Georgia Institute of Technology College of Sciences

Mean-field control with common noise and filtering problems naturally lead to second-order PDEs on Wasserstein space. In this talk, we analyze a class of such equations in which the second-order operator is finite-dimensional in nature. We establish comparison principles and apply them to obtain particle convergence rates in mean-field control. The talk is based on joint work with Erhan Bayraktar, Ibrahim Ekren, and Xihao He.

Artificial intelligence (AI) has become a transformative force in scientific discovery---known as AI for Science---with profound impact on computational molecular design, as highlighted by the 2024 Nobel Prize in Chemistry. Due to their remarkable capability in analyzing complex structures, message-passing neural networks and diffusion- and flow-based generative models stand out as effective tools for molecular property prediction and structure generation. However, message-passing neural networks struggle to efficiently integrate multiscale molecular features and complex 3D geometry for accurate property prediction, and (2) the generative processes of generative models are often computationally intensive and error-prone. 

In this talk, I will present our recent advances toward overcoming these limitations: (1) multiscale graph representations and message-passing architectures for efficient and accurate molecular learning, and (2) one-step flow-based generative models that enable high-fidelity molecule generation with dramatically reduced computational cost.

Suppose $f$ is a univariate polynomial with integer coefficients of absolute value at most $H$, exactly $t$ monomial terms, and degree $d$. Suppose also that $p,q$ are integers of absolute value at most $H$. Then one can determine the sign of $f(p/q)$ in time $(d^3 \log H)^{2+\epsilon}$, by combining work of Liouville from about 170 years ago, work of Mahler from about 60 years ago, work of Neff, Reif, and Pan from about 30 years ago, and more recent refinements.

However, when $t$ is small, one can do much better: When $t=2$, one can determine the sign of $f(p/q)$ in time $\log^5(dH)$, via work of Baker from about 55 years ago. Koiran asked, around 2016, about the $t=3$ case, after proving new bounds on the minimal separation of complex roots of univariate trinomials.

We make progress on Koiran's question by giving a new, dramatically faster algorithm for solving univariate trinomials over the real numbers. The key innovation is a new family of non-hypergeometric series for the roots of $f$ when $f$ is close to having a degenerate root. This is joint work with Emma Boniface and Weixun Deng.

We characterize global minimizers below the so-called first critical field of the inhomogeneous version of the Ginzburg-Landau energy functional in a three-dimensional setting. Minimizers of this functional describe the behavior of type-II superconductors exposed to an external magnetic field, which is characterized by the presence of codimension 2 singularities called vortices where superconductivity is locally suppressed. We will talk about how to adapt the results from the standard Ginzburg-Landau theory into an inhomogeneous framework and present results from a recent work in collaboration with Carlos Roman (Pontificia Universidad Catolica de Chile).