" The recent tragedies of Hurricanes Harvey and Irma, together with earlier extreme events such as Hurricanes Katrina and Sandy, has raised the question whether the apparent increasing severity of such events can be attributed to the human influence on greenhouse gas warming. Dr. Emanuel will review the growing consensus that the incidence of the…
Sums-of-squares certificates define a hierarchy of relaxations for polynomial optimization problems which are parametrized with the degree of the polynomials in the sums-of-squares representation. Each level of the hierarchy generates a lower bound on the true optimal value, which can be computed in polynomial time via semidefinite programming, and these lower bounds converge to the…
As one of the oldest nonlinear PDE systems, the compressible Euler equations has been studied by many outstanding mathematicians. However, some basic questions, such as the global existence of classical solution v.s. finite time blowup, are still open even in one space dimension. In this lecture, we will report our recent progress in this direction, including a complete understanding on…
This tutorial will provide an introduction to nonlinear optimization. We will begin by presenting a general constrained nonlinear program, defining concepts like local versus global optimality, and examining the differences between convex and nonconvex problems. We will then present the Karush-Kuhn-Tucker optimality conditions. Students who have taken OR 706 (Nonlinear Programming) will find this tutorial…
Earlier Helme Guizon and Rong have defined chromatic graph homology complex for a graded commutative algebra, and it is easy to extend the definition to graded commutative DG algebra. One of important applications, considered earlier in our joint work with Radmila Sazdanovic, is to the case of an algebra computing cohomology of a manifold, such…
Algebraic geometry and combinatorics play an important role in the analysis of phylogenetic models. I will give a brief overview of toric geometry. Then I will introduce a particular phylogenetic model, called the Cavander-Farris-Neyman model with a molecular clock, and I will discuss how we can study this model from the point of view of…
Why do atherosclerotic plaques only occur at specific sites in the arteries? Does the surface geometry of the brain affect the way waves move through the cortex? These questions and many others in the physiological sphere contain implicitly a real difficulty for modellers. How do we contend with the multiple scale lengths. Plaques are quite large compared to cells making…
Retailers typically use assortment planning to maximize store profits given product characteristics. We study the manufacturers' price-setting interactions and how these can be manipulated by the retailer's assortment strategy. We show that constraining the breadth of the assortment has two main effects on retailer profits: first, a larger assortment may intensify competitive pressure and decrease…
Circadian (~24-hour) rhythms offer one of the clearest examples of the interplay between different levels of nervous system organization, with dynamic changes in gene expression leading to daily rhythms in neural activity, physiology, and behavior. The main output signal of the master circadian clock in mammals has long been believed to be a simple day/night…
Heegaard Floer theory is a suite of invariants for studying low-dimensional manifolds. In the case of punctured torus, for instance, this theory constructs a particular algebra. And, the invariants associated with three-manifolds having (marked) torus boundary are differential modules over this algebra. This is structurally very satisfying, as it translates topological objects into concrete algebraic…
Investigating through mathematical modelling the complex chemistry in cells has grown in the research community over the past ten years. However trying to understand the relationship between cellular (microscale) and larger scale lengths such as the vasculature upstream of the capillary bed has proved a more difficult task. The tutorial (if you want to call…
Semilinear control systems in infinite dimensional Banach spaces are the natural framework for the description of several control problems governed by PDEs. In many models, some constraints for the state of the system may be present. In this talk, a Mayer problem associated to such systems will be discussed. In particular, a simple proof of…
The numerical range has been studied extensively in linear algebra and analysis. We will define and discuss properties of the numerical range, then show every numerical range invariant under rotation is associated with a matrix with nice structure.
The $n$-associahedron is a well-known $n$-dimensional polytope whose vertices are labeled by triangulations of an $(n+3)$-gon with edges given by diagonal flips. The $n$-cyclohedron is defined analogously using centrally-symmetric triangulations of a $(2n+2)$-gon, or, modding out by the symmetry, triangulations of an $(n+1)$-gon with one orbifold point. The polytopes can be realized in such a…
The pharmaceutical industry has increasingly embraced mathematical modeling as a method to predict drug-body interactions. In particular, issues of drug toxicity and drug-drug interactions impact the drug development and approval process. Initially begun as a private-public partnership, DILIsym Services is the leading software platform to help inform issues relating to drug-induced liver injury (DILI) by…
Self-organized behaviors are commonly observed in nature and human societies, such as bird flocks, fish swarms and human crowds. In this talk, I will present some celebrated mathematical models, with simple small-scale interactions which lead to the emergence of global behaviors: aggregation and flocking. I will discuss the models in different scales: from microscopic agent-based…
In many practical control problems (e.g. in the power grid/traffic flow) agents are not able to effectively coordinate their actions. One classical method proposed by economists for solving such decentralized control problems is known as the ficticious play algorithm. This talk will discuss some recent work which establishes converge rates for ficticious play. In particular,…
We develop a novel, data-driven approach to deal with appointment sequencing and scheduling in a multi-server system, where both customer punctuality and service times are stochastic. Our model is calibrated using a data set of unprecedented resolution, gathered at a large-scale outpatient oncology practice. This data set combines real-time locations, electronic health records and appointments…