affiliation  Duke University, Mathematics Dept. [map] 
title  Assistant Research Professor 
mentors  Eric VandenEijnden (PhD Advisor), Jonathan Mattingly, Mike Reed 
interests  probability theory, functional analysis, numerical analysis, applications to biology, physics, chemistry, etc. 
research area  analytical and numerical aspects of large deviation theory 
curriculum vitae  [pdf] 
research statement  [pdf] 
publication list  [pdf] 
recent work  Minimum Action Curves for Degenerate Finsler Metrics  Existence and Properties 
Rare Transition Events in Nonequilibrium Systems with StateDependent Noise: Application to Stochastic Current Switching in Semiconductor Superlattices  
heymann [at] math [dot] duke [dot] edu 
year  title 
occasion  download 
2010  Computation, Existence and Properties of Maximum Likelihood Transition Curves, TU Dresden  job talk  [slides engl.] [slides ger.] 
2009  The Geometric Minimum Action Method  A Least Action Principle on the Space of Curves, IPAM, UCLA: Rare events in highdimensional systems  conference  [poster] 
2007  Ph.D Thesis Defense Talk, New York University  thesis defense  [slides] 
year  title  area  download 
2010  Minimum Action Curves for Degenerate Finsler Metrics  Existence and Properties, Springer Lecture Notes of Mathematics  geometry / prob. theory  (at Springer) 
I study a class of action functionals on the space of unparameterized oriented rectifiable curves in R^{n}. The local action is a degenerate type of Finsler metric that may vanish in certain directions, thus allowing for curves with positive Euclidean length but zero action. Given two sets A_{1} and A_{2}, I develop criteria under which there exists a minimum action curve leading from A_{1} to A_{2}. I then study the properties of these minimizers, and I prove the nonexistence of minimizers in some situations. Applied to a geometric reformulation of the quasipotential of large deviation theory, my results can prove the existence and properties of maximum likelihood transition curves between two metastable states in a stochastic process with small noise. See my research statement for more details.  
2010  Rare Transition Events in Nonequilibrium Systems with StateDependent Noise: Application to Stochastic Current Switching in Semiconductor Superlattices  theo. phys. / num. math.  [paper] 
Using recent mathematical advances, a geometric approach to rare noisedriven transition events in nonequilibrium systems is given, and an algorithm for computing the maximum likelihood transition curve is generalized to the case of statedependent noise. It is applied to a model of electronic transport in semiconductor superlattices to investigate transitions between metastable electric field distributions. When the applied voltage V is varied near a saddlenode bifurcation at V_{th} , the mean life time <T> of the initial metastable state is shown to scale like log<T> ~ V  V_{th}^{3/2} as V →V_{th} .  
2009  The sources of rare transitions in continuoustime Markov jump processes, (in preparation)  prob. theory  
Chemical reactions in a test tube or in a biological cell in which one wants to keep track of the concentrations of various types of molecules are often modelled as a continuoustime Markov process. In the limit as the total number of molecules goes to infinity, transitions from one stable state into another become rare, and large deviation theory tells us the transition path of maximum likelihood. Observe that if every reaction occurs at its expected rate then the system stays in its stable state, and thus transitions can only occur if at least one reaction occurs at a frequency that is different from its expected rate. In this paper I develop a method for finding those reactions that occur at frequencies that are most likely to differ the most from the expected reaction rates. These reactions are thus most likely to be responsible for the transition and can therefore be considered the weakest link in the reaction network. See my research statement for more details.  
2008  The geometric minimum action method for computing minimum energy paths, Jour. Chem. Phys. 128, 061103, 2008  chem. phys. / num. math. 
[paper] 
This short paper introduces the gMAM algorithm to the chemistry community. As an application, we compute the maximum likelihood transition curve of the Müller potential (a twodimensional SDE).  
2008  The Geometric Minimum Action Method: A Least Action Principle on the Space of Curves, Comm. Pure Appl. Math. 61.8, 10521117, 2008  prob. theory / num. math. 
[paper] 
FreidlinWentzell theory of large deviations for the description of the effect of small random perturbations on dynamical systems is exploited as a numerical tool. Specifically, a numerical algorithm is proposed to compute the quasipotential in the theory, which is the key object to quantify the dynamics on long time scales when the effect of the noise becomes ubiquitous: the equilibrium distribution of the system, the pathways of transition between metastable states and their rate, etc. can all be expressed in terms of the quasipotential. We propose an algorithm to compute these quantities called the geometric minimum action method (gMAM) which is a blend of the original minimum action method (MAM) and the string method. It is based on a reformulation of the large deviations action functional on the space of curves which allows one to easily perform the double minimization of the original action required to compute the quasipotential. The theoretical background of the gMAM in the context of large deviations theory is discussed in detail, as well as the algorithmic aspects of the method. The gMAM is then illustrated on several examples: a finitedimensional system displaying bistability and modeled by a nongradient stochastic ordinary differential equation; an infinitedimensional analogue of this system modeled by a stochastic partial differential equation; and an example of a bistable genetic switch modeled by a Markov jump process. This is joint work with Eric VandenEijnden.  
2007  Pathways of maximum likelihood for rare events in nonequilibrium systems, Application to nucleation in the presence of shear, Phys. Rev. Letters 100.14, 140601, 2007  physics / num. math. 
[paper] 
This short paper introduces the gMAM algorithm to the physics community. As an application, the gMAM is applied to predict the transition pathway in a bistable stochastic reactiondiffusion equation (an SPDE) in the presence of a shear flow. We also analyze how the shear intensity influences the mechanism and rate of the transition.  
2007 
PhD Thesis: The Geometric Minimum Action Method: A Least Action Principle on the Space of Curves 
prob. theory / num. math. 
[thesis] [talk] 
The core of my thesis is the CPAM paper above. Additional material can be found in Chapters 5 and 6: In Chapter 5 it is shown how to adjust the gMAM algorithm to deal with constraints or penalty terms associated to the endpoint of the curve (with an application to finance: the valuation of equity index options). In Chapter 6 we give an introduction to the field of synthetic biology, with special focus on possible uses of large deviations theory; and we develop a tool to detect the sources of instabilities in (genetic) networks.  
2002  A model for the online time of network users  statistics  [paper] 
A model is proposed to describe individual behavior of internet users, and using a maximum likelihood method, the model parameters are fitted to a dataset consisting of 2 million connections to the Bell Labs network. The model can then be used to generate artificial internet user traffic that simulates human behavior, for example to test new designs of server mechanisms. This is joint work with Mark Hansen.  
2002  A new set of sound commands for R; Sonification of the HMC algorithm, ASA Proceedings 2002, Statistical Computing Section, 14391443  statistics  [paper] 
In this paper that accompanied my talk at the ASA conference, I present the R sound package to the public. As a fun example, we show how we use these commands to tune a parameter in the Hybrid Monte Carlo algorithm. This is joint work with Mark Hansen.  
2002  The R sound package: A new set of commands for using sound under R  programming  [manual] [package] 
The statistical programming and data analysis language R, the popular freeware clone of the commercial language SPLUS, is by itself not suited for using sound files in any way. With this package I provide the R community with a set of additional commands that enables them do load, save, listen to, and manipulate wave files in several ways. The package can be downloaded from CRAN, the official website of the the R development team (click on "CRAN", select a mirror page, then click on "packages", and look for the package "sound"). This work was done under the supervision of Mark Hansen and David James.  
2002  The Stieltjes convolution and a functional calculus for nonnegative operators  func. anal.  [paper] 
In this paper I present an approach to the multidimensional distributional Stieltjes transform that allows me to define a
convolution operation on a class of Stieltjestransformable distributions. As an application, I develop a functional calculus for nonnegative operators with which one can easily prove a variety of operator equations, even for noncommuting operators and on noncomplex Banach spaces. Two representation theorems that identify my classes of distributions as finite sums of derivatives of functions are essential throughout this paper.  
2001 
Diploma thesis: Fractional powers of operators, and their applications  funct. anal.  [paper] 
In my diploma thesis I developed a functional calculus for nonnegative operators on a Banach space, and show that it can be used to define fractional powers of such operators, and a family of bounded operators that approximate these fractional powers. This family of operators can then be used to generalize a result on the Abelian mean ergodic theorem from the class of generators of bounded semigroups to general nonnegative operators. 

 
Please
download the poster of my exhibition for my students' linear
algebra semester project "3D graphics".
[pdf] In 2005 I was nominated for the NYUwide "Dean's Outstanding Graduate Teaching Award". 