Chains in CR geometry as geodesics of a Kropina metric
Vladimir Matveev, Friedrich Schiller University in Jena, Germany September 28, 2018
11:20-12:20 pm
ATG 121

With the help of a generalization of the Fermat principle in general relativity, we show that chains in CR geometry are geodesics of a certain Kropina metric constructed from the CR structure. We study the projective equivalence of Kropina metrics and show that if the kernel distributions of the corresponding 1-forms are non-integrable then two projectively equivalent metrics are trivially projectively equivalent. As an application, we show that sufficiently many chains determine the CR structure up to conjugacy, generalizing and reproving the main result of [J.-H. Cheng, 1988]. The talk is based on a joint paper with J.-H. Cheng, T. Marugame, and R. Montgomery.

Let $\Omega$ be a $C^2$-smooth bounded pseudoconvex domain
in $\mathbb{C}^n$ for $n\geq 2$ and let $\varphi$ be a holomorphic
function on $\Omega$ that is $C^2$-smooth on the closure of $\Omega$.
We prove that if the Hankel operator $H_{\overline{\varphi}}$ is in Schatten
$p$-class for $p\leq 2n$ then $\varphi$ is a constant function. As a corollary,
we show that the $\overline{\partial}$-Neumann operator on $\Omega$
is not Hilbert-Schmidt. This is joint work with N\.{i}hat G\”{o}khan
G\”{o}\u{g}\”{u}\c{s}

Temple-Rutgers Global Analysis Seminar: Escobar-Yamabe compactification for Poincare-Einstein manifolds and related rigidity theorems Fang Wang, Shanghai Jiao Tong University April 20, 2018
4-4:50 pm
BSB 116

I will talk about two types of Escobar-Yamabe compactification for Poincare-Einstein manifolds and derive some inequalities between the Yamabe constants for the conformal class of compactified metric and the Yamabe constant for the conformal infinity. Based on thoses inequalities, I will also give some rigidity theorems. This work was motivated by recent work of Gursky-Han, and collaborated with M. Lai and X. Chen.

Starting with the observation that every continuous complex-valued function on the unit circle can be approximated by rational combinations of one function and polynomial combinations of two functions, we will discuss analogous approximation phenomena for compact manifolds of higher dimensions. On a related note, we will discuss some questions regarding the minimum embedding (complex) dimension of real manifolds within the context of polynomial convexity. In the special case of even-dimensional manifolds, we will present a technique that improves previously known bounds. This is joint work with Rasul Shafikov.

Couplings, metrics and contraction rates for Langevin diffusions Andreas Eberle, University of Bonn, Germany
March 23, 2018
11:20 am-12:20 pm
Armitage 121

Carefully constructed Markovian couplings and specifically designed Kantorovich metrics can be used to derive relatively precise bounds on the distance between the laws of two Langevin processes. In the case of two overdamped Langevin diffusions with the same drift, the processes are coupled by reflection, and the metric is an L1 Wasserstein distance based on an appropriately chosen concave distance function. If the processes have different drifts, then the reflection coupling can be replaced by a „sticky coupling“ where the distance between the two copies is bounded from above by a one-dimensional diffusion process with a sticky boundary at 0. This new type of coupling leads to long-time stable bounds on the total variation distance between the two laws. Similarly, two kinetic Langevin processes can be coupled using a particular combination of a reflection and a synchronous coupling that is sticky on a hyperplane. Again, the coupling distance is contractive on average w.r.t. an appropriately designed Wasserstein distance. This can be applied to derive new bounds of kinetic order for convergence to equilibrium at the borderline between the overdamped and the underdamped regime. Similar approaches are also useful when studying mean-field interacting particle systems, McKean-Vlasov diffusions, or diffusions on infinite dimensional state spaces. (joint work with Arnaud Guillin and Raphael Zimmer).

“Metastability of the Nonlinear Wave Equation” Katherine Newhall, UNC
February 23, 2018
11:20 am- 12:20 pm
Armitage 121

I will discuss the long-time dynamics of infinite energy solutions to a wave equation with nonlinear forcing. Of particular interest is when these solutions display metastability in the sense that they spend long periods of time in disjoint regions of phase-space and only rarely transition between them. This phenomenon is quantified by calculating exactly via Transition State Theory (TST) the mean frequency at which the solutions of the nonlinear wave equation with initial conditions drawn from its invariant measure cross a dividing surface lying in between the metastable sets. Numerical results suggest a regime for which the dynamics are not fundamentally different from that observed in the stochastic counterpart in which random noise and damping terms are added to the equation, as well as a regime for which successive transitions between the metastable sets are correlated and the coarse-graining to a Markov chain fails.

“Pricing Uber’s Marketplace”
James Davis, Senior Data Scientist at Uber & a 2010 graduate of Rutgers–Camden Mathematics
February 16, 2018
11:20 am- 12:20 pm
Penn 401 (Note special location)

Dynamic pricing at Uber is a key part of what makes the services so reliable. We will see how dynamic pricing works in an up-close way. First, we will walk through pricing during a typical day in Philadelphia, Then, after we explain the key concepts behind dynamic pricing, we will show these concepts in action after the Eagles Superbowl win. We will show just how the Uber market evolved and how pricing responded. If there is enough time we will discuss some big upcoming challenges for Uber’s pricing system.

“Kinetics of Particles with Short-range Interactions” Miranda Holmes-Cerfon, NYU Courant
February 9, 2018
11:20 am- 12:20 pm
Armitage 121

Particles in soft-matter systems (such as colloids) tend to have very short-range interactions, so traditional theories, that assume the energy landscape is smooth enough, will struggle to capture their dynamics. We propose a new framework to look at such particles, based on taking the limit as the range of the interaction goes to zero. In this limit, the energy landscape is a set of geometrical manifolds plus a single control parameter, while the dynamics on top of the manifolds are given by a hierarchy of Fokker-Planck equations coupled by “sticky” boundary conditions. We show how to compute dynamical quantities such as transition rates between clusters of hard spheres, and then show this agrees quantitatively with experiments. The framework may also be used to more efficiently ask questions about programmable self-assembly.

“Stochastic Simulation of Multiscale Reaction-Diffusion Models via First Exit Times” Lina Meinecke, Department of Mathematics at UC Irvine
January 26, 2018
11:20 am- 12:20 pm
Armitage 121

Mathematical models are important tools in systems biology, since the regulatory networks in biological cells are too complicated to understand by biological experiments alone. However, analytical solutions can be derived only for the simplest models and hence numerical simulations are often necessary.
This talk focuses on the mesoscopic simulation level, which captures both, space dependent behavior by diffusion and the inherent stochasticity of cellular systems. Space is partitioned into compartments by a mesh and the number of molecules of each species in each compartment gives the state of the system. We first examine how to compute the jump coefficients for a discrete stochastic jump process on unstructured meshes from a first exit time approach guaranteeing the correct speed of diffusion. Furthermore, we analyze different methods leading to non-negative coefficients by backward analysis and derive a new method, minimizing both the error in the diffusion coefficient and in the particle distribution.
The second part investigates macromolecular crowding effects. A high percentage of the cytosol and membranes of cells are occupied by molecules. This impedes the diffusive motion and also affects the reaction rates. Most algorithms for cell simulations are either derived for a dilute medium or become computationally very expensive when applied to a crowded environment. Therefore, we develop a multiscale approach, which takes the microscopic positions of the molecules into account, while still allowing for efficient stochastic simulations on the mesoscopic level. Finally, we compare on- and off-lattice models on the microscopic level when applied to a crowded environment.