In this talk, we explore adaptations of semidefinite programming relaxations for solving PDE problems. Our approach transforms a high-dimensional PDE problem into a convex optimization problem, setting it apart from traditional non-convex methods that rely on nonlinear re-parameterizations of the solution. In the context of statistical mechanics, we demonstrate how a mean-field type solution for an interacting particle Fokker-Planck equation can be provably recovered without resorting to non-convex optimization. 

A frame (x_j) for a Hilbert space H allows for a linear and stable reconstruction of any vector x in H from the linear measurements (<x,x_j>).  However, there are many situations where some information of the frame coefficients is lost.

When is it possible to project two sets of labeled points lying in a pair of projective planes to the same points on a projective line? Here one answer: such projections exist if and only if the two 2D point sets are themselves images of a common point set in 3D projective space. Furthermore, when the two sets of points are in general position, it is possible to give a complete description of the loci of pairs of projection centers.

Many problems in computer vision are represented using a parametrized overdetermined system of polynomials which must be solved quickly and efficiently. Classical methods for solving these systems involve specialized solvers based on Groebner basis techniques or utilize randomization in order to create well-constrained systems for numerical techniques. We propose new methods in numerical linear algebra for solving such overdetermined polynomial systems and provide a complete error analysis showing that the numerical approach is stable.

The study of fragile economic systems is important in identifying systems that are vulnerable to a dramatic collapse. For instance, complex systems like supply chains are at risk of being fragile because they require many parts to work well simultaneously. Even when each individual firm has a small susceptibility to a shock, the global system may still be at great risk. A recent survey by Matthew Elliot and Ben Golub review fragile economic systems from the point of view of networks.

To gain their unique biological function, plant cells regulate protein biosynthesis through gene activation and repression along with multiple mRNA mechanisms. The subcellular localization of mRNAs has been reported as a complementary regulatory mechanism of the biology of fungi, yeast, and animal cells. However, studies comprehensively reporting the impact of mRNA localization in plant cells are lacking.

Modern complex systems often involve multiple interacting agents in a shared environment, e.g., transportation systems, power systems, swarm robotics, and human-robot interactions. Controlling these multi-agent systems (MASs) requires the characterization of agents’ interactions to account for their interdependent self-interests and coupled agents’ constraints such as collision avoidance and/or limited shared resources.

Group synchronization is a mathematical framework used in a variety of applications, such as computer vision, to situate a set of objects given their pairwise relative positions and orientations subjected to noise. More formally, synchronization estimates a set of group elements given some of their noisy pairwise ratios.

Diffusion model is an emerging generative modeling technique, achieving the state-of-the-art performances in image and video synthesis, scientific simulation, inverse problems, and offline reinforcement learning. Yet, existing statistical analysis of diffusion models often requires restrictive theoretical assumptions or is suboptimal.

Diffusion-based generative models have become the standard for image generation. ODE-based samplers and flow matching models improve efficiency, in comparison to diffusion models, by reducing sampling steps through learned vector fields. However, the theoretical foundations of flow matching models remain limited, particularly regarding the convergence of individual sample trajectories at terminal time - a critical property that impacts sample quality and being critical assumption for models like the consistency model.