PhD proposal

Ling-Chen Bu

January 28, 2016

Abstract

text

Quote Samuelson and Hemingway here. 1

1We will keep updating this proposal script in http://www-levich.engr.ccny.cuny.edu/

webpage/lingchen/

1

Contents

1 Introduction 61.1 On modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.2 The big picture: math, physics and others . . . . . . . . . . . . . 6

1.2.1 example: contagion process . . . . . . . . . . . . . . . . . 71.3 Mathematical physics . . . . . . . . . . . . . . . . . . . . . . . . 71.4 Mathematical physics . . . . . . . . . . . . . . . . . . . . . . . . 71.5 Before Descartes and Newton/Leibniz . . . . . . . . . . . . . . . 71.6 Complex system . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.7 research plans/interests/proposals/briefs . . . . . . . . . . . . . . 81.8 Rules/Equations and Data/Observables . . . . . . . . . . . . . . 81.9 Outline of the proposal . . . . . . . . . . . . . . . . . . . . . . . . 81.10 geometry and physics . . . . . . . . . . . . . . . . . . . . . . . . 81.11 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.12 modeling and stochastic dynamical system . . . . . . . . . . . . . 8

2 Dynamical system 92.1 existence, uniqueness of ODE(PDE), dynamical system . . . . . 102.2 Stability theory of dynamical system . . . . . . . . . . . . . . . . 10

2.2.1 Local result around fixed points . . . . . . . . . . . . . . . 102.2.2 global result, Lyapunov function(energy function?) . . . . 10

2.3 Geometric/Topological theory of dynamical system . . . . . . . . 102.4 Conley Index Theory, fundamental theorem of dynamical system 102.5 Ergodic theory of dynamical system . . . . . . . . . . . . . . . . 102.6 Topological entropy . . . . . . . . . . . . . . . . . . . . . . . . . . 122.7 DS and (equilibrim? non equilibrim) statistical mechanics . . . . 132.8 LaSalle invariant principle . . . . . . . . . . . . . . . . . . . . . . 132.9 Hilbert sixth problem . . . . . . . . . . . . . . . . . . . . . . . . 132.10 chaotic dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.10.1 Quantum Chaos . . . . . . . . . . . . . . . . . . . . . . . 132.10.2 SRB measure, and Lyapunov exponents . . . . . . . . . . 132.10.3 chaos control(OGY) . . . . . . . . . . . . . . . . . . . . . 132.10.4 Existence of Lorenz Attractor . . . . . . . . . . . . . . . . 132.10.5 numerical problems of chaotic dynamic systems . . . . . . 13

2.11 global attractor of infinite dimension dynamical system/PDE(PDEproject, review/report) . . . . . . . . . . . . . . . . . . . . . . . . 14

2.12 Random perturbation of dynamical system, stochastic dynamicalsystem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.13 Embedding of Dynamical System: work of Robinson and Stark . 14

3 Inverse Problems of Dynamical Systems (Time series analysisincludes more stuff, statistics, information theory of DS, trans-fer entropy) 143.1 Reconstruction of Dynamical Systems, Equation Free, Takens

Embedding Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 143.2 Reconstruction of Vector Fields of Dynamical Systems . . . . . . 143.3 State space reconstruction; embedding theorem; nonlinear time

series analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2

3.4 Parameter estimation/identification/reconstruction/reverse engi-neering of dynamical system; or state space reconstruction . . . . 163.4.1 for discretizing approach: optimization . . . . . . . . . . . 16

3.5 dynamical system and harmonic analysis(Bourgain) . . . . . . . 173.6 Ergodic theory(dynamical system) and number theory/combinatoric

number theory(TaoSzemerdi) . . . . . . . . . . . . . . . . . . . . 173.7 DS and Teichmuller space(curtis @harvard and Avila),CUNY a

big group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.8 Reverse problem of dynamical system; reconstruction of vector

fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.9 Information theory of dynamical system(”applied ergodic theory

of dynamical system”) . . . . . . . . . . . . . . . . . . . . . . . . 183.9.1 Information theory . . . . . . . . . . . . . . . . . . . . . . 18

3.10 control theory (on network) . . . . . . . . . . . . . . . . . . . . . 18

4 Causality 19

5 From atomist to : ABM(agent based model) or IBM(individualbased model) versus EBM(equation based model) or state vari-able/space model or kinetic model 205.1 Coupled Map Lattice and universality . . . . . . . . . . . . . . . 205.2 modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215.3 ”agent-based model” of hydrodynamics . . . . . . . . . . . . . . 215.4 ecology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215.5 transmission disease . . . . . . . . . . . . . . . . . . . . . . . . . 215.6 neuroscience . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215.7 Data-driven coupled particle-continuum model . . . . . . . . . . 21

6 Statistical Physics: Some Topics 226.1 Kinetic theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226.2 Ising model, Maximum Entropy and Graphical Models . . . . . . 226.3 Statistical mechanics and phase transition; catastrophe and at-

tractor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226.4 Boltzmann distribution, Gibbs measure . . . . . . . . . . . . . . 226.5 notes on statistical physics . . . . . . . . . . . . . . . . . . . . . . 226.6 notes on large deviation . . . . . . . . . . . . . . . . . . . . . . . 226.7 notes on stochastic /process and dynamical system; theory and

time series analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 226.8 coarse grain/master equation/mean field . . . . . . . . . . . . . . 226.9 master equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236.10 mean field approach . . . . . . . . . . . . . . . . . . . . . . . . . 23

7 Information theory and statistical mechanics 23

8 Dynamical System and Statistical Physics 23

9 Phase transition, collective dynamics 23

3

10 Complex Network or Network Science 2410.1 Spatial networks and temporal networks . . . . . . . . . . . . . . 2410.2 scale free distribution v.s. network structure(always imply fractal

structure?) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

11 Transmission disease, and more general contagion dynamics;epidemic spreading; spreading dynamics; contagion process 2511.1 in Neuroscience/brain . . . . . . . . . . . . . . . . . . . . . . . . 2511.2 Compare percolation and transimission disease on network . . . . 2511.3 Applications: transmission of information; in stock market; in

brain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

12 Notes on Monte Carlo methods 2612.1 inverse problem of Ising model by Monte Carlo . . . . . . . . . . 2612.2 review of history of Monte Carlo . . . . . . . . . . . . . . . . . . 2612.3 Latin hypercube sampling(LHS) and applications in ODEs, mas-

ter project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2612.4 ref . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

13 Complex network(random graph) 2713.1 community structure detection . . . . . . . . . . . . . . . . . . . 2713.2 reaction diffusion process on network . . . . . . . . . . . . . . . . 2713.3 Ising model on complex network . . . . . . . . . . . . . . . . . . 2713.4 Spatial networks . . . . . . . . . . . . . . . . . . . . . . . . . . . 2713.5 Network induced chaos . . . . . . . . . . . . . . . . . . . . . . . . 2713.6 Proposal: origin of scale-free complex network (evolution) . . . . 27

14 Compressive sensing 2814.1 geometry and embedding . . . . . . . . . . . . . . . . . . . . . . 2814.2 RIPless approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 2814.3 Statistical Physics and Compressive Sensing . . . . . . . . . . . . 28

15 Graphical model(statistics, statistical mechanics, machine learn-ing) 2915.1 graphical lasso Inference . . . . . . . . . . . . . . . . . . . . . . . 29

16 information theory and maximum entropy method 2916.1 numerical algorithms: monte carlo(Metropolis algorithm) . . . . 29

17 Neuroscience and brain 3017.1 degeneracy and redundancy . . . . . . . . . . . . . . . . . . . . . 3017.2 Theoretical or computational neuroscience . . . . . . . . . . . . . 3017.3 network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

18 human system as a complex system: economics and finance;spatial economics(urban economics; ecogeography) 3118.1 review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3118.2 a complex system perspective . . . . . . . . . . . . . . . . . . . . 3118.3 stock market as particle system . . . . . . . . . . . . . . . . . . . 3118.4 phase transition of economic system: from lower level to higher

level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4

18.5 Econophysics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

19 Discussion and Conclusion/Summary 32

20 Appendix Projects 3320.1 (Global)attractor of dynamical system(PDE, with Prof Parshad) 3320.2 Parameter estimation of (nonlinear) dynamical systems (by comp-

pressive sensing) . . . . . . . . . . . . . . . . . . . . . . . . . . . 3320.2.1 Ambiguity of parameters and network structure; Kro-

necker product and hierarchy structure? . . . . . . . . . . 3320.3 Convex geometry(probably related with compressive sensing), un-

dergraduate thesis . . . . . . . . . . . . . . . . . . . . . . . . . . 3320.4 Shanghai University thesis: dynamical system of transimission

disease/reaction-diffusion process(just ODE, PDE also major branch) 3420.5 Mathematics(dynamical systems) of transmission disease . . . . . 3420.6 introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3420.7 LaSalle invariant set principle . . . . . . . . . . . . . . . . . . . . 3420.8 persistence of dynamical system . . . . . . . . . . . . . . . . . . . 3420.9 project: graph approach of construction of global Lyapunov func-

tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3420.10Parameter sensitivity analysis of ODE system . . . . . . . . . . . 34

21 Appendix A: Notes of some fundamental and significant com-mon sense of mathematics(keep updating) 3521.1 calculus and calculus(ordinary differential) equations . . . . . . . 3521.2 measure theory, integral and differentiation . . . . . . . . . . . . 3521.3 common senses . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

22 Appendix B: Brief personal statement 3622.1 Family . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3622.2 Academic experiences . . . . . . . . . . . . . . . . . . . . . . . . 3622.3 Courant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3622.4 Teaching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3622.5 Courses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3622.6 GPA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

23 notes on probability, ?random / stochastic process), statisticsand information theory 3723.1 entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

24 Notes based on/inspired by lectures/seminars 37

5

1 Introduction

2

from Hamiltonian(Laplacian) mechanics to...,Cartan’s work on exterior algebra(?) differential form, general relativity,fundamental research may have applications in elsewhere.. for example

Whitney embedding theorem in 1930’s, been applied to state space(or phasespace) reconstruction of dynamical system.

This proposal is centered on(?) dynamical system and related topics. mathstructuredynamical system: long time evolution ofthe discussion may be somewhat idiosyncratic.graph of chapters/parts:

dynamical system; statistical physics; complex network(statistical physics of(random) graphs);

try to be cutting-edge, meanwhile make it readable for broadest(more) au-diences, try to make it self-contained, put the(my own) notes/projects intoappendix.

We try to make balance between fundamental issues and more recent devel-opments.

from the philosophy or epistemology, all math models: connections, geome-try, rigid body.

in connections, agent based model: statistical mechanics, the statistical me-chanics approaches are introduced in collective dynamics(2010, Chialvo),

agent based modelA unified mathematical framework of agent based model and state variable

model will inspire/ applicationdeeper/ understanding of real world.

state variable model of statistical mechanics?Klein, continue transformation group,Lie, 1983,Langlands conjecture

1.1 On modelling

rigid body movement, fluid movement,on different motivations

1.2 The big picture: math, physics and others

math and others: different kinds of math:

inner connection of math and physics: random matrix theory and Riemannconjecture, Montgomery, Freeman Dyson.

2hh

6

1.2.1 example: contagion process

quantum physics, wave functionsemigroup: PDE, stochastic process,

1.3 Mathematical physics

Roughly speaking, mathematical physics is mathematical understanding of physics;phycists

The remarriage of mathematics and physics, JaffeInternational Association of Mathematical Physics was founded in 1976.1964 Haag algebraic quantum field theory,1965, Communications in Mathematical Physics: The mission of Communi-

cations in Mathematical Physics is to offer a high forum for works which aremotivated by the vision and the challenges of modern physics and which at thesame time meet the highest mathematical standards.

1.4 Mathematical physics

Jaffe(Harvard), Haag’s theorem, quantum field theory,MacKay(Havard), Spencer(IAS),3

geometry of time, Einstein, Lorentz transformation,...string theory 4

In 1926, SchrodingerMathematics,Representation Theory1994,Seiberg-Witten equation, developed gauge theory,expositorymainly in chronological order, but highlight

1.5 Before Descartes and Newton/Leibniz

1637, Descartes established analytic/Coordinate geometry,1707, birth of Euler,Poincare,

we would avoid a big part of mathematics, modern number theory, algebraicgeometry,

the mathematical foundation of physics,Spencer’s research on constructive quantum physics.Mackay, Harvard, equation, symmetry?Weyl, gauge theory, geometry and electromagnetic theory,1950’s Yang-Mills, Gauge theory,in the last forty year, quantum field theory, quantum mechanics, string the-

ory, super symmetry, mirror symmetry,

3Communications in Mathematical Physics, from 19654Partly inspired by Juan Maldecena’s lecture at City College of New York in October 2015.

7

1.6 Complex system

Bak: How nature worksLess is more(anderson)Thom(math), Smale

1.7 research plans/interests/proposals/briefs

(cognitive) neuroscience/brain related,mathematical or neurological explanation of deep learning()machine learning),

ABM v.s. EBM: hydrodynamics from statistical point view(1954 Morrey;recently Ruelle);

information theory of dynamical system from information theory of statisti-cal physics?

how the change/perturbation of microstate change the structural stabilityof corresponding dynamical system,

the topology of dynamical system, geometrical theory of dynamical system;on the other side, the evolution of a dynamical system on a geometric vari-

able, for example, the curvature,

1.8 Rules/Equations and Data/Observables

1.9 Outline of the proposal

This proposal is structured as follows:

1.10 geometry and physics

Yau(1949-), 1969, Berkeley, 1970 general relativity;Calabi-Yau manifold:Hilbert’s axioms, 1899

1.11 Topology

Poincare’s paper Analysis Situs in 1984historical notes of topology, for example Dieudonne(1989) ending on 1960,

and (1994) ending on 1950, James(1999)[10]

1.12 modeling and stochastic dynamical system

8

Figure 1: long

2 Dynamical system

invention of calculus by Newton and Leibniz may be the start point of modernmath.

we will discuss some topics theoretical, applied and computational aspectsof dynamical system. (numerical method of chaotic system by Yorke and etc.)

”We maintain a wide scope, but given the wealth of material on this subject,we obviously cannot aim for completeness. ” Takens, 2010 [4].

one dimension:

there are essential differences between continuous and discrete dynamicalsystems, discrete: Logistic map, could be chaotic,

continuous case: due to Poincare-Bendixson Theorem, there’s no chaoticdynamics in dimension

textbooks:[1](1978, Abraham, Marsden), reviewed by Sternberg[20] mathematical struc-ture underlying mechanics and areas of pure mathematics stimulated by prob-lems in mechanics,

vector fields as dynamical systems; vector fields as differential operators,From 1950’sentropyLyapunov exponentdifferent kinds of dynamical system (Strogatz category figure here),Pesin’s identity(1977)

9

It’s impossible here to illustrate the broad area of dynamical system andrelated topic, we discuss with some focus and try to make( essential connection,) .

”The world is about changing(variants) and connecting(connections).” Max-ist philosophy.

Poincare section: photograph is (part of) P.s of real worldgeneric propertyrandom perturbations of dynamical system: [13](1988, Kifer) [8](2012, Frei-

dlin and etc.),

2.1 existence, uniqueness of ODE(PDE), dynamical sys-tem

math v.s. model, whether well/-defined/posed, ill-posed,take example of DS project for example (bucket) 5

PDE example(textbook, fudan PDE course)

2.2 Stability theory of dynamical system

1982, Lyapunov,1902, Poincare,

2.2.1 Local result around fixed points

by manifold theorems,for arbitrary ODE system

2.2.2 global result, Lyapunov function(energy function?)

2.3 Geometric/Topological theory of dynamical system

Transversality, differential topology,geometric theory(and operator) of DS1982, PalisAbraham, Marsden, [1]: vector fields as... and ....

2.4 Conley Index Theory, fundamental theorem of dy-namical system

1978 Conley,advocated by Norton as the fundamental theorem of dynamical system,

(1995, Norton)

2.5 Ergodic theory of dynamical system

1931 von Neumann and Birkhoff1948 Wiener, Wintner, Harmonic analysis and ergodic theorem.1931, Birkhoff Ergodic Theorem [?].1941, Wiener, [26],

5the material here is based on/ mainly a project in DS course, http link

10

Ergodic theory of dynamical system: review paper by Young [28](1995)given a dynamical system, (X,φ, ρ), where the invariant measure ρ(x),6

information entropy: 1948, Shannon, a mathematical theory of communica-tion,1949, Shannon, The mathematical theory of communication,

in [16](1971), Ruelle and Takens first coined the term strange attractor,”Laudau-Lipschitz theory must be modified”

[15](2006, Ruelle)Young reviewed ergodic theory of dynamical systems [27](1994 ICM) [28](1995).E, Sinai and etc. discussed invariance measure for Burgers equation with

stochastic forcing [25](2000).

6the notes by Tim Austin(NYU) and ...(Warwick)

11

2.6 Topological entropy

We referred a brief review [2](2008) by Adler and etc. .

12

2.7 DS and (equilibrim? non equilibrim) statistical me-chanics

2.8 LaSalle invariant principle

and it’s application in transmission disease.

2.9 Hilbert sixth problem

polynomial vector field, (Smale 1998)

2.10 chaotic dynamics

2.10.1 Quantum Chaos

2.10.2 SRB measure, and Lyapunov exponents

2.10.3 chaos control(OGY)

7

2.10.4 Existence of Lorenz Attractor

2.10.5 numerical problems of chaotic dynamic systems

Gregori, Yorke, PRL, Physica D. series papers around from 1985 to 1995.

7this part is based on one project of dynamical system course(graduate level)

13

2.11 global attractor of infinite dimension dynamical sys-tem/PDE(PDE project, review/report)

Intuitively, imagine one trivial case, say wave equation dynamics on the planewith finite boundary(or similar but more illustrative )

8

Temam[23](1988)

2.12 Random perturbation of dynamical system, stochas-tic dynamical system

Intuitively and unrigorously, we could consider the dynamicsthe solution of stochastic dynamical system is a stochastic process,stochastic version of Takens’s embedding theorem

2.13 Embedding of Dynamical System: work of Robinsonand Stark

motivation of embedding among/between dynamical systems,one significant application of embedding of dynamical systems is Takens

embedding theorem and other versions of it. We will discuss in 3.1

3 Inverse Problems of Dynamical Systems (Timeseries analysis includes more stuff, statistics,information theory of DS, transfer entropy)

3.1 Reconstruction of Dynamical Systems, Equation Free,Takens Embedding Theorem

as discussed in

3.2 Reconstruction of Vector Fields of Dynamical Systems

observables of dynamical system,information theory of dynamical system;entropy(transfer, causation...),embedding theoremthey all deal with time series.coherent of low-dimensional dynamics and stochastic, high-dimensional dy-

namics. [7](1991, Casdagli)[6](1991, Casdagli) [5](1991, Casdagli)embedding between (phase spaces of) dynamical systems,motivation of Robinson’s embedding work is by Taken’s embedding theorem?Robinson:

all dissipative dynamical system could be embedded into three dimension dy-namical system.

8This part is mainly based on advanced PDE course by Prof Parshad in Clarkson Universityin spring 2014.

14

3.3 State space reconstruction; embedding theorem; non-linear time series analysis

1979, Cruthfield, thesis; 1980, PRL, Cruthfield and etc. [14] 1981, Takens.[21] [22] 1992, Sauer, Yorke 1999 and 2003 Stark, 2005, James Robinson,delayembedding theorem for infinite-dimensional dynamical system, 2011, Sugihara,And very recently 2015, Sejnowski,

2015 Gutman,problems: for piece-wise smooth dynamical system? sometimes, the dynam-

ical rules may change by stages, for stochastic dynamical system?choose embedding dimensionnumerical calculation of Lyapunov exponent,Timmer’s review (other papers?)[12](2004,Kantz and Schreiber)relationship among: embedding theorem, (applied) ergodic theory of dynam-

ical systems(ergodic theorem),1979, Cruthfield, undergraduate thesis, 9

1981(1980), Takens, proved... 1981, Aeryl, SIAM, 1991, Sauer, Yorke, 2003(2002),Stark, Huke, Huke’s note(2006?)

2011, Deyle, Sugihara, non-consecutive2015, Sejnowski, Delay differential analysis.

9We don’t get to see the thesis, this is due to May and Sugihara in 2011,cite.

15

3.4 Parameter estimation/identification/reconstruction/reverseengineering of dynamical system; or state space recon-struction

at least two methods:1, discretize the left hand side of differential equations/derivatives in differentialequations, the problem then would be an optimization problem for the generatedlinear system, we will discuss related optimization in appendix, one section, orhere?2, Takens’s embedding theorem, or lagged variables of a single time series,

more work of Aeyls, Sontag...Yu and etc. [29](2007)

3.4.1 for discretizing approach: optimization

application in Sporns PNAS paper(the first one that compares structural andfunctional network of brain ROIs)

16

3.5 dynamical system and harmonic analysis(Bourgain)

3.6 Ergodic theory(dynamical system) and number the-ory/combinatoric number theory(TaoSzemerdi)

Diophantine approximation,

3.7 DS and Teichmuller space(curtis @harvard and Avila),CUNYa big group

dynamics of geodesic flow,geometric problem,dynamical system and geometry

17

3.8 Reverse problem of dynamical system; reconstructionof vector fields

3.9 Information theory of dynamical system(”applied er-godic theory of dynamical system”)

Markov partition,probabilistic distribution

3.9.1 Information theory

1955, D. ter Haar(statistical physicists, Leiden): peopel are abusing concept ofinformation

3.10 control theory (on network)

Sontag,

Li, Guan-Rong Chen, around 1989.

Yang-Yu Liu, Barabasi, review of modern physics, aug, 2015.Timme’s series papers.question:

time variant/dependent systems? Say: x = A(t)x(t) rather than x = Ax(t).2015 YY Liu, Barabasi,RMP.

18

4 Causality

I would rather discover one causal law than be King of Persia. Democritus(460-370 B.C.)

Berkeley’s problem(2012,Sugihar, May),We referred [18] ()Scholarpedia, Anil Seth),Granger, 1969, 1980,Causation entropy2015, Sun, Bollt, SIAM,2015, Sun, Cafaro, Bollt, Entropy (above there’s section about information

entropy...transfer entropy...)

19

5 From atomist to : ABM(agent based model)or IBM(individual based model) versus EBM(equationbased model) or state variable/space model orkinetic model

mathematically/formally: cts.: functionif we assign network structure on nodes, agents, or particles,no matter the network structure is static with more specific networkdepend on the audience. 10

different extra structures are assigned to the naive particle system/agentbased system, say spatial structure

notice that it’s different with continuous models and discrete models, forexample, we may get

Network dynamics, for example, coupled map lattices, can present spatiallyextended systems, Actually if we consider this way: xn(t) could be written asx(n, t),

uncover the relationshipspatiotemporal patterns,particle system, say gases, can be considered as network whose interaction

is instantaneous.Proposal:

study the transition of11

motivations of comparing IBM and EBM:by comparing IBM and EBM, we can get deeper understanding of modellingprocess(or the objects we are dealing with);

know: when to use IBM or EBM, and how.

information perspective of IBM and EBM?multi-agent system: Ising model and statistical mechanics. (chaos of multi-

agent system?)originally, how differential equations were introduced into biology(ecology

and epidemiology), consider the number is a variable, (metaphysical)master equation, from statistical physics,

5.1 Coupled Map Lattice and universality

Kaneko [11]

10Private communication with Prof James Yorke.11The word ”agent” comes from academic fields such as biology or artificial intelligence and

control theory(say multi-agent system), while the word ”particle” may come from phyiscalsystems with less ”intelligence”; we will use different terminology under different contexts,but one goal is to analyze (all) in the same general framework, in the general case we will use”particle”. In Vicsek-type particles, there’s also ”intelligence”.(2013 RMP)

20

5.2 modelling

There’s no randomness in mathematics. This means: (consider) the descriptionby probability distribution as a space of all possible states/configurations, thenassign values on the configurations.

The logic of kinetic model, taking population dynamics or mathematicalepidemiology for example, we observe the micro ;

[17]

5.3 ”agent-based model” of hydrodynamics

1954, Morrey, On the derivation of the equations of hydrodynamics from statis-tical mechanics.

5.4 ecology

IBM in ecology usually also concerns spatial movement, interaction and distri-bution.

1988, Huston,...1999, Grimm, ten years of agent based modelling.

5.5 transmission disease

2000, JTB, Keeling.Yau, Charles Morrey and transmission disease on network.

5.6 neuroscience

models of neurons(ABM): in a short time window, we can consider the activi-ties/functions/dynamics of neurosystem as discrete impulse signals.

random variables(Ising model, Hopfield neural network model): (ABM) in-teracting particle system; moreover: what’s Langevin equation?(EBM of particlesystem?)

using approaches of Yau(1998) (Morreay 1954) to derive EBM of neurosys-tem from system of neurons(ABM).(what can we do with such model? Hydrid of ABM and EBM?Data? Timeseries analysis?)

5.7 Data-driven coupled particle-continuum model

21

6 Statistical Physics: Some Topics

12

Maxwell, Boltzmann, Gibbs,statistical physics v.s. statistical mechanics,statistical physics is from statisticsGibbs established statistical physics, start from such simple assumption:

every state in the phase space has same probability(of appearance?)Glauber dynamics, JMP, 1963Kawasaki, Phase Transitions and Critical Phenomena, 1972,

6.1 Kinetic theory

6.2 Ising model, Maximum Entropy and Graphical Mod-els

Ising model is one kind of ... particle system,formally we could define Ising model as follows:every spin is mathematically a random variable with a probabilistic distri-

bution,

6.3 Statistical mechanics and phase transition; catastro-phe and attractor

6.4 Boltzmann distribution, Gibbs measure

Boltzmann distribution is a probability distribution (wikipedia)

F (state)∞ e− E

kBT (1)

where kB is the Boltzmann constant.In a more general mathematical setting, Gibbs measure:

P (X = x) =1

Z(β)exp(−βE(x)) (2)

where x is the state in configuration space X, E(X) is the energy of correspond-ing state x. Z(β) is the partition function(which is made for the consistency ofthe probability distribution).

6.5 notes on statistical physics

6.6 notes on large deviation

6.7 notes on stochastic /process and dynamical system;theory and time series analysis

6.8 coarse grain/master equation/mean field

on neuroscience: from neuron population to continuous variable

12The discussion in this section intensively referred [19](2011,Shen).

22

6.9 master equation

Mathematical models in biology, population, transmission disease could be con-sidered as master equation of corresponding interacting particle system (?)

6.10 mean field approach

7 Information theory and statistical mechanics

Definition of entropy,ref:

1957 Jaynes.2008,Adom Giffin,PhD thesis,MAXIMUM ENTROPY: THE UNIVERSAL METHODFOR INFERENCE.

entropy of particle system(statistical physics) (MaxEnt) and entropy of dy-namical system?

8 Dynamical System and Statistical Physics

recall Birkhoff theorem:

Theorem 1. ghgh

9 Phase transition, collective dynamics

Viscek,Kelum, Bollt: differential geometry approach/curvature change to detect

phase transition.

23

10 Complex Network or Network Science

dynamical system could be useful/used for example in systhetic biology(Elowitz),More detailed in neuroscience?

10.1 Spatial networks and temporal networks

10.2 scale free distribution v.s. network structure(alwaysimply fractal structure?)

24

11 Transmission disease, and more general con-tagion dynamics; epidemic spreading; spread-ing dynamics; contagion process

the process of contact-reaction process is a very general process in nature andsociety.

We will discuss applications of contagion process in neuroscience and finan-cial system.

11.1 in Neuroscience/brain

Network diffusion accurately models the relationship between structural andfunctional brain connectivity networks

2015 Cell, Sporns, Cooperative and Competitive Spreading Dynamics on theHuman Connectome.

run percolation in brain?(BJ Qiongge)

11.2 Compare percolation and transimission disease onnetwork

11.3 Applications: transmission of information; in stockmarket; in brain

25

12 Notes on Monte Carlo methods

12.1 inverse problem of Ising model by Monte Carlo

algorithm:

1, initialize Ising model on network:A is the adjency matrix of given network, notice that A could be(is) asymmetric

12.2 review of history of Monte Carlo

discuss/mention important papers.Monte carlo is rarely efficient.(Newman), one motivation of Newman’s book

is about how to use Monte Carlo efficiently.

12.3 Latin hypercube sampling(LHS) and applications inODEs, master project

12.4 ref

1999, Newman, Barkema,Monte Carlo Methods in Statistical Physics.

26

13 Complex network(random graph)

interaction of network structure and dynamic processes, some scenarios, staticnetwork,

different communities have different focus/taste,community of physics(statistical physics): percolation, phase transition on

network,community of computer science: algorithm(graph): community structure

detection,mathematics: JAMS paper on complex network.from other applied fields: neuroscience, biology(systems biology).control community(engineering).since 1998 Watts, Strogatz and 1999 Albert Barabasi paper,one achievement is the explanation of network evolution rule. the gravity

model: where network growsproposal: stability, (global)stability of the system of or on large random

network(complex network), catastrophe,

13.1 community structure detection

NP hard? (paper)

13.2 reaction diffusion process on network

a very general frame/model for particle system or agent based model.what’s left: synchronization,

13.3 Ising model on complex network

13.4 Spatial networks

review on spatial network [3] (2011, Barthelemy)on urban system, neural network(section neuroscience),

13.5 Network induced chaos

label of thethe one to one map between arbitrary(?) network structure and point be-

tween [0, 1] or [a, b], without loss of generality, consider [0, 1], the hierarchicalstructure

characterize the dynamics of network growth,

13.6 Proposal: origin of scale-free complex network (evo-lution)

information perspectiveminimum energy rule2000’s research on river network

27

14 Compressive sensing

Sparse approximation13

14.1 geometry and embedding

14.2 RIPless approach

14.3 Statistical Physics and Compressive Sensing

13According to the review [9](2008, Holtz), Temlyakov cite located this first example ofsparse approximation.

28

15 Graphical model(statistics, statistical mechan-ics, machine learning)

Jordan(intro of graphical model): graphical models have roots in statistics,artificial intelligence, statistical mechanics...

Pearl(1982, BP: belief propagation)1957 Jaynes work shows that the probabilistic distribution in statistical me-

chanics such as Ising models could be derived independently from informationtheory in form.

together with the summary by giffin(2008),2006 Bialek’s series papers showed that we can assume the brain network as

Ising model(guess, not sure, read papers)

15.1 graphical lasso Inference

2008(2007) Biostatistics, Friedman,

16 information theory and maximum entropymethod

1957, Jaynes papers.

16.1 numerical algorithms: monte carlo(Metropolis algo-rithm)

put into appendix.applications of above theoritical discussioin.

29

17 Neuroscience and brain

introduce basic neuroscience and brain knowledges, and related mathematicalmodels.

Structual network, functional network,Eliquiz, 2005, PRL,Then naturally we ask:

how the models(for example, HH model of neurons) were established?are those models working fine? when will they be not working?

ideas/methods from math and physics,energy, information(flow), entropy(in the sense of both information theory

and dynamical system)network dynamics,coarse brain approach; ABM vs state space models; master equations,Ising modeldeep learning

17.1 degeneracy and redundancy

Measures of degeneracy and redundancy in biological networks, [24](1999,Tononi,Spornsand Edelman)

Yao Li,how brain evolves, development,

17.2 Theoretical or computational neuroscience

17.3 network

spatial network [3] (2011, Barthelemy)there are normally three kinds of ”neural networks”: one is the actually

network of neurons in neuroscience, one is the dynamical system model,i.e. H-Hmodel, of neuron(?) dynamics, one is each node represents a random variable,which is essentially a probabilistic graph model(cite PearlJordan, section graphmodels),

which model is better? We suggest that we should try to avoid interests ofspecific communities, say mathematics, physics or statistics.

30

18 human system as a complex system: eco-nomics and finance; spatial economics(urbaneconomics; ecogeography)

metastability, phase transition,

18.1 review

18.2 a complex system perspective

combine human socioeconomic systems and urban economics, agent-based model,

18.3 stock market as particle system

notice one essential difference of stock market and ...

collective behavior of agents/particles,

reconstruction of dynamical system by time series,

18.4 phase transition of economic system: from lower levelto higher level

starting point:1, consider human socioeconomic system as a particle system:the

2, how to consider the acceleration of technology improvement macroscopicobservable variables?

18.5 Econophysics

1998 Sornette D., Johansen A., A hierarchical model of financial crashes, PhysicaA

2000 Mantenga R.N., Stanley H.E., An Introduction to Econophysics. Cor-relations and Complexity in Finance,

Price Dynamicsfinancial system stability, resilience Farmer(oxford)

31

19 Discussion and Conclusion/Summary

prospective projects/research plans:

32

20 Appendix Projects

20.1 (Global)attractor of dynamical system(PDE, with ProfParshad)

20.2 Parameter estimation of (nonlinear) dynamical sys-tems (by comppressive sensing)

20.2.1 Ambiguity of parameters and network structure; Kroneckerproduct and hierarchy structure?

20.3 Convex geometry(probably related with compressivesensing), undergraduate thesis

33

20.4 Shanghai University thesis: dynamical system of tran-simission disease/reaction-diffusion process(just ODE,PDE also major branch)

20.5 Mathematics(dynamical systems) of transmission dis-ease

20.6 introduction

20.7 LaSalle invariant set principle

20.8 persistence of dynamical system

20.9 project: graph approach of construction of globalLyapunov function

20.10 Parameter sensitivity analysis of ODE system

34

21 Appendix A: Notes of some fundamental andsignificant common sense of mathematics(keepupdating)

Those notes are not supposed to be cutting edge, but we think those are (es-sentials...),

21.1 calculus and calculus(ordinary differential) equations

Implicit function theorem.

Existence and uniqueness theorem of ODE initial value problem(IVP)

21.2 measure theory, integral and differentiation

21.3 common senses

compact implies existence of extreme values.

Lebegue integral/measure theory is established to solve the problem of order-exchange of operation on limits of function series sum(mulitiply?) and operationof differential/integral.

complex number and fundamental theorem of algebra.

analysis: key ideas/concepts and theorems in analysis(real and functional)calculus equation: function version fix point theorem, existence and unique-

ness of solutions of ODE initial problem.differential form, stokes theorem(general form): multi-linear algebra

35

22 Appendix B: Brief personal statement

I’m somehow prepared in several ways...

22.1 Family

22.2 Academic experiences

top player of China Mathematics Olympics in the city,

22.3 Courant

my experience in Shanghai (Shanghai University,Fudan Unversity and CDC)an environment with both intensive (traditional) compartmental model(statevariable/ state space model) and discrete (ABM agent based model) complexnetwork model

Collective Behavior: Macroscopic versus Kinetic Descriptionsstudied embedding problems

22.4 Teaching

recommendation from Prof Parshad.2013 Fall Semester:

Calculus I,2014 Spring Semester:

Ordinary differential equations,

22.5 Courses

In case you may want to know my explanation on grades on some courses espe-cially dynamical system courses.

PDE(audit) at Fudan University by Prof ... Zhou(PhD of Courant),

22.6 GPA

about my GPA.no point to make excuses but in case it’s wondered my side explanations on

my GPA:1,master in shanghai University, China:I’ve decided to pursue PhD abroad(US), considered to drop the degree; alsowent to Fudan to Pang-Ting several courses and seminars.

2,Clarkson University:took care of my ex-fiancee in Boston(Harvard).

36

23 notes on probability, ?random / stochasticprocess), statistics and information theory

23.1 entropy

appendix:show from maximum problem

maxPS[P ] such that all possible constraints from data are satisfied,(3)

24 Notes based on/inspired by lectures/seminars

37

References

[1] R. Abraham and J.E. Marsden. Foundations of Mechanics. AMS Chelseapublishing. AMS Chelsea Pub./American Mathematical Society, secondedition edition, 1978.

[2] R. Adler, T. Downarowicz, and M. Misiurewicz. Topological entropy. Schol-arpedia, 3(2):2200, 2008. revision 91878.

[3] Marc Barthelemy. Spatial networks. Physics Reports, 499(1?3):1 – 101,2011.

[4] Henk Broer and Floris Takens. Dynamical systems and chaos, volume 172.Springer Science & Business Media, 2010.

[5] Martin Casdagli. Chaos and deterministic versus stochastic non-linear mod-elling. Journal of the Royal Statistical Society. Series B (Methodological),pages 303–328, 1992.

[6] Martin Casdagli. A dynamical systems approach to modeling input-outputsystems. In SANTA FE INSTITUTE STUDIES IN THE SCIENCES OFCOMPLEXITY-PROCEEDINGS VOLUME-, volume 12, pages 265–265.ADDISON-WESLEY PUBLISHING CO, 1992.

[7] Martin Casdagli. Nonlinear forecasting, chaos and statistics. In Modelingcomplex phenomena, pages 131–152. Springer, 1992.

[8] Mark I Freidlin, Joseph Szucs, and Alexander D Wentzell. Random per-turbations of dynamical systems, volume 260. Springer Science & BusinessMedia, 2012.

[9] Olga Holtz. Compressive sensing: a paradigm shift in signal processing.arXiv preprint arXiv:0812.3137, 2008.

[10] I.M. James. History of topology. In I.M. James, editor, History of Topology,pages v –. North-Holland, Amsterdam, 1999.

[11] Kunihiko Kaneko. Lyapunov analysis and information flow in coupled maplattices. Physica D: Nonlinear Phenomena, 23(1):436–447, 1986.

[12] Holger Kantz and Thomas Schreiber. Nonlinear time series analysis, vol-ume 7. Cambridge university press, 2004.

[13] Yuri Kifer. Random perturbations of dynamical systems, volume 16.Springer Science & Business Media, 1988.

[14] Norman H Packard, James P Crutchfield, J Doyne Farmer, and Robert SShaw. Geometry from a time series. Physical review letters, 45(9):712,1980.

[15] David Ruelle. What is a strange attractor? Notices of the AMS, 53(7),2006.

[16] David Ruelle, Floris Takens, et al. On the nature of turbulence. Commun.math. phys, 20(3):167–192, 1971.

38

[17] Thomas Schreiber. Interdisciplinary application of nonlinear time seriesmethods. Physics reports, 308(1):1–64, 1999.

[18] A. Seth. Granger causality. Scholarpedia, 2(7):1667, 2007. revision 91329.

[19] Hui-Chuan Shen. Statistical Mechanics, in Chinese. 2011.

[20] Shlomo Sternberg. Review: Ralph abraham and jerrold e. marsden, foun-dations of mechanics. Bull. Amer. Math. Soc. (N.S.), 2(2):378–387, 031980.

[21] Floris Takens. Detecting strange attractors in turbulence. In Dynamicalsystems and turbulence, Warwick 1980, pages 366–381. Springer, 1980.

[22] Floris Takens. Detecting strange attractors in turbulence. Springer, 1981.

[23] Roger Temam. Infinite-dimensional dynamical systems in mechanics andphysics, volume 68. Springer Science & Business Media, 1988.

[24] Giulio Tononi, Olaf Sporns, and Gerald M Edelman. Measures of degen-eracy and redundancy in biological networks. Proceedings of the NationalAcademy of Sciences, 96(6):3257–3262, 1999.

[25] E Weinan, K Khanin, A Mazel, and Ya Sinai. Invariant measure for burgersequation with stochastic forcing. Annals of Mathematics-Second Series,151(3):877–960, 2000.

[26] Norbert Wiener and Aurel Wintner. Harmonic analysis and ergodic theory.American Journal of Mathematics, pages 415–426, 1941.

[27] Lai-Sang Young. Ergodic theory of attractors. In Proceedings of the Inter-national Congress of Mathematicians, pages 1230–1237. Springer, 1995.

[28] Lai-Sang Young. Ergodic theory of differentiable dynamical systems. InReal and complex dynamical systems, pages 293–336. Springer, 1995.

[29] Wenwu Yu, Guanrong Chen, Jinde Cao, Jinhu Lu, and Ulrich Parlitz.Parameter identification of dynamical systems from time series. PhysicalReview E, 75(6):067201, 2007.

39