After a brief historical introduction, I will present some recent developments in the theory of minimal surfaces in Euclidean spaces which have been obtained by complex analytic methods. The emphasis will be on results pertaining to the global theory of minimal surfaces including Runge and Mergelyan approximation, the conformal Calabi-Yau problem, properly immersed and embedded minimal surfaces, and a new result on the Gauss map of minimal surfaces.

How neuronal variability impacts neural codes is a central question in systems neuroscience, often with complex and model dependent answers. Most population models are parametric, with tacitly assumed structure of neuronal tuning and population variability. While these models provide key insights, they cannot inform how the physiology and circuit wiring of cortical networks impact information flow. In this work, we study information propagation in spatially ordered neuronal networks. We focus on the effects of feedforward and recurrent projection widths relative to columnar width, as well as attentional modulation. We show that narrower feedforward projection width increases the saturation rate of information. In contrast, the recurrent projection width with spatially balanced excitation and inhibition has small effects on information. Further, we show that attention improves information flow by suppressing the internal dynamics of the recurrent network.

We consider point-to-point directed paths in a random environment on the two-dimensional integer lattice. For a general independent environment under mild assumptions we show that the quenched energy of a typical path satisfies a central limit theorem as the mesh of the lattice goes to zero. The proofs rely on concentration of measure techniques and some combinatorial bounds on families of paths. This is joint work with Christian Gromoll and Leonid Petrov.

We will study an obstacle problem coming from consumer search in a product market. I will discuss several properties concerning the geometry of the free boundary. The difficulty is to determine how the geometry depends on the dimension d. This is a joint work with T. Tony Ke, Wenpin Tang and J. Miguel Villas-Boas.

Accurate RNA structural prediction remains challenging, despite its increasing biomedical importance. Sampling secondary structures from the Gibbs distribution yields a strong signal of high probability base pairs. However, identifying higher order substructures requires further analysis. Profiling (Rogers & Heitsch, NAR, 2014) is a novel method which identifies the most probable combinations of base pairs across the Boltzmann ensemble. This combinatorial approach is straightforward, stable, and clearly separates structural signal from thermodynamic noise.

The behavior of the eigenvalues of large random matrices is generally very predictable, on multiple scales. Macroscopically, results like the semi-circle law describe the overall shape of the eigenvalue distributions. Indeed, for many natural ensembles of random matrices, we can describe in great detail the way the distribution of the eigenvalues converges to some limiting deterministic probability measure. On a microscopic scale, we often see the phenomenon of eigenvalue rigidity, in which individual eigenvalues concentrate strongly at predicted locations. I will describe some general approaches to these phenomena, with many examples: Wigner matrices, Wishart matrices, random unitary matrices, truncations of random unitary matrices, Brownian motion on the unitary group, and others.

Neuronal variability is a reflection of recurrent circuitry and cellular physiology, and its modulation is a reliable signature of cognitive and processing state. A pervasive yet puzzling feature of cortical circuits is that despite their complex wiring, population-wide shared spiking variability is low dimensional with all neurons fluctuating en masse. Previous model cortical networks are at loss to explain this variability, and rather produce either uncorrelated activity, high dimensional correlations, or pathologically network behavior. We show that if the spatial and temporal scales of inhibitory coupling match known physiology then model spiking neurons naturally generate low dimensional shared variability that captures in vivo population recordings along the visual pathway. Further, top-down modulation of inhibitory neurons provides a parsimonious mechanism for how attention modulates population-wide variability both within and between neuronal areas, in agreement with our experimental results. Our theory provides a critical and previously missing mechanistic link between cortical circuit structure and realistic population-wide shared neuronal variability.

Stacking a few layers of 2D materials such as graphene and molybdenum disulfide, for example, opens the possibility of tuning the electronic and optical properties of 2D materials. One of the main issues encountered in the mathematical and computational modeling of 2D materials is that lattice mismatch and rotations between the layers destroys the periodic character of the system.
Basic concepts like mechanical relaxation, electronic density of states, and the Kubo-Greenwood formulas for transport properties will be formulated and analyzed in the incommensurate setting. New computational approaches will be presented and the validity and efficiency of these approximations will be examined from mathematical and numerical analysis perspectives.

A bilevel optimization problem is a sequence of two optimization problems where the constraint region of the upper level problem is determined implicitly by the solution set to the lower level problem. It can be used to model a two-level hierarchical system where the two decision makers have different objectives and make their decisions on different levels of hierarchy. Recently more and more applications including those in machine learning have been modelled as bilevel optimization problems. This talk will discuss issues, challenges and discoveries for solving bilevel optimization problems.