In range counting problems we are given a set of objects, and the goal is to preprocess them, so that given any query range, the number of objects that fall into that range is computed efficiently. For example, the objects are points corresponding to the coordinates of several cities, and we would like to compute how many cities lie within a certain region. In many cases, it is too expensive to process the entire set of objects, which raises the necessity to resort to approximate range counting. Motivated by this problem, relative approximations have become a central tool Computational Geometry. In fact, they were introduced by Har-Peled and Sharir, who derived them from the seminal work of Li, Long, and Srinivasan in the theory of Machine Learning. Informally, a relative approximation is a sample of the input objects that guarantees a bounded relative error for any given range when comparing its exact count (that is, with respect to the entire set of objects) to its approximate count (that is, confined to the sample). In this talk I will discuss the properties of relative approximations and their relation to the so-called "Epsilon-nets". I will also present improved upper bounds for the size of relative approximations, which eventually yields better performance for approximate range counting. Our approach is probabilistic, where we apply the constructive version of the general Local Lemma of Lovasz. Last but not least, I will present a few applications to Sensor Networking and Machine Learning.

The mean curvature flow evolves hypersurfaces in time: the velocity at each point is given by the mean curvature vector. It is the most natural geometric evolution equation for submanifolds and shares many interesting features with Hamilton's Ricci flow for Riemannian manifolds. In this talk, I will start by explaining and motivating the equation. In particular, I will explain that the mean curvature flow can be viewed as geometric heat equation and thus improves the geometry. To illustrate this, I will describe Huisken's classical result that any convex hypersurface evolves into a round sphere. Finally, I will discuss the formation of singularities in the flow of mean convex hypersurfaces. This is based on fundamental results due to White and Huisken-Sinestrari, and on my recent joint work with Cheeger-Naber and Kleiner.

Cyclotrons have been a successful tool for accelerating beams of charged particles that are then used for basic science experiments, and for nuclear security and medical applications. As new scientific needs emerge, a consistent trend is to push the limits of cyclotron designs, either by increasing the intensity of the beams or by decreasing the size of the cyclotrons. In order to design such high performance machines, it is crucial to understand and be able to control the detailed dynamics of intense charged particle beams. In this talk, we will present several mathematical models and numerical methods to study large ensembles of particles subject to electromagnetic fields. Focusing on the particular case of the cyclotron, we will then show how starting from a kinetic description we can derive a set of coupled fluid equations describing vortex dynamics in charged beams. We will finish by discussing the consequences of this vortex motion on beam control in cyclotrons.

Cancelled due to weather delays for speaker

In many physical problems, energy minimization leads to the formation of microstructure. Examples include the wrinkling of a stretched elastic membrane, the formation of domains in a magnetic material, and the twinning produced by martensitic phase transformation. The mathematical heart of the matter involves nonconvex variational problems regularized by higher-order singular perturbations. I will discuss through selected examples (some old, some new) how the analysis of such problems has opened a new chapter in the calculus of variations.

Stein's method is a semi-classical tool for establishing distributional convergence with explicit rates, particularly effective in problems involving complex dependencies. Currently there are several approaches in using Stein's method for Gaussian Central Limit theorems, involving dependency graphs, exchangeable pairs, zero biasing, size biasing and others. In this talk I will briefly describe the main idea behind the method and the existing approaches for normal approximation, and finally explain a new approach for applying the method with examples. No knowledge beyond basic probability will be assumed.

Analog-to-digital (A/D) conversion is typically a deterministic operation. A signal comes in, a sequence of bits comes out. The output bits are constrained to yield accurate reconstruction of the incoming signal via some pre-specified method. Nevertheless, various notions of randomness play important roles in the design and analysis of A/D algorithms. We'll see some classical and modern examples.

The landscape of computational methods is currently undergoing a dramatic change. Modern computing affords an abundance of floating point operations, but, by comparison, little data motion. This has the potential to affect anyone using computers to carry out moderate-to-large-scale experiments, and making use of these advances has the potential to necessitate changes to algorithms, methods and tools. After working out the basic changes in computer architecture, we will be examining an adaptation of an existing high-order finite element method to massively parallel many-core architectures as a motivating example. Working forward from this, we will discuss the arising challenges and some potential solutions, in the form of computational tools that support and generalize the approach taken. All of the tools I will be describing are embedded in the high-level programming language `Python' and aim to support a seamless transition from prototype to full-scale application. They provide a range of capabilties from low-level machine access, through flexible parallel primitives, to parallelizing code generators for array-based codes in a polyhedral framework.

A dispersive partial differential equation is one where different frequency components propagate at different velocities. On non-compact domains (like R^d), dispersion is a mechanism for (conservative or non-dissipative) decay and ultimately stability. This is not the case on compact domains where conservation laws prohibit any form of decay, and different frequency components have no escape from "overlapping" with each other all the time. Consequently, the mathematical analysis of such equations is considerably more delicate and, more importantly, the asymptotic behavior is very rich. A mathematically and physically important example of a compact domain is that of a box, in which case one can retain the power of Fourier analysis. Having the lattice $\Z^d$ as Fourier space, the analysis ultimately requires input from number theory, a connection first explored by Bourgain in the early nineties. I will try to present some instances of this rather beautiful interaction between these two, seemingly distant, fields of mathematics, both at the level of linear and nonlinear problems.