Georgia Institute of Technology College of Sciences

We discuss a novel class of swarm-based gradient descent (SBGD) methods for nonconvex optimization. Each agent in the swarm is characterized by its position and mass. 

The dynamics combines two mechanisms: persistent transfer of mass from agents positioned on “higher ground” to those with lower objective values, and a mass-dependent time-stepping protocol. This coupling creates a dynamic distinction between “leaders” and “explorers.” Heavier agents act as leaders, use small time steps to refine promising regions near local minima, while light agents take larger steps, exploring the landscape for lower objective values. The swarm dynamics adaptively balances exploitation of local refinement with global exploration. 

We present convergence results and numerical experiments illustrating the effectiveness of SBGD for global optimization.

(second of two talks; the first was on Sep. 22)

A random geometric graph (RGG) is generated by first sampling $n$ latent points independently and uniformly from the unit sphere in $R^d$, and then connecting each pair of points if their inner product exceeds a threshold. We study the sharp detection threshold---the largest dimension at which the RGG can be statistically distinguished from the Erdős--Rényi graph with the same edge density $p$. This threshold is conjectured to be $d \asymp (n h(p))^3$, where $h(p)$ is the binary entropy function. Previous works proved this conjecture for dense graphs with constant $p$ and, up to polylogarithmic factors, very sparse graphs with constant average degrees. In this series of two talks, I will discuss a resolution of this conjecture. This is based on joint work with Hang Du, Nike Sun, Yihong Wu, and Jiaming Xu.

It has been observed that the relationship of ribbon concordance induces monotonic behavior across several link invariants. 

Resolving a famous open question of Gordon, Agol unified these observations by showing that ribbon concordance induces a partial order on knots in the 3-sphere.

Agol's argument can be extended to show that ribbon concordance likewise induces a partial order on arbitrary links in the 3-sphere.

Perhaps motivated by the slice-ribbon conjecture, it is natural and interesting to ask which knots and links are minimal with respect to this partial order. 

Leveraging recent developments in link Floer homology, I will share new examples of ribbon minimal knots and links. 

Some of this work is joint with Jaewon Lee and Alessio Di Prisa.

The multipermutohedral Chow ring was introduced in a series of papers by Clader, Damiolini, Eur, Huang, Li, and Ramadas to study moduli spaces with cyclic symmetry. It generalizes Chow rings of permutohedral varieties and type-B Coxeter arrangements. In this talk, we establish the combinatorial structure of the multipermutohedral Chow ring through an explicit Gröbner basis, yielding a Feichtner-Yuzvinsky-type monomial basis and a formula for the Hilbert series. Using this formula, we refine the palindromicity of the Hilbert series. From a representation-theoretic perspective, we also compute the equivariant Hilbert series under two natural group actions and construct combinatorial maps that establish equivariant unimodality and palindromicity.