树状结构的一些例子及其它

吴耀琨

上海交通大学数学系

应用数学和自然科学春季研讨会上海交大，2009/1/18

自然科学与数学 I

科学的基本任务之一是研究结构形成的法则。

科学中的（应用）数学：1 严格推演关于提出的法则的假

设模型的推论（常微，偏微，差分方程，元胞自动机，微分几何，拓扑，计算等）2 理解和表示这些结构的各种

特征（各种记帐工具，傅立叶分析，线性代数，群论，组合数学等）

Andreas DressResearch Center for Interdisciplinary Studies on Structure Formation, Univ. of Bielefeld

Max-Planck Institute for Mathematics in the SciencesCAS-MPG Partner Institute for Computational Biology

自然科学与数学 IIMathematics is biology’s next microscope, only better; biology is mathematics’ next physics, only better。J.E. Cohen， PLOS Biology (2004)

自然科学与数学 III

To deal with their own problems, the applied areas will use more and more heavily mathematics. Without the help from the mathematics community, they will develop the necessary mathematics on their own. This will lead to a separation of mathematics with the rest of sciences. … Nothing is more damaging to applied mathematics than isolating itself from the applications. 鄂维南, Mathematics and Sciences, 2002.

自然科学与数学 IV

As a mathematical discipline travels far from its empirical source, or still more, if it is a second and third generation only indirectly inspired by ideas coming from ``reality’’, it is beset with very grave dangers. … In any event, whenever this stage is reached, the only remedy seems to me to be the rejuvenating return to the source: the reinjection of more or less directly empirical ideas. -- John von Neumann, The Mathematician, 1947.

目录

一 树与树状结构

二 图模型

三 系统发生组合学

四 代数统计,热带代数几何与赋值拟阵

五 总结

目录

一 树与树状结构

二 图模型

三 系统发生组合学

四 代数统计,热带代数几何与赋值拟阵

五 总结

无圈有向图（因果网络）

颜色的分类树

元胞自动机状态空间

Barnes-Hut Multibody Algorithm

Modular Decomposition

A Ehrenfeucht,T. Harju,G. Rozenberg, The Theory of 2-Structures:A Framework for Decomposition and Transformation of Graphs, World Scientific, 1999.

数学家家谱

弦图

G

这是树么？

弦图的树表示

G representation of G on tree T

弦图的等价刻画

不含长度大于3的无弦圈

每一个极小点分离集为团

任何顶点诱导子图都有单纯顶点

图中极大团可以被布置在有连接性质的树上

每一个素子图都是团

每一个与该图邻接阵有相同非零模式的部分半正定矩阵都有半正定完备

拥有相同非零模式的任何矩阵都可以在执行高斯消去法过程中没有零元变为非零元。

补图的边理想有线性分解

相应的环面理想被2次多项式生成

……

树状结构的其它各种表现

零调，Cohen-Macaulay性质，可压缩复形（代

数组合，拓扑组合）

小树宽， 有界树宽（参数复杂性理论）

无圈超图（数据库理论中的信息可保持分解,离散

多元分析中的列联表的可压缩性）

…

树状结构的观念

一个貌似复杂的对象也许在合适的尺子的度量下有

较低的维数,在合适的角度下可以被较好的认知。

树状结构是自然科学与数学中自然出现的结构。它

泛指某种驯顺的，可认知的，低复杂度的，可以从

局部推断整体的，可以分解的对象。复杂性，可解

性往往与树状程度有关。

Though we could not formulate our ideas on the comparison of algorithms and homotopies in a precise way, yet we believe that our point of view is worth to be explored. B. Chen, S-T. Yao, Y.-N. Yeh, Graph homotopy and Graham homotopy, Discrete Mathematics, 241 (2001), 153-170.

目录

一 树与树状结构

二 图模型

三 系统发生组合学

四 代数统计,热带代数几何与赋值拟阵

五 总结

什么是图模型 ?

(概率)图模型就是一个概率分布且该分布可以按照一

个图的结构来作分解。

顶点： 随机变量;无向边：变量之间相关性;有向边：

变量之间因果关系

三种图模型：1 Markov Random Field (Undirected Graphical Model)2 Bayesian Network (Causal Network, Belief Network, Directed Graphical Model)3 Chain Graph (一个无圈有向图中把每个点替换为一个连通无向图)

S.E. Fienberg: Algebraic Statistics and Contingency Tables: Old Wine in New Bottles?

马尔可夫随机场

马尔可夫随机场＝无向图＋每个团上的势函数

贝叶斯网络

贝叶斯网络＝无圈有向图＋各个顶点处的条件概率表

结构＋局部机制 图模型的所有信息

正问题：计算条件概率，边缘概率和各种统计量

反问题：模式选择（结构？），参数推断（局部机制？）

正问题与反问题

Lauritzen (1996). Graphical Models. Oxford UP.

系统发生树

贝叶斯网络的应用：E. Horvitz ( Microsoft )的两个发明

Paperclip

Microsoft Outlook Mobile Manager （OMM）:

OMM has the ability to learn from you to continue to enhance its behavior

马尔可夫性质：图的连通性与随机变量独立性

整体马尔可夫性质： 图中的分离性质<=>随机变

量的条件独立性

局部马尔可夫性质：一个随机变量的取值在给定

其邻居的取值后与其它变量取值无关

如果一族随机变量联合分布中每个状态都有非零

概率，则分解性质<=>局部马尔可夫性质<=>整体马尔可夫性质

马尔可夫性质：图的连通性与随机变量独立性

整体马尔可夫性质： 图中的分离性质<=>随机变

量的条件独立性

局部马尔可夫性质：一个随机变量的取值在给定

其邻居的取值后与其它变量取值无关

如果一族随机变量联合分布中每个状态都有非零

概率，则分解性质<=>局部马尔可夫性质<=>整体马尔可夫性质

如果图结构为弦图，该条件可去。

Hammersley-Clifford定理

马尔可夫随机场

J. Moussouris, Gibbs and Markov random systems with constraints, Journal of Statistical Physics, 1974.P. Clifford, Markov random fields in statistics,in: G.R. Grimmett, D.J.A. Welsh (Eds.), Disorder in Physical Systems, A Volume in Honor of John M. Hammersley, pp. 19—32, Oxford University Press, 1990.

Lauritzen-Verma-Pearl定理

贝叶斯网络

S.L. Lauritzen, et al., Independence properties of directed Markov fields, Networks, 20 (1990) 491—505.

T. Verma, J. Pearl, Causal networks: Semantics and expressiveness, in: R.D. Shachter, T.S. Lewitt, L.N. Kanal, J.F. Lemmer (Eds.), Proceedings of the Fourth Annual Conference on Uncertainty in Artificial Intelligence Uncertainty in Artificial (1988), 69 -- 78

North-Holland, 1990

马尔可夫等价类

刻画相同条件独立关系的图被称为Markov等价类。

无向图模型的Markov等价类只含一个元素，而有

向图模型就没有这么简单。确定Markov等价类有

利于降低模式识别算法的复杂度。

Each Markov-equivalence class of an ADG is uniquely determined by a single chain graph whose chain components are chordal.

S.A. Andersson, D. Madigan, M.D. Perlman, A characterization of Markov equivalence classes foracyclic digraphs, The Annals of Statistics, 25 (1997), 505—541.

Causal Independence

N. Ay, D. Polani, Information flows in causal networks, Advances in Complex Systems, 11 (2008) 17—41.

强相关＝＝》结构

B. Steudel, N. Ay, Inferring common causes from high multi-information, 2008

树与概率计算

Important inference problems in statistical physics, computer vision, error-correcting coding theory, and artificial intelligence can all be reformulated as the computation of marginal probabilities on factor graphs. The belief propagation algorithm is an efficient way to solve these problems that is exact when the factor graph is a tree, but only approximate when the factor graph has cycles. – J. Yedidia, W. Freeman, Y. Weiss, Constructing free-energy approximations and generalized belief propagation algorithms, IEEE Trans. On Inform. Theory, 51 (2005) 2282—2312.

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Secondary Structure/Junction Tree• multi-dim. random variables• joint probabilities

(potentials)

Bayesian Network• one-dim. random variables• conditional probabilities

abd

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ad ae ce

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…

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连接树

Message Passing

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b c

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abd

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1. For each cluster C and sepset S;

φC ←1 , φS ←1

2. For each vertex u in the BN select a parent cluster C s.t. C⊆ fa(u). Include the conditional probability P( Xu | Xpa(u) ) into φC ;

φC ← φC · P( Xu | Xpa(u) )

弦图与精确推断Group random variables which are “fully connected”.

Connect group-nodes with common members: the “junction graph”.

Every node only needs to “communicate”with its neighbors.

If the junction graph is a tree (namely the junction graph is chordal), there is a “message passing”protocol which allows exact inference.

Survey Propagation Algorithm

M. Mézard, G. Parisi, R. Zecchina, Analytic and Algorithmic Solution of Random Satisfiability Problems, Science, 2002.

E. Maneva, et. al., A new look at survey propagation and its generalizations, Journal of the ACM, 2007.

Message passing procedures on a locally tree-like graph

局部计算

…local computation requires two things. The joint probability distribution with which we are working must factor into functions each involving a small set of variables. And these sets of variables must form a hypertree. -- G.R. Shafer, P.P. Shenoy, Probability propagation, Annals of Mathematics and Artificial Intelligence 2 (1990) 327—352.

两篇综述

F.R. Kschischang, et al., Factor graphs and the sum-product algorithm, IEEE Trans. Information Theory 47 (2001).H-A. Loeliger, An introduction to factor graphs, IEEE Signal Processing Magazine, 2004.

“Graphical models are a marriage between probability theory and graph theory.

They provide a natural tool for dealing with two problems that occur throughout applied mathematics and engineering – uncertainty and complexity – and in particular they are playing an increasingly important role in the design and analysis of machine learning algorithms.

Fundamental to the idea of a graphical model is the notion of modularity – a complex system is built by combining simpler parts.

乔丹语录 I

Probability theory provides the glue whereby the parts are combined, ensuring that the system as a whole is consistent, and providing ways to interface models to data.

The graph theoretic side of graphical models provides both an intuitively appealing interface by which humans can model highly-interacting sets of variables as well as a data structure that lends itself naturally to the design of efficient general-purpose algorithms.

Many of the classical multivariate probabilistic systems studied in fields such as statistics, systems engineering, information theory, pattern recognition and statistical mechanics are special cases of the general graphical model formalism -- examples include mixture models, factor analysis, hidden Markov models, Kalman filters and Ising models.

乔丹语录 II

The graphical model framework provides a way to view all of these systems as instances of a common underlying formalism.

This view has many advantages -- in particular, specialized techniques that have been developed in one field can be transferred between research communities and exploited more widely.

Moreover, the graphical model formalism provides a naturalframework for the design of new systems.“

--- Michael Jordan, 1998.

乔丹语录 III

目录

一 树与树状结构

二 图模型

三 系统发生组合学

四 代数统计,热带代数几何与赋值拟阵

五 总结

系统发生学

Phylogenetics: Use the present to predict the past; Reconstruct the Darwinian trees of species。

As we have no record of the lines ofdescent, the pedigree can be discovered only by observing thedegree of resemblance between the beings to be classed. － C. Darwin, The descent of man, and selection in relation to to sex, 1871.

达尔文在《物种起源》中绘制的进化树

系统发生组合学的两本专著

A.W.M. Dress, K.T.Huber, J.Koolen, V. Moulton， Basic Phylogenetic Combinatorics, Cambridge University Press, 2009.

C. Semple, M. Steel, Phylogenetics, Oxford University Press, 2003.

Charles Darwin （2/12/1809-4/19/1882）

Darwin Day is a global celebration of science and reason held on or around Feb. 12, the birthday anniversary of evolutionary biologist Charles Darwin. This year marks the 200th anniversary of Charles Darwin's birth. www.darwinday.org/

达尔文日

May 5, DARWIN DAY at PICB

Mike Steel Andreas Dress

Quartet

Character

Convexity: An Example

Biological Justification for Convexity

Character Compatiblity

Partition Intersection Graph

F: a set of characters

Int(F): partition intersection graph

顶点： (f,A): f ∈F, A=f^{-1}(i)边： (f,A) ～(g,B) iff A∩B is nonempty

Restricted Chordal Completion

称图G是图H的三角剖分(triangulation, chordalization),如果G为弦图，G与H的顶点集合

一致，并且前者的边集包含后者边集。

若G为Int(F)的三角剖分，且不含形如(f,A) ～(f,B)的边，称其为Int(F) 的restricted chordalcompletion。

Mike Steel的一个定理

F is compatible if and only if int(F) has a restricted chordal completion.

If F is compatible, then int(F) has treewidth at most #F.

上世纪70年代末, Andreas Dress与诺贝尔化学奖得主Manfred Eigen共同研究如何将某20种t-RNA的相互演变关系尝试用一棵树表现出来. 由此,Andreas Dress 和其合作者 终发展出T-理论,即树(Tree)的理

论. 这一理论在系统发生网络重建,在在线算法分析,在离散度量空间,相似性与不相似性的数学量度等等方面发挥了重要作用. Andreas Dress也由于此理论及其它对计算生物学的贡献,被马普协会和中科院

任命为上海的中德合作计算生物学伙伴研究所所长,以及被邀请在

1998年国际数学家大会作邀请报告.

Bernd Sturmfels(美国数学会副主席)和 Josephine Yu利用T-理论中核心的tight span概念成功地分类了6个点上的所有离散度量空间. Sturmfels等人与生物学家合作发展出的代数统计学,将T-理论与热带代数几何学相结合,取得很大成功.

T-理论

The Wife of Bath‘s Prologue (58 manuscripts)

The Network of Human mtDNA

A Network of Fungi

Tight Span:距离数据的图表示

(1) representing the various objects and their mutual distance relationships by canonically associating some big high-dimensional “cell complex” to a given distance table, and then

(2) visualising these distance relationships in terms of “sensible” 2-dimensional projections of the “1-skeleton” of that complex.

B. Sturmfels, Can Biology Lead to New Theorems?

树度量的等价刻画

Andreas Dress,The Tight Span of Metric Spaces andits Uses in Phylogenetic Analysis,Lecture Notes.

树度量的等价刻画

Tight Span短期课程（闵行校区）

The Tight Span of Metric Spaces and its Uses in Phylogenetic Analysis (Nov. 21,22,23,28,29,30, 2008)

树度量和弦图

树度量和弦图都有如此丰富的刻画，但他们看起

来是用不同的语言写出来的。既然他们都是对树

性的刻画，似乎可以猜测在这两大块研究之间应

当存在一座沟通的桥梁。

树度量的刻画容易将参数放松而得到树状度量的

某种定义。是否由此能翻译出某种拟弦图的定

义，并由此扩大对图模型应用有好算法的图类？

Gromov(1987)定义了delta-hyperbolic metric(delta度量该metric与 tree metric的局部偏离). 0-hyperbolic metric =

tree metric

目录

一 树与树状结构

二 图模型

三 系统发生组合学

四 代数统计,热带代数几何与赋值拟阵

五 总结

M. Drton, B. Sturmfels, S. Sullivant, Lectures on Algebraic Statistics, Springer, 2009.

L. Pachter, B. Sturmfels, (Eds.) Algebraic Statistics for Computational Biology, Cambridge University Press, 2005.

G. Pistone, E. Riccomagno, H. Wynn, Algebraic Statistics: Computational Algebra in Statistics. CRC Press, 2000.

三本代数统计专著

两篇奠基性文章

G. Pistone, H.P. Wynn, Generalisedconfounding with Grobner bases, Biometrika, 83 (1996), 653—666.

P. Diaconis, B. Sturmfels, Algebraic algorithms for sampling from conditional distributions, Annals of Statistics, 26 (1998), 363—397.

这个词是Pistone和 Wynn造出来的。他们利用Gröbner基来构造好的实验设计并为相关方向取名。

什么是代数统计?

Man V. M. Nguyen: Computational Algebraic Statistics (CAS)

Bruno Buchberger: A Historic Introductionto Gröbner Bases

P.J. Cameron,Sudoku, Mathematics and Statistics

数独 (Sudoku)，代数统计

数独的要求可以被一组多项式方程来表示

Graphical Models and the Geometry -- Statistics Dictionary

• Computational foundations: Graphical models• Sum product algorithm -- inference• Polytope propagation -- parametric inference

• Mathematical foundations: Algebraic geometry• Algebraic varieties -- the model• Amoebas -- the model in log probabilities• Tropical varieties -- parametric MAP inference

maximum a posteriori

Tropical sum-product

General Mathematical Framework

• Statistical models are algebraic varieties.

• Algebraic varieties can be tropicalized.

• Tropicalized models are useful for MAP inference in statistics.

L. Pachter and B. Sturmfels, Tropical Geometry of Statistical Models, Proceedings of the National Academy of Sciences, Volume 101:46 (2004), p 16132--16137.

L. Pachter and B. Sturmfels, Parametric Inference for Biological Sequence Analysis, Proceedings of the National Academy of Sciences, Volume 101:46 (2004), p 16138--16143.•

Patcher,Sturmfels ,Algebraic Statistics for Computational Biology

图模型的代数观点

代数：

代数簇（多项式组的零点） －－－ 代数簇的参数表示

图模型：

马尔可夫性质 －－－ 概率的分解表示

热带半环

These semirings were baptized tropical semirings by Dominique Perrin in honour of the pioneering work of our brazilian colleague and friend Imre Simon, but are also commonly known as (min,+)-semirings. －－ Jean-Eric Pin，Tropical semirings, in: Idempotency (Bristol, 1994), J. Gunawardena (Ed.) (1998) 50-69.

热带半环是各种动态规划算法的合适的记帐工具！

热带几何 I

Tropical Geometry first appeared as a subject of its own in 2002, while its roots can be traced back at least to Bergman’s work on logarithmic limit sets. .. It is intertwined with algebraic and symplectic geometry, geometric combinatorics, integrable systems, and statistical physics. Tropical Geometry can be viewed as a sort of algebraic geometry with the underlying algebra based on the so-called tropical numbers. – I. Itenberg, G. Mikhalkin, E. Shustin, Tropical Algebraic Geometry, Birkhauser, 2007.

热带几何 II

One could think of tropical geometry as beinga shadow of classical algebraic geometry, whichcarries enough information to shed some light onthe classical objects but which at the same timeis light enough to be easier to deal with, or betterto allow the application of tools from otherareas of mathematics. – T. Markwig, Tropical

Geomerty, 2008.

赋值拟阵与热带几何H. Herrmann, et al., HOW TO DRAW TROPICAL PLANES, 2008

Dress教授赋值拟阵短期课程（May-June,2007，徐汇校区）

Beginning with the ground-breaking work by Hassler Whitney in the 1930s, it was observed that certain properties relating to linear-dependence and linear-independence relationships within subsets of vector spaces can be treated successfully in an abstract fashionhighlighting the essentially purely combinatorial character of these relationships.

The resulting investigations culminated in the ``Theory of Matroids with Coefficients" that we will be outlined in these lectures based on the series of joint papers with Walter Wenzel and Werner Terhalle which appeared in the 1990s.

Special emphasis will be given to the special case of Valuated Matroids, also treated in: ``K. Murota, Matrices and Matroids for Systems Analysis, Algorithms and Combinatorics 20, Springer Verlag, 2000." and, in a disguised form, in ``D. Speyer, B. Sturmfels, The tropical Grassmannian, Adv. in Geom. 4, 2004, 389--411."

代数几何

There are two overlapping and intertwining paths to understanding algebraic geometry. The first leads through sheaf theory, cohomology, derived functors and caterories, and abstract commutative algebra – and these are just the prerequisites! … Rather, we will focus on specific examples and limit the formalism to what we need for these examples. Indeed, we will emphasize the strand of the formalism most useful for computations: We introduce Grobner bases early on and develop algorithms for almost every technique we describe. The development of algebraic geometry since the mid 1990s vindicates this approach. – B. Hassett, Introduction to Algebraic Geometry, Cambridge, 2007.

系统发生代数几何

J.P.S. Kung, Combinatorics and Nonparametric Mathematics

Human tctctggttagtttgtaacatcaagtacttacCTCATTCAGCATTTTTCTTTCTTTAATAGACTGGGTCAChimp tctctggttagtttgtaacatcaagtacttacCTCATTCAGCATTTTTCTTTCTTTAATAGACTGGGTCAMouse tcccagatcagttcgt---atcaggtacccacCACATTCAGAAGTCTTCTTTCTTGGATAGACCGGACCARat tccgggattagtctgt---atgaggtacccacCACACTCAGAAGTTTTCTTTCTTGGATAGACTTGATCADog tttctgattcgtttgtaacattgagtacctacCTCATCTAGTATCTTTCTTTCTTTAATAGACTGGGTTA

* * * ** ** ** **** *** ** ** * ********* ****** * *

Comparative Genomics

Petersen graph parametrizestrees on 5 taxa.

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幂等数学 I

Idempotent mathematics is a new branch of mathematical sciences, rapidly developing and gaining popularity over the last

decade. It is closely related to mathematical physics. Tropical mathematics is a very important part of idempotent mathematics. The literature on the subject is vast and includes numerous books and

an all but innumerable body of journal papers. -- G.L. Litvinov, V.P. Maslov, S.N. Sergeev (Eds.), IDEMPOTENT AND TROPICAL MATHEMATICS AND PROBLEMS OF MATHEMATICAL PHYSICS, 2007.

幂等数学 II

There exists a heuristic correspondence between interesting useful and important constructions and results over the field of real or complex numbers and similar constructions and results over idempotent semirings in the spirit of N. Bohrs correspondence principle in Quantum Mechanics. So Idempotent Mathematics can be treated as a classical shadow or counterpart of the traditional Mathematics over fields. –G.L. Litvinov, V.P. Maslov, A.N. Sobolevski, Idempotent mathematics and interval analysis, Preprint ESI 632, The Erwin Schrodinger International Institute for Mathematical Physics, 1998.

学术活动 I

Algebraic Statistics is a new field in which ideas from combinatorics, geometry of polytopes, computational algebra and algebraic geometry contribute to the formulation, interpretation, and solution of statistical problems. Many of the motivating problems in algebraic statistics arise from problems in computational biology: sequence analysis and reconstruction of phylogenetictrees from sequence data, to name two. Workshop on Algebraic Statistics and Computational Biology, Clay Mathematics Institute, 2005.

学术活动 II

Roughly speaking, algebraic statistics is concerned with those problems that lie at the intersection of algebra, combinatorics, statistics, and their applications. The field is still relatively new (less than ten years old) so the full scope of applications is emerging.

Algebraic Statistics: Theory and Practice, Harvard University, 2006http://www.math.harvard.edu/~seths/AMS.html

学术活动 III

Algebraic statistics is a maturing discipline focused on the applications of algebraic geometry and its computationaltools in the study of statistical models. Initial results in the area were related to specific problems in categorial data analysis and experimental design, however a flurry of activity during the past several years has greatly increased the scope of the subject. Areas of interest now include graphical models, maximum likelihood estimation and Bayesian methods. Moreover, a strong connection has developed to applications in the physical and biological sciences. The field draws its tools not only from computational algebraic geometry but also from tropical, convex, and information geometry. Moreover, research in algebraic statistics has led to new directions in those fields. Algebraic Statistics, The Mathematical Sciences Research Institute, 2008.

学术活动 IV

Computational Algebraic Statistics, Theories and Applications, 2008, Kyoto University.

学术活动 V

In recent years, methods from algebra, algebraic geometry, and discrete mathematics have found new and unexpected applications in systems biology as well as in statistics, leading to the emerging new fields of "algebraic biology" and "algebraic statistics." Furthermore, there are emerging applications of algebraic statistics to problems in biology. This year-long program will provide a focus for the further development and maturation of these two areas of research as well as their interconnections. 2008-09 Program on Algebraic Methods in Systems Biology and Statistics, Statistical and Applied Mathematical Sciences Institute

学术活动 VI

This workshop will concentrate on tropical methods in Combinatorics and Algebra. Some of the topics we expect to explore are:

Tropical ideas in combinatorial linear algebra, such as tropicalconvexity, tropical linear spaces and oriented matroids, tropical matrix algebra and its applications. Tropical methods in combinatorial representation theory, including both discovery of new formulas and improved understanding of oldones. Computational issues, including both how to compute tropicalobjects and how to use tropical tools in other computational settings. Applications of tropical methods in algebraic statistics.Tropical Geometry in Combinatorics and AlgebraOctober 12, 2009 to October 16, 2009

目录

一 树与树状结构

二 图模型

三 系统发生组合学

四 代数统计,热带代数几何与赋值拟阵

五 总结

Something to Take Home

好的研究对象会有许多侧面让人欣赏，会与许多

别的对象相关联。

树状结构里头还有很多可以挖掘的新数学，也与

许多科学和技术领域相关联。

各种记帐工具：非参数化《＝》参数化。

希望能够组织一个讨论班，共同学习图模型，系

统发生组合学与代数统计。

参考文献

“GOOGLE大学”：http://google.com

谢 谢！

David Mumford: What Is Mathematics I

First, I want to quote a denition of what is mathematics due to Davis and Hersh in their very penetrating book ``The Experience of Mathematics" (Davis-Hersh, 1980, p.399): `The study of mental objects with reproducible properties is called mathematics.' -- David Mumford, The Dawning of the Age of Stochasticity, in Mathematics: Frontiers and Perspectives (V. I. Arnold et al., eds.). Amer. Math. Soc., 2000, pp. 197–218.

David Mumford: What Is Mathematics II

I love this definition because it doesn’t try to limit mathematics to what has been called mathematics in the past but really attempts to say why certain communications are classified as math, others as science, others as art, others as gossip. Thus reproducible properties of the physical world are science whereas reproducible mental objects are math.Art lives on the mental plane (the real painting is not the set of dry pigments on the canvas nor is a symphony the sequence of sound waves that convey it to our ear) but, as the post-modernists insist, is reinterpreted in new contexts by each appreciator. As for gossip, which includes the vast majority of our thoughts, its essence is its relation to a unique local part of time and space.

数学与艺术

The more I study the interrelations of the arts the more I am convinced that every man is in part an artist. Certainly as as artist he shapes his own life, and moves and touches other lives. I believe that it is only as an artist that man knows reality. Reality is what he loves, and if his love is lost it is his sorrow. -- Marston Morse, Mathematics and the Arts, in： R.G. Ayoub, Editor, Musings of the Masters, The Mathematical Association of America, 2004.

丢勒, 忧郁, 铜版画 1514

Melencolia I, 1514Albrecht Dürer(German, 1471–1528)http://www.metmuseum.org/toah/hd/durr/ho_43.106.1.htm#

因果分析 I

Admissions to Berkeley by department

因果分析 II

因果分析 III

The explanation turned out to be that women tended to apply to competitive departments with low rates of admission even among qualified applicants (such as English), while men tended to apply to less-competitive departments with high rates of admission among qualified applicants (such as engineering). The conditions under which department-specific frequency data constitute a proper defense against charges of discrimination are formulated in Pearl (2000). Judea Pearl. Causality: Models, Reasoning, and Inference, Cambridge University Press, 2000

Algorithmic applications of simplicialdecompositions to problems in statistics are considered by Lauritzen and Speigelhalter(1988). Decompositions of chordal graphs into their cliques have applications to problems in areas as diverse as measure theory and database schemes; see Lauritzen, Speed and Vijayan (1984) and Beeri et al. (1981). －－R. Diestel, Graph decompositions: A study in infinite graph theory, Oxford, 1990.

Most of our results are just ``translations’’ of results from other areas. It is somewhat technical to establish the connection between graphical models and decomposable models. In fact, in our opinion these results are of a purely graph theoretic nature …－－J.N. Darroch, S.L. Lauritzen, T.P. Speed, Markov fields and log-linear interaction models for contingency tables, The Annals of Statistics 8 (1980), 522—539.

…the application of graph theory to loglinearmodels of contingency tables provides a rich, unexpected connection between two rather diverse areas. Increased awareness of this connection can be expected to lead to mutual benefits for both areas. – H.J. Khamis, T.A. McKee, Chordal graph models of contingency tables, Computers Math. Applic. 34 (1997), 89—97.