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.

Contact geometry has a long history, with connections to many areas of physics and mathematics. I will begin with some history and motivation for contact geometry. I will then discuss the development of contact geometry in dimension three, and the central role knot theory has played in that development. We will end by considering the beautiful structure of special knots in contact manifolds and their use in classifying contact structures on three-manifolds. 

In many applications of mathematical optimization, one may wish to optimize an objective function without access to its derivatives. These situations call for derivative-free optimization (DFO) methods. Among the most successful approaches in practice are model-based trust-region methods, such as those pioneered by M.J.D Powell. These methods rely on function approximations via low degree polynomials and carefully adapt the local geometry of interpolation points to balance exploration and exploitation.  While relatively complex to implement, these methods are now available in standard scientific computing platforms, including MATLAB and SciPy. However, theoretical analysis of their computational complexity lags behind practice. In particular, it is important to bound the number of function evaluations required to achieve a desired level of accuracy. Using concepts from Lagrangian interpolation and linear algebra we systematically derive complexity bounds for classical model-based trust-region methods and their modern variations. We establish, for the first time, that these methods can have the same worst case complexity than any other known DFO method.

The Birkhoff Ergodic Theorem describes typical behaviors and averaged quantities with respect to an invariant measure. In this talk, I will focus on "observable" events, equating observability with positive Lebesgue measure. From this observational viewpoint, "typical" means typical with respect to Lebesgue measure. This leads immediately to issues for attractors, where all invariant measures are singular. I will present highlights of developments in smooth ergodic theory that address these questions. The theory of physical and SRB measures applies to dynamical systems that are deterministic as well as random, in finite and infinite dimensions (where observability has to be interpreted differently). This body of ideas argue in favor of convergence of ergodic averages for typical orbits. But the picture is a little more complicated: In the last part of the talk, I will discuss some recent work that shows that in many natural settings (e.g. reaction networks), it is also typical for ergodic averages 
to fluctuate in perpetuity due to heteroclinic-like behavior.