We will discuss how to solve algebraic equations using symbolic, numerical, and combinatorial methods.
Soot particles are major pollutants emitted from propulsion and power generation systems. In turbulent combustion, soot evolution is heavily influenced by soot-turbulence-chemistry interaction. Specifically, soot is formed during combustion of fuel-rich mixtures and is rapidly oxidized before being transported by turbulence into fuel-lean mixtures. Furthermore, different soot evolution mechanisms are dominant over distinct regions of mixture fraction. For these reasons, a new subfilter Probability Density Function (PDF) model is proposed to account for this distribution of soot in mixture fraction space. At the same time, Direct Numerical Simulation (DNS) studies of turbulent nonpremixed jet flames have revealed that Polycyclic Aromatic Hydrocarbons (PAH), the gas-phase soot precursors, are confined to spatially intermittent regions of low scalar dissipation rates due to their slow formation chemistry. The length scales of these regions are on the order of the Kolmogorov scale (i.e., the smallest turbulence scale) or smaller, where molecular diffusion dominates over turbulent mixing irrespective of the large-scale turbulent Reynolds number. A strain-sensitivity parameter is developed to identify such species. A Strain-Sensitive Transport Approach (SSTA) is then developed to model the differential molecular transport in the nonpremixed “flamelet” equations. These two models are first validated a priori against a DNS database, and then implemented within a Large Eddy Simulation (LES) framework, applied to a series of turbulent nonpremixed sooting jet flames, and validated via comparisons with experimental measurements of soot volume fraction.
In complex dynamics, the main objects of study are rational maps on the Riemann sphere. For some large subset of such maps, there is a way to associate to each map a marked torus. Moving around in the space of these maps, we can then track the associated tori and get induced mapping classes. In this talk, we will explore what sorts of mapping classes arise in this way and use this to say something about the topology of the original space of maps.
The field of complex dynamics melds a number of disciplines, including complex analysis, geometry and topology. I will focus on the influences from the latter, introducing some important concepts and questions in complex dynamics, with an emphasis on how the concepts tie into and can be enhanced by a topological viewpoint.
In this talk, we consider a quasi-periodically forced system arising from the problem of chemical reactions. For we demonstrate efficient algorithms to calculate the normally hyperbolic invariant manifolds and their stable/unstable manifolds using parameterization method. When a random noise is added, we use similar ideas to give a streamlined proof of the existence of the stochastic invariant manifolds.
A random array indexed by the paths of an infinitely-branching rooted tree of finite depth is hierarchically exchangeable if its joint distribution is invariant under rearrangements that preserve the tree structure underlying the index set. Austin and Panchenko (2014) prove that such arrays have de Finetti-type representations, and moreover, that an array indexed by a finite collection of such trees has an Aldous-Hoover-type representation.
Motivated by problems in Bayesian nonparametrics and probabilistic programming discussed in Staton et al. (2018), we generalize hierarchical exchangeability to a new kind of partial exchangeability for random arrays which we call DAG-exchangeability. In our setting a random array is indexed by N^{|V|} for some DAG G=(V,E), and its exchangeability structure is governed by the edge set E. We prove a representation theorem for such arrays which generalizes the Aldous-Hoover representation theorem, and for which the Austin-Panchenko representation is a special case.
A brief meeting to discuss the plan for the semester, followed by an informal discussion over lunch (most likely at Ferst Place).
Great News Everyone! The cartoon series Futurama is packed with science jokes. Adopting my Professor Farnsworth Alterego, I will explain some of these mathematical jokes with stills and clips from the series.
Residential crime is one of the toughest issues in modern society. A quantitative, informative, and applicable model of criminal behavior is needed to assist law enforcement. We have made progress to the pioneering statistical agent-based model of residential burglary (Short et al., Math. Models Methods Appl., 2008) in two ways. (1) In one space dimension, we assume that the movement patterns of the criminals involve truncated Lévy distributions for the jump length, other than classical random walks (Short et al., Math. Models Methods Appl., 2008) or Lévy flights without truncation (Chaturapruek et al., SIAM J. Appl. Math, 2013). This is the first time that truncated Lévy flights have been applied in crime modeling. Furthermore (2), in two space dimensions, we used the Poisson clocks to govern the time steps of the evolution of the model, rather than a discrete time Markov chain with deterministic time increments used in the previous works. Poisson clocks are particularly suitable to model the times at which arrivals enter a system. Introduction of the Poisson clock not only produces similar simulation output, but also brings in theoretically the mathematical framework of the Markov pure jump processes, e.g., a martingale approach. The martingale formula leads to a continuum equation that coincides with a well-known mean-field continuum limit. Moreover, the martingale formulation together with statistics quantifying the relevant pattern formation leads to a theoretical explanation of the finite size effects. Our conjecture is supported by numerical simulations.
Please Note: Please note different day and room.
In this talk, I will describe joint work with Maximilien Péroux on understanding Koszul duality in ∞-topoi. An ∞-topos is a particularly well behaved higher category that behaves like the category of compactly generated spaces. Particularly interesting examples of ∞-topoi are categories of simplicial sheaves on Grothendieck topologies. The main theorem of this work is that given a group object G of an ∞-topos, there is an equivalence of categories between the category of G-modules in that topos and the category of BG-comodules, where BG is the classifying object for G-torsors. In particular, given any pointed space X, the space of loops on X, denoted ΩX, can be lifted to a group object of any ∞-topos, so if X is in addition a connected space, there is an equivalence between objects of any ∞-topos with an ΩX-action, and objects with an X-coaction (where X is a coalgebra via the usual diagonal map). This is a generalization of the classical equivalence between G-spaces and spaces over BG for G a topological group.
