Georgia Institute of Technology College of Sciences

The 2025 Atlanta Science Festival Mathapalooza! will take place in an art gallery at the opening of an exhibition, The Art of Math.  The show will begin with magic by Matt Baker, followed by hands-on art construction, circus acts, "Math Court" skits, and 4-dimensional dance, before ending with a performance by Tracy Woodard of Bach's Crab Canon, with a discussion of its symmetry algebra.  Find out more and get tickets at https://atlantasciencefestival.org/events-2025/1113-mathapalooza-at-the-gallery/

Recent advances in deep learning have introduced numerous innovative frameworks for solving high-dimensional mean-field games (MFGs). However, these methods are often limited to solving single-instance MFGs and require extensive computational time for each instance, presenting challenges for practical applications.

In this talk, I will present our recent work on a novel framework for learning the MFG solution operator. Our model takes MFG instances as input and directly outputs their solutions in a single forward pass, significantly improving computational efficiency. Our method offers two key advantages: (1) it is discretization-free, making it particularly effective for high-dimensional MFGs, and (2) it can be trained without requiring supervised labels, thereby reducing the computational burden of preparing training datasets common in existing operator learning methods. If time permits, I will also explore connections between this framework and in-context learning, highlighting its broader implications and potential for further advancements.

Modern Artificial Intelligence (AI) systems, such as ChatGPT, rely on artificial neural networks (ANNs), which are historically inspired by the human brain. Despite this inspiration, the similarity between ANNs and biological neural networks is largely superficial. For instance, the foundational McCulloch-Pitts-Rosenblatt unit of ANNs drastically oversimplifies the complexity of real neurons.Recognizing the intricate temporal dynamics in biological neurons and the ubiquity of feedback loops in natural networks, we suggest reimagining neurons as feedback controllers. A practical implementation of such controllers within biological systems is made feasible by the recently developed Direct Data-Driven Control (DD-DC). We find that DD-DC neuron models can explain various neurophysiological observations, affirming our theory.

We discuss the proof of the following Theorem

Assume $E$ is a $C^{p}$ real Banach manifold, and $f:E\circlearrowleft$, $f\circ f=f$ is a $C^{p}$ retraction, where $1\leq p\leq +\infty$. Then the retract $f(E)$ is a $C^{p}$ sub Banach manifold of $E$.

If time allows, we will also discuss how this fact is related to the study of smoothness and structural stability of attractors, along the intersection of topology and dynamics. We will be focusing on the proof and perspective of Oliva 1975, who was interested in Banach manifolds as phase-spaces of delay equations.