Georgia Institute of Technology College of Sciences

The theory of $p$-adic differential equations first rose to prominence after Dwork used them to prove the rationality of zeta functions of a positive characteristic variety in 1960. Since then, there has been growing interest in the category of convergent $F$-isocrystals and the subcategory of overconvergent $F$-isocrystals due to this subcategory having good cohomology theory with finiteness properties. Recent work by Grubb, Kedlaya and Upton examines when a convergent $F$-isocrystal is overconvergent by restricting to smooth curves on the scheme under a mild tameness assumption (measured by the Swan conductor). In my talk, I will introduce the above categories and talk about work in progress about bounding the Swan conductor of an overconvergent $F$-isocrystal in terms of data associated with the corresponding convergent $F$-isocrystal.

Complex fluids are abundant in our daily life. Unlike traditional solids, liquids and the diluted solutions, the model equations for complex fluids continue to evolve with the new experimental evidences and emerging applications. Most of these important properties are due to the coupling and competition between effects from different scales or even from different physical origins/principles. The energetic variational approaches (EnVarA), motivated by the seminal works of Onsager and Rayleigh, are designed to study such systems. In this talk, I will discuss several complex fluid systems, and the associated mathematical issues.

One of the fundamental problems in contact topology is to classify contact structures on a given 3-manifold. In particular, classifying contact structures on surgeries along a given knot has been very poorly studied. The only fully understood case so far is that of the unknot  (lens spaces); for all other knots we have only partial results, or none at all. Several topological and algebraic tools have been developed to attack this problem. In this talk, we discuss recent developments and the strategy for classifying tight contact structures on surgeries along torus knots. This is joint work with John Etnyre, Bülent Tosun, and Konstantinos Varvarezos.

I will present a series of papers in which we constructed multi-soliton solutions to three-dimensional ($L^2$-subcritical) and four-dimensional ($L^2$-critical) Hartree equations. In these solutions, the soliton centers evolve according to an effective N-body system. Our work generalized and improved the 2009 result of Krieger–Martel–Raphaël, which constructed two-soliton solutions for the three-dimensional Hartree equation. In four dimensions, our results further yield the existence of multi-point pseudo-conformal blow-up via the pseudo-conformal symmetry.

We discuss a general philosophy that loosely states that if the matrix representation of a linear operator is concentrated on its diagonal, then the operator’s compactness is characterized by the decay of its matrix representation along the diagonal. We formulate rigorous versions of this idea and apply them to study the compactness of (bi-parameter) Calderón-Zygmund operators, pseudodifferential operators, and Fourier integral operators. These applications recover and unify various earlier works and provide new results.

Modern homology theories have given many knot invariants with the following useful properties: they are additive with respect to connected sum, they give a lower bound for a knot's slice genus, and this lower bound is equal to the slice genus for torus knots. These invariants, called slice-torus invariants, include the Ozsváth–Szabó $\tau$ and Rasmussen $s$ invariants. We discuss how, on a large class of knots, the value of a slice-torus invariant is fully determined by these properties, and can be computed without reference to the homology theory. We also discuss results that follow from the existence of slice-torus invariants, and a potential connection to the smooth 4-dimensional Poincaré conjecture.

A well known result in graph theory states that a graph is connected if and only if the second eigenvalue of its Laplacian matrix is positive. In fact, the larger the second eigenvalue, the more connected the graph is. By varying the weights on edges, one can in general increase the second eigenvalue which in turn affects many graph properties such as expansion, mixing times of random walks etc.

In this talk, I will introduce conformally rigid graphs, which are those unweighted undirected graphs in which one cannot increase the second eigenvalue or decrease the largest eigenvalue by changing
weights. This notion turns out to be deeply connected to graph embeddings, semidefinite programming and other ideas in geometry, optimization and combinatorics.

Joint work with Joao Gouveia and Stefan Steinerberger

The capillary energy functional is used to model the equilibrium shape of a liquid drop meeting a substrate at a prescribed interior contact angle. We will discuss a rigidity theorem for volume-preserving critical points of the capillary energy in the half-space: among all sets of finite perimeter, every such critical configuration corresponding to a prescribed contact angle between 90 degrees and 120 degrees must be a finite union of spheres and spherical caps with the correct contact angle. Assuming that the tangential part of the capillary boundary is $\mathcal{H}^n$-null, this rigidity extends to the full hydrophobic range of contact angles between 90 degrees and 180 degrees. We will also present an anisotropic counterpart, establishing rigidity under suitable lower density assumptions. This talk is based on joint work with A. De Rosa and R. Resende.