We are now able to solve many partial differential equation problems that were well beyond reach when I started in academia. Some of this success is due to computer hardware but much is due to algorithmic advances. 

I will give a personal perspective of the development of computational methodology in this area over my career thus far. 

Large Language Models (LLMs) have demonstrated impressive achievements. However, recent research has shown that their deeper layers often contribute minimally, with effectiveness diminishing as layer depth increases. This pattern presents significant opportunities for model compression. 

In the first part of this seminar, we will explore how this phenomenon can be harnessed to improve the efficiency of LLM compression and parameter-efficient fine-tuning. Despite these opportunities, the underutilization of deeper layers leads to inefficiencies, wasting resources that could be better used to enhance model performance. 

The second part of the talk will address the root cause of this ineffectiveness in deeper layers and propose a solution. We identify the issue as stemming from the prevalent use of Pre-Layer Normalization (Pre-LN) and introduce Mix-Layer Normalization (Mix-LN) with combined Pre-LN and Post-LN as a new approach to mitigate this training deficiency.

Kinetic and mean-field equations are central to understanding complex systems across fields such as classical physics, engineering, and the socio-economic sciences. Efficiently solving these equations remains a significant challenge due to their high dimensionality and the need to preserve key structural properties of the models. 

In this talk, we will focus on recent advancements in deterministic numerical methods, which provide an alternative to particle-based approaches (such as Monte Carlo or particle-in-cell methods) by avoiding stochastic fluctuations and offering higher accuracy. We will discuss strategies for designing these methods to reduce computational complexity while preserving fundamental physical properties and maintaining efficiency in stiff regimes. 
Special attention will be given to the role of these methods in addressing multi-scale problems in rarefied gas dynamics and plasma physics. Time permitting, we will also touch on emerging techniques for uncertainty quantification in these systems.

We discuss the trace problem and stable rank for tracialy complete C*-algebras

In the category of operator systems, identification comes via complete order isomorphisms, and so an operator system can be identified with a C*-algebra without itself being an algebra. So, when is an operator system a C*-algebra? This question has floated around the community for some time. From Choi and Effros, we know that injectivity is sufficient, but certainly not necessary outside of the finite-dimensional setting. In this talk, I will give a characterization in the separable nuclear setting coming from C*-encoding systems. This comes from joint work with Galke, van Lujik, and Stottmeister.