenses. If a user looks through the lens at a data set, the rendered image is modified by the applied transformation. The interesting aspect of this is that several lenses can be put in sequence, to obtain compositions, or can be reversed to get the inverse operation. This allows multiple users to explore a meta-structure of related mathematical expressions, a task that gets confusing in a 2D medium.
- J. Lanier, V. Mateevitsi, K. Rathinavel, L. Shapira, J. Menke, P. Therien, J. Hudman, G. Speiginer, A. S. Won, A. Banburski, X. B. Palos, J. A. Fernandez, J. P. Lurashi, W. Chang, "The RealityMashers: Augmented Reality Wide Field-of-View Optical See-Through Head Mounted Displays," ISMAR 2016
- A. Banburski, L.Q. Chen, "A simpler way of imposing simplicity constraints," arXiv:1512.05331 [gr-qc], accepted by Physical Review D for publication
- A. Banburski, L.Q. Chen, L. Freidel, J. Hnybida, "Pachner moves in a 4d Riemannian holomorphic Spin Foam model," Physical Review D 92, 124014 (2015), arXiv:1412.8247 [gr-qc]
- A. Banburski, L. Freidel, "Snyder Momentum Space in Relative Locality," Physical Review D 90, 076010 (2014), arXiv:1308.0300 [gr-qc]
- A. Banburski, "Twisting loops and global momentum non-conservation in Relative Locality," Physical Review D 88, 076012 (2013), arXiv:1305.7289 [gr-qc]
- A. Banburski, P. Schuster, "The Production and Discovery of True Muonium in Fixed-Target Experiments," Physical Review D 86, 093007 (2012), arXiv:1206.3961 [hep-ph]