Georgia Institute of Technology College of Sciences

Complex fluids are abundant in our daily life. Unlike traditional solids, liquids and the diluted solutions, the model equations for complex fluids continue to evolve with the new experimental evidences and emerging applications. Most of these important properties are due to the coupling and competition between effects from different scales or even from different physical origins/principles. The energetic variational approaches (EnVarA), motivated by the seminal works of Onsager and Rayleigh, are designed to study such systems. In this talk, I will discuss several complex fluid systems, and the associated mathematical issues.

The theory of $p$-adic differential equations first rose to prominence after Dwork used them to prove the rationality of zeta functions of a positive characteristic variety in 1960. Since then, there has been growing interest in the category of convergent $F$-isocrystals and the subcategory of overconvergent $F$-isocrystals due to this subcategory having good cohomology theory with finiteness properties. Recent work by Grubb, Kedlaya and Upton examines when a convergent $F$-isocrystal is overconvergent by restricting to smooth curves on the scheme under a mild tameness assumption (measured by the Swan conductor). In my talk, I will introduce the above categories and talk about work in progress about bounding the Swan conductor of an overconvergent $F$-isocrystal in terms of data associated with the corresponding convergent $F$-isocrystal.

Given a simple graph on $n$ vertices and a parameter $k$, the triangle-densest-$k$-subgraph problem is known to be computationally hard in the worst case. To circumvent the computational hardness, we study an average-case model where a triangle-dense subgraph on $k$ vertices is planted in an Erd\H{o}s--R\'enyi random graph on $n$ vertices. For the recovery of the planted subgraph, we propose a simple spectral algorithm and a semidefinite program, both of which use a graph matrix whose entries are local signed triangle counts. Theoretical guarantees for these algorithms are established through spectral analysis of the graph matrix. Finally, we provide evidence showing a statistical-to-computational gap analogous to that for the planted clique problem. The computational threshold in terms of the subgraph size $k$ is at least $\sqrt{n}$ in the framework of low-degree polynomial algorithms, while the information-theoretic threshold is at most logarithmic in $n$. Joint work with Cheng Mao and Benjamin McKenna.

Spin glasses are disordered statistical physics system with both ferromagnetic and anti-ferromagnetic spin interactions. The Gibbs measure belongs to the exponential family with parameters, such as inverse temperature $\beta>0$ and external field $h\in R$.  A fundamental statistical problem is to estimate the system parameters from a single sample of the ground truth. In 2007, Chatterjee first proved that under reasonable conditions, for spin glass models with $h=0$, the maximum pseudo-likelihood estimator for $\beta$ is $\sqrt{N}$-consistent. This is in contrast to the existing estimation results for classical non-disordered models. However, Chatterjee's approach has been restricted to the single parameter estimation setting.  The joint parameter estimation of $(\beta,h)$ for spin glasses has remained open since then. In this talk, I will introduce a new idea to show that under some easily verifiable conditions,  the bi-variate maximum pseudo-likelihood estimator is jointly $\sqrt{N}$-consistent for a large collection of spin glasses, including the Sherrington-Kirkpatrick model and its diluted variants. Based on joint work with Wei-Kuo Chen, Arnab Sen.