My research interests and experiences lie in the areas of Geometry, Mathematical Physics and Statistical Learning. 

 

Geometry: 

My interests to Degenerate Complex Monge Amp`ere equations (DCMA) has its roots in my PhD thesis where I had studied a class of singular complex manifolds arising from singularities of Ricci flat metrics.

 

A closely related long standing open problem in the area of complex geometry is the characterization of the behavior of solutions to DCMAs. This highly important problem had been remained unsolved since the pioneering work by S-T Yau.

I have recently resolved it: https://arxiv.org/abs/2302.14106.

 

Statistical Learning

One of the open problems in the area of variational Bayesian approximation consists of uncertainty quantification.

The method named Mean Field Variational Bayesian Approximation (MFVBA) is now considered as one of the most  effective approachess to approximate the posterior probability distribution in many Bayesian inference problems like Gaussian mixture models (GMM). The method suffers meanwhile from lack of mathematically rigorous tools to study the errors and to quantify their corresponding uncertainty.

In a recent work with my PhD student we have completely resolved this problem for the case of Gaussain Mxture Model. The method combines recent developments of the theory of optilmal transport and  technics from statistical mechanics,

 

Some More interests: I am interested in the interaction  between deep learning,  geometry  topology and statistical physics and their application for understanding and optimizing the methods in deep learning.