Georgia Institute of Technology College of Sciences

A question going back to Serre asks which groups arise as fundamental groups of smooth, complex projective varieties, or more generally, compact Kaehler manifolds.  The most basic examples of these are surface groups, arising as fundamental groups of 1-dimensional projective varieties.  We will survey known examples and restrictions on such groups and explain the special role surface groups play in their classification. Finally, we connect this circle of ideas to more general questions about surface bundles and mapping class groups. 

In the past decade, the catalog of algorithms available to combinatorial optimizers has been substantially extended to settings which allow submodular objective functions. One significant recent result was a tight (1-1/e)-approximation for maximizing a non-negative monotone submodular function subject to a matroid constraint. These algorithmic developments were happening concurrently with research that found a wealth of new applications for submodular optimization in machine learning, recommender systems, and algorithmic game theory.

The related supermodular maximization models also offer an abundance of applications, but they appeared to be highly intractable even under simple cardinality constraints and even when the function has a nice structure. For example, the densest subgraph problem - suspected to be highly intractable - can be expressed as a monotonic supermodular function which has a particularly nice form. Namely, the objective can be expressed by a quadratic form $x^T A x$ where $A$ is a non-negative, symmetric, 0-diagonal matrix. On the other hand, when the entries $A(u,v)$ form a metric, it has been shown that the associated maximization problems have constant factor approximations. Inspired by this, we introduce a parameterized class of non-negative functions called meta-submodular functions that can be approximately maximized within a constant factor. This class includes metric diversity, monotone submodular and other objectives appearing in the machine learning and optimization literature. A general meta-submodular function is neither submodular nor supermodular and so its multi-linear extension does not have the nice convexity/concavity properties which hold for submodular functions. They do, however, have an intrinsic one-sided smoothness property which is essential for our algorithms. This smoothness property might be of independent interest.

In the realm of Laplacians of Riemannian manifolds, nodal domains have been the subject of intensive research for well over a hundred years. 

Given a Riemannian manifold M, let f be an eigenfunctions f of the Laplacian with respect to some boundary conditions.  A nodal domain associated with f is the maximal connected subset of the domain M  for which the f does not change sign.

Here we examine the discrete cases, namely we consider nodal domains for graphs. Dekel-Lee-Linial shows that for a Erdős–Rényi graph G(n, p), with high probability there are exactly two nodal domains for each eigenvector corresponding to a non-leading eigenvalue.  We prove that with high probability, the sizes of these nodal domains are approximately equal to each other. 

Back in the year 2000, Christ and Kiselev introduced a useful "maximal trick" in their study of spectral properties of Schro edinger operators.
The trick was completely abstract and only at the level of basic functional analysis and measure theory. Over the years it was reproven,
generalized, and reused by many authors. We will present its recent application in the theory of restriction of the Fourier transform to
surfaces in the Euclidean space.