Tuesday, January 22, 2008 at 3:00 PM in HA 330
Van Savage, Harvard Medical School
New mathematical models for understanding physiological and ecological systems
It has long been known that metabolic rate, heart rate, and lifespan scale in a systematic and inter-related way with body size and temperature. These scaling relationships hold over an astronomical range in body size (~21 orders of magnitude) and across taxonomically diverse organisms that live in a myriad of environments. Moreover, these relationships for body mass are usually well approximated by power laws with exponents that are simple multiples of 1/4, and for body temperature by exponential Boltzmann-Arrhenius factors. I will describe a model to explain these relationships that focuses on the cardiovascular system and the kinetics of biochemical reactions. I will also discuss how finite-size corrections and asymmetric branching can refine the original model’s predictions. I will then present my work that shows how these scaling relationships can be used to examine critical physiological and ecological processes. At the physiological level, I will discuss models to explore, for example, tumor growth dynamics, cell size, and why an elephant sleeps much less than a mouse. At the ecological level, I will outline a trait-based framework to investigate the effects of fluctuating environments on ecosystems and the effects of temperature on predator-prey interactions. Together, these have the potential to gauge the impact of climate change on ecosystem dynamics and stability.
Van Savage is a math candidate. An online version of his CV can be found at: http://fontana.med.harvard.edu/www/Documents/VanSavage/Van%20Savage/cv.htm
Wednesday, January 23, 2008 at 3:00 PM in HA 330
Hongchao Zhang, IMA, University of Minnesota
An Active Set Algorithm for Box Constrained Optimization
This talk focuses on large-scale optimization algorithms for box constraint optimization. Such problems arise in various fields including optimal control, variational inequalities, multiplier methods as well as in many real applications. In this talk, an overview of recent developments on both theoretical and computational results in this field is given. In particular, a new active set algorithm (ASA) with strong local and global convergence properties is introduced. In addition, a brief review of the nonlinear conjugate gradient methods is also included. An
implementation of ASA based on the recent conjugate gradient algorithm CG_DESCENT and a cyclic Barzilai-Borwein algorithm is given. Numerical experiments are presented using box constrained problems in the CUTEr and MINPACK-2 test problem libraries.
Refreshments at 2:30 pm in HA 243.
Dr. Zhang is a candidate for the General Math position.Friday, February 1, 2008 at 4:00 PM in HA 330
Clayton Webster, Sandia National Labs
A Dimension-Adaptive Sparse Grid Stochastic Collocation Technique for Partial Differential Equations with High-Dimensional Random Input Data
This talk will propose and analyze a dimension-adaptive (anisotropic) sparse grid stochastic
collocation method for solving partial differential equations with random coefficients and forcing
terms (input data of the model). These methods have proven to have dramatic impact on several
application areas, including statistical mechanics, financial mathematics, bioinformatics, and other
fields that must properly predict certain model behaviors. The method consists of a Galerkin
approximation in the space variables and a collocation, in probability space, on anisotropic sparse
tensor product grids utilizing either Clenshaw-Curtis or Gaussian knots. Even in the presence of
nonlinearities, the collocation approach leads to the solution of uncoupled deterministic problems,
just as in sampling-based methods, such as Monte Carlo. This talk includes both a priori and a
posteriori approaches to adapt the anisotropy of the sparse grids to each given problem.
This talk will also provide a rigorous convergence analysis of the fully discrete problem and
demonstrate strong error estimates for the solution using Lq norms. In particular, our analysis
reveals at least an algebraic convergence with respect to the total number of collocation points.
The derived estimates are then used to compare the efficiency of the method with other ensemble-
based methods. Numerical examples illustrate the theoretical results and are used to compare this
approach with several others, including the standard Monte Carlo. In particular, for moderately
large dimensional problems, the sparse grid approach with a properly chosen anisotropy is very
efficient and superior to all examined methods.Tuesday, February 5, 2008 at 3:00 PM in HA 335
Molly Fenn, University of Massachusetts Amherst
Platonic Solids and Four Color Theorem
Graph theory is an area of mathematics that requires very little introduction before some cool problems can be understood and solved. We will start from the basic definitions and examine proofs (and false proofs!) of two historically interesting problems, classifying all regular polyhedra and coloring maps with as few colors as possible.
Molly Fenn is a candidate for Teaching Assistant Professor Position.Thursday, February 7, 2008 at 3:00 PM in HA 335
Alina Duca, Vassar College
The abc Conjecture
It is often the case in number theory that a reasonable question
is very easy to ask yet extremely difficult or even impossible
to answer. The most famous example, of course, is Fermat’s Last Theorem, the proof of which eluded mathematicians for more than 300 years.
In recent years a problem has arisen for which the search for
a proof might turn out to be as turbulent as Fermat’s Last Theorem. The abc conjecture was formulated in 1985 by J.Oesterle and D.Masser. It is very easy to state, yet nonetheless has far-reaching implications throughout number theory, and it is probable that if a proof is found, it too will have deep
consequences beyond the conjecture itself.
In mathematics it is often possible to translate a problem from
one area to another, in the hope that the resulting question is
easier to tackle and offers insight for the original. One of
the most fruitful analogies in mathematics is that between
the integers and the ring of polynomials. We will discuss first
the abc conjecture for polynomials, then we will see how this
theorem can be translated into the abc conjecture about
ordinary integers.Friday, February 8, 2008 at 2:30 PM in HA 335
Edward Richmond, UNC Chapel Hill
Recursive structures in the cohomology of flag varieties
Let Gr(r,n) denote the Grassmannian of r-dimensional subspaces in some fixed n-dimensional complex vector space. In this talk, we will look at some results about Schubert varieties in Gr(r,n) and their connections to cohomology. In particular, we ask under what conditions is a product of Schubert classes nonvanishing in cohomology (sometimes called Schubert calculus). The answer to this question can be given in terms of a recursive algorithm known as Horn recursion. We will also look at applications of Schubert calculus to enumerative geometry and some generalizations of Horn recursion to partial flag varieties.Thursday, May 1, 2008 at 4:00 PM in HA 307
Blaine Lawson, SUNY Stony Brook
Dirichlet Duality and the Non-linear Dirichlet Problem
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