Georgia Institute of Technology College of Sciences

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.

Primitive integral Apollonian circle packings are fractal arrangements of tangent circles with integer curvatures.  The curvatures form an orbit of a 'thin group,' a subgroup of an algebraic group having infinite index in its Zariski closure.  The curvatures that appear must fall into a restricted class of residues modulo 24. The twenty-year-old local-global conjecture states that every sufficiently large integer in one of these residue classes will appear as a curvature in the packing. We prove that this conjecture is false for many packings, by proving that certain quadratic and quartic families are missed. The new obstructions are a property of the thin Apollonian group (and not its Zariski closure), and are a result of quadratic and quartic reciprocity, reminiscent of a Brauer-Manin obstruction. Based on computational evidence, we formulate a new conjecture.  This is joint work with Summer Haag, Clyde Kertzer, and James Rickards.  Time permitting, I will discuss some new results, joint with Rickards, that extend these phenomena to certain settings in the study of continued fractions.

Dynamical systems exhibiting some degree of hyperbolicity often admit “fractal" invariant objects.  However, extra symmetries or “randomness” in the system often preclude the existence of such fractal objects.

I will give some concrete examples of the above and then discuss problems and results related to random dynamics and group actions on surfaces.  I will especially focus on questions related to absolute continuity of stationary measures.

The totally non-negative Grassmannian is the set of points in a real Grassmannian such that all Plucker coordinates have the same sign (some can be zero). I will show how points in totally non-negative Grassmannians arise from the spaces of polynomials in one variable whose Wronskian has only real roots. Then I will discuss a similar result for the spaces of quasi-exponentials.

The main statements of this talk should be understandable to an undergraduate student. Somewhat surprisingly, the proofs use the theory of quantum integrable systems related to $GL(n)$. I will try to explain the logic of such proofs in a gentle way.

This talk is based on a joint work with S. Karp and V. Tarasov.