# Research

### Algebra ⟶ non-linear statistics

I build statistical models for discrete relational data that capture more complex behavior than traditional models, study them through the algebrogeometric lens, prove their interpretability in practice, and develop scalable model/data fit testing methodologies using a blend of combinatorial, algebraic, probabilistic, and Bayesian algorithms.

### Randomness ⟶ non-linear algebra

I study randomized algorithm approaches to computational algebra problems whose expected runtimes are much lower then the well-known worst-case complexity bounds, develop probabilistic models to study average and extreme behavior of algebraic objects, and use machine learning to predict and improve behavior of algebraic computations.

### Algebraic statistics for network models

Non-asymptotic goodness-of-fit tests based on Markov bases;

Existence and complexity of MLE;

Dynamic combinatorially-inspired data-oriented algorithms for model fitting;

Application to (large) sparse network data;

Time-evolving networks.

### Sponsors:

**DARPA** FA9550-12-1-0392 (2012-2013), **AFOSR** FA9550-14-1-0141 (2014-2017).

### Randomness and learning for non-linear algebra

Stochastic non-linear algebra;

Solving systems of multivariate equations;

Fast randomized algorithms for large structured systems;

Randomized structures in algebra with applications;

Machine learning for computational algebra.

### Sponsors:

**NSF** DMS-1522662 (2015-2019) joint with UCDavis.

**Simons** **Foundation**, through a Collaboration Grant for Mathematicians 854770 (2021-2026).

### Statistical models in human sciences

Computational challenges in occupational health psychology:

using statistics to model dynamic worker well-being.

### Sponsors:

2018 **CISC** seed grant with Mahima Saxena, formerly with IIT Psychology, and Lulu Kang, IIT Applied Math.

Curious? Check out research summary slides from April 2017.