Research Interests
- Mathematical Physics: Schramm-Loewner evolutions, Statistical Mechanics;
- Random Matrix Theory;
- Study of the properties of Random Riemann Zeta functions, Random Dirichlet L-functions, classes of Random L-functions,- probabilistic analogues of (generalized) Riemann Hypothesis type problems.
- Rough Path Theory;
- (Biased) Random walks on random graphs and their scaling limits;
- Interactions with Machine Learning and Data Science;
News: For the [
REU 2024] program I will be joined by a student from Georgia Tech and one from Columbia University in the city of New York. Thank you everybody for your applications and interest in the project! I hope we connect in the future.
Research Snapshot:
1)[Update: 6th of October 2024] Modern Tools in the context of the Riemann Hypothesis type problems and study of the structure of the primes
A recent direction of research consists of the study of Random Riemann Zeta functions, Random Dirichlet L-functions, classes of random L-functions, and their properties.
Potential
probabilistic analogues of the Riemann hypothesis type problems in this context involves the study of existence of the set of zeros/number of zeros in a region of the Random Riemann Zeta functions for a given model of pseudo-primes. In contrast, one can study the zero-free regions of the Random Riemann Zeta functions for various models of pseudo-primes, and the interactions of their complements with the famous Re s=1/2 line. Similarly, for the random Dirichlet L- functions and multiplicative functions.
Given that these studies are based on probabilistic models of primes, this gives a link also towards the use of modern tools of Machine Learning in the study of these problems. As we are experiencing significant growth in these areas, we expect that these tools will help us in this study. In addition, the accuracy of these psuedo-primes models can be tested against the predictions on the zero-free regions of the Riemann Zeta function or other known estimates for the Riemann Zeta function. Similarly, for Dirichlet L- functions and multiplicative functions.
For more details, see paper (16).
2) Random Matrices and Schramm-Loewner Evolutions
My research journey started with two projects in Random Matrix Theory (RMT). RMT has applications ranging from Physics, Biology to Neural Networks and it is an area of active research!
Later, I moved to a different area and started studying Schramm-Loewner Evolutions Theory (SLE), another great and active area of research, which I studied with techniques from Rough Path Theory, with Stochastic Analysis tools, and recently (see paper (11) below) with deterministic tools.
Although both SLE and RMT have been thriving areas of Probability Theory, these two fields have developed relatively independently.
We recently introduced a toolbox that connects the two areas and provides many potential new research directions and a platform for using techniques of one field to the other.
The toolbox consists of seeing the right-hand side of the multiple SLE dynamics with Dyson Brownian motion driver as a normalized trace of a resolvent of a certain Random Matrix. This goes first by seeing the right-hand side of the multiple Loewner equation as a Stieltjes transform using the empirical measure on the drivers and then using the relation between this transform and the normalized trace of the resolvent of a Random Matrix. For more details see paper (12). See paper (8) also.
One of the advantages of this is that the hard critical parameters κ=4 and κ=8 in the SLE theory correspond to the β=2 and β=1 which are some of the nice well-studied cases in Random Matrix Theory.
3) Statistical Mechanics, Long-Range Models : In a different direction, I explore models from Statistical Mechanics, in particular the ones with long-range interaction. For more details, see papers (9) and (14). In a different direction, see also the recent work in paper (15).
Background
Tenure-Track Assistant Professor at the University of North Carolina at Charlotte (2023-)
Visiting Assistant Professor (via Burnett Meyer Fellowship) at the University of Colorado at Boulder (2022-2023)
Postdoctoral Fellow at NYU Shanghai (2019-2022)
DPhil (PhD) student in Mathematics, University of Oxford (2015-2019), under the supervision of Prof. Dmitry Belyaev and Prof. Terry Lyons.
MSc in Mathematics, ETH Zürich (between 2013-2015), under the supervision of
Prof. Antti Knowles.
BSc in Mathematics, University of Bucharest (between 2010-2013), under the supervision of Prof. Victor Vuletescu.