Math 361-D1 --Introduction to probability-Fall2003


 

Time:  MWF 11-12 pm

Location:  Altgeld Hall 447

Instructor: Marius Junge            Office hours:  TBA, Altgeld 363
 
 

Course e-mail: math34402@math.uiuc.edu
 

E-mail of the grader: glin2@uiuc.edu

Adrress problems with grading first to him-I am willing to find a solution if you two together don't.
 
Books:  We will use

Sheldon Ross: A first course in probability
Remark: The book contains more material than we will cover in class. The students are invited to build their own mind by reading the presentation in the book which will definitely differ from the presentation in class.

Grading:  30% HW (submission in pairs), 40 % 2 midterm, 25% final, 5% contribution in class

Remark:  The students are expected to share their responsibilities towards the homework and
                    to discuss the material among themselves. Discussing between students is an important part of the learning process.

Every student will explain an example or the solution of a homework in class!!!
 

Course description:

This course is intended as an introduction to probability theory. This includes many interesting examples, the right definitions and some little  easy proofs. Indeed, the theoretical part including proofs are essential in getting a grip on the main concepts.
A good example for a probability space is a finite collection of outcomes and every outcome has a certain probability (for examples a fixed number of colored balls in a container). The probability for an event is the sum of all the probabilities of all outcomes with a certain property (the sum of probabilities of green balls). This counting procedures require some combinatorics.
Plan:

I) Combinatorics: Products, permutations, drawing with and without replacement, binomial coefficients

II) What is probability?  Drawing balls: possible outcomes and probabilities, repeating experiments. What happens for many repetitions?
Definition of a probability space, the sigma algebra of events and the probability function. Simple and less simple properties (additivity, monotony, union of events, probability of increasing unions and intersections). Examples.
 

III) Conditional probability and independence: Definition,, imposing extra evidence, Bayes formula, gold and silver drawer,  Polya's urn model. Definition of independence, example for mutual independent not independent events, continuous example.
Examples with crimes and industry.

Borell-Cantelli Lemma
 

IV) Random variables: Definition, example: random time of a complete set,
Discrete random variables: binomial (+/-) (Banach match), hypergeometric, poisson distribution.
expected value, transformation, Markov-Chebychev inequality, Borel-Cantelli lemma, variance and correlation.
Independent random variables,  infinite sequence of Bernoulli trials, strong law of large numbers (using finite variance).

V) Continuous random variables: Expectation and Variance, transformation, proposition 2.1, uniform random variable, normal random variable, approximation of the binomial distribution, gamma distribution

Crash course on integration
 

VI) Joint distribution: Definition, sums of independent random variables, examples, normal, gamma, multivariate distributions, conditional expectation, calculating expectations by conditioning.

VII) Limit theorems: Formulation and illustrations, sketch of proofs.
 
 
 

Exam2-material
 

Homework

Exams




                                   Math 468-Banach spaces and applications -Fall 03


 
 
 
 
 

Time:  MWF 2-3 pm

Location:  Altgeld Hall 241

Instructor: Marius Junge            Office hours:  Thursday 5-6, Altgeld 363
 
 

Book:  Wojtaszczyk: Banach spaces for analysts, Cambridge studies in advanced mathematics 23

Grading:   50% HW (submission in pairs), 50 % presentation or final

Plan:

I:    Introduction to Banach spaces, some tools in functionalanalysis (Baire category thm, closed graph thm for F-spaces, review of the HB-thm, weak topologies)

II:   Isomorphism and basis: Wojtaszczyk 35-43

III:  Absolutely summing operators
       a) Definition and factorization property
       b) An extrapolation result
       c) Definition of cotype for Banach spaces
       d) Khintchine's inequality
       e) p-summing maps and  cotype 2 space, in particular  L(c_0,X)=P_2(c_0,X) for X cotpye 2
       f)  the little Grothendieck inequality
         g) Grothendieck's thm
        f) A second proof of Grothendieck's theorem using harmonic analysis
        h) some applications

IV:   Rademacher-Menchof results
        a) Some conditions for matrices, examples
        b) Maurey's theorem for L(c_0,X) with X cotpye q
        c) Maximal inequality for Riesz summation methods for unconditional sequences in Lp

V:   Factorization results (see Wojtaszcztyk 257-276)
 
 


Homework

Exams: Presentation




Math 347-C1 --Introduction to higher analysis: real variables -Spring 2004


 
 
 

Time:  MWF 10-11 pm

Location:  Altgeld Hall  441

Instructor: Marius Junge            Office hours: Friday 2-3pm  TBA, Altgeld 363
 
 
 
 

E-mail of the graders:  Cheongju Lee   and  Hua Tao

Please contact them first if you have any doubts about the grading-I am only willing to change points after you have contacted and communicated with the grader.
 

Books:  We will use

Kenneth A. Ross: Elementary Analysis: The theory of calculus
 

Remark: We will cover the material in a different order. The theory of metric spaces gives a better start and we have gained enough strenght in the progress to go into the deeper anlysis of the real numbers.  I expect from the students to read the book as additional information and obtain a more profund understanding from
comparing different sources.

Grading:
Homework (30%)-individual submission (always Mondays).  Mark one problem  you wish to be  discussed in class.
team work (10%)-discuss and produce proofs in groups of 3-4 students, one student has to write a protocol.        2 Midterm exams  (40%)
Final exam  (20%)
Course description:
In this course you have to learn proofs, proofs and real analysis.
Plan:

    Part 1: Metric spaces

I-IV: Metric spaces  Definition, topology, continuous functions, completeness.

V Real numbers as completion of the rationals: Archimedian principle, density of the rationals.

VI Sequences and series for real numbers: limsup and liminf and applications to series     (Notes)

VII Compactness:Continuous functions on compact sets attain the maixumum, the asbstract characterization of compactness, Bolzano Weierstrass.  (Notes)

   Part 2: Analysis

I  Continuous functions and sequences

II  Differentiation

III Integration
 
 


 




Homework

Exams

Projects





Math 361-D1 --Introduction to probability -Spring 2004


 
 
 
 
 

Time:  MWF 11-12 pm

Location:  Altgeld Hall  345

Instructor: Marius Junge            Office hours:  TBA, Altgeld 363

Email of the grader:  Guixian Lin

More information  can be found on the webboard(e-mail me if you cannot login)
 
 
 
 
 

Adrress problems with grading first to him-I am willing to find a solution if you two together don't.
Books:  We will use

Sheldon Ross: A first course in probability
Remark: The book contains more material than we will cover in class. The students are invited to build their own mind by reading the presentation in the book which will definitely differ from the presentation in class.
 
 

Grading:

Homework (25 %)-individual submission (always Mondays), group work is recommded (2-4 students),  but indicate joint work. Mark one problem  you wish to be   discussed in class.
Question of the day/joint projects (15%): With the help of the webboard we will a daily or weakly  question related to the material. Moreover, you will prepare 2 'reports' in a group of 2 students either explaining a theoretical exercise or a practical problem. You are invited to find interesting exercises yourself. The essay will be posted on the web.
        2 Midterm exams  (40%)
        Final exam  (20%)
 

Course description:

This course is intended as an introduction to probability theory. This includes many interesting examples, the right definitions and some little  easy proofs. Indeed, the theoretical part including proofs are essential in getting a grip on the main concepts.
A good example for a probability space is a finite collection of outcomes and every outcome has a certain probability (for examples a fixed number of colored balls in a container). The probability for an event is the sum of all the probabilities of all outcomes with a certain property (the sum of probabilities of green balls). This counting procedures require some combinatorics.
Plan:
 

Unit 1: Axioms and elementary properties

-What is probability? Axioms of probablitity.  Examples from combinatorics (counting).
 -Conditional probability and properties.  Independence.  Recursive calculations and everydays problems.

Scetch of unit1

Questions: (Write explanations understandable for your neighbor:)
 -When does an infinite series of positive numbers converge?
 -Let S be a finite set. Why is it impossible to find a strictly increasing sequence  of subsets? (Here strictly increasing means
E_1 contained in E_2 but not equal,  E_2 contained in E_3 but not equal, and so on..
-If we have a number of light balls working simulatneously. Why should we assume that there failure is independent?
 
 

Unit 2: Discrete random variables

-What are random variables?  Discrete probablity spaces and discrete random variables, distribution, Expectation, Variation and standard deviation. Moment generating function
-Bernoulli, binomial, gemetric, negative binomial, hyergeometric, Poisson.

Unit 3: Continuous random variables

-Why do we need sigma-algebras? A minimum of measure and integration. Distribution and density function, expectation, variation and moment generating functions,
-Uniform, normal, exponential, Gamma, Beta, Cauchy distribution
 

Unit 4: Joint distribution, independence and conditional expectation

-How do we handle two variables simutaneously? Joint distribution. What is a conditional expectation (discrete and continuous case)? How can we use conditional expectations?    Building random variables on others.
-Expcetations and cacluating expectations using conditional expectation, best prediction.

Unit 5: Limit theorems

-Central limit theorem and law of large numbers,  statement and use, proof using moment generating functions.

Crash course on integration


 



Homework

Exams

Material



Math 540 --Introduction to real analysis -Fall 2004


 
 
 

Time:  MWF 3-4 pm

Location:  Altgeld Hall  347

Instructor: Marius Junge      Course email   Office hours: Friday 4-5pm
 
Grader:  Shivi Bansal           Jian Yang

Course discription
:

Part I: Metric spaces
      1) R
      2) sequences  in R
      3) definition of metric spaces, open closed sets
      4) compact and complete sets
      5) continuous  functions
      6)  unique extension proinciple
      7)  Arzela-Ascoli
       8) Baire Category thm
       9) completion of metric spaces

Part II: Measurable sets and measures
      1) sigma-algebras, borel algebras
        2) measurses, probability measures
        3) outer measures and extension from algebras
        4) Lebesgue measure, Lusin's theorem
         5) Cantor set

Part III:  Integration theory
   
1) Integrable functions and convergence theorems
     3) Connection to the Riemann integral
     3) Banach spaces
     4) Hilbert spaces and the Radon-Nikodym Theorem
 
Part IV: Integration on IR

    1) Absolutely continuous functions
     2) A covering lemma
     3) Differentiation of monotone functions
     4) Jensen's inequality

Books:
H. L. Royden: Real Analysis, Prentive Hall

Remark: the notes in Part IV are based on P. Loeb's lecture notes for real analysis 2003
Grading:
Homework (1/3-individual submission (always Mondays),
2 Midterm exams  (1/3)

Final exam  (1/3)

Homework

Exams

Material





Math 595 --Transition course for graduate studies-Fall 2004


 
 
 

Time:  MWF 1-2 pm

Location:  Altgeld Hall  341

Instructor: Marius Junge   Course email        Office hours: Wednesday 2-3 or by appointment
 
Intention of the course :  We have realized that graduate students with very different backgrounds come to our university. This concerns in particular the ability to work with abstract concepts, formal proofs and  basic knowledge in analysis and linear algebra
(which might formally qualify as 'undergraduate material'). This course is not a review course. To the contrary we will treat interesting,
but fundamental material, on a  level  which  is appropriate for graduate students in pace and complexity. We will also encourage projects where students are encouraged to fill gaps in their knowledge by independent research (also in peer team work). A particular focus of this course is the interaction of analytic and algebraic concepts.


Course discription :

Part I: Metric spaces (script including compact spaces)
      1) definition of metric spaces,  space of continuous functions,
      2) complete metric spaces, existence of the completion, unique extension of continuous functions,  three proofs
      3) compactness, equivalent conditions (sequentially compact, totally bounded and complete),  continuous
          functions attain their maximum, Heine-Borel Theorem,
      4) contraction mapping principle with application to  Picard-iteration,
      5) definition of topological spaces and connected sets.


Part II: Vector spaces and topological properties
  1) definition of vectors spaces over  R and C, linear  maps and spaces of linear maps,
  2) basic properties of the minimal polynomial (in comparison with the characteristic polynomial),
  3) eigenvector and generalized eigenspaces for linear maps, matrices and change of basis, characterization of diagonalizable maps in terms of
      the minimal polynomial,
  4) Discussion of the Jordan normal form of a linear map and the form of the blocks, sketch of proof,
  5)  the definition of topological vector spaces, in particular  normed linear spaces, the space of continuous linear maps and completeness,         characterization of completeness in terms of absolutely convergent series, uniform convergence of powe series,
  6)  differentiable functions between normed linear spaces,  differentiation of power series, solution of f'(t)=A(f(t))  for  bounded linear maps A,    calculating  e^{tA},  cos(A)  and sin(A)  (1-A)^{-1}  using the Jordan normal form, applications to systems of linear differential equations,
  7) proof of the inverse function theorem.

Part III:  Elementary Geometry in Hilbert spaces
 
  1) the scalar product and the Cauchy-Schwarz inequality, parallelogram equality,
  2) characterization of minima of convex functions in terms of directional derivatives,
  3) applications to least norm approximations, the existence of orthogonal  projections,
  4) existence of orthogonal basis, easy version of Bessel's inequality,  illustration for basis of eigenvalues for selfadjoint matrices.


Books:

Juergen Jost: Postmodern Analysis, Springer 1988,Sections 6-10.

Charles W. Curtis:  Linear Algebra-An Introductory approach,
Springer 1984, Sections 2 and 7.

Grading:
Homework (30%)-individual submission (always Mondays), projects (10%)
2 Midterm exams  (30%)

Final exam  (30%)


Homework

Exams     Takehome

Material



                            Math 595 --Introduction to Banach spaces-Spring 2005



 
 
 

Time:  MWF 12-12.50pm

Location:  Altgeld Hall  441

Instructor: Marius Junge      Course email   Office hours: TBA
 

Course discription
:

Part I: Introduction
      1) Introduction
      2) Hahn-Banach
      3) Weak topologies and locally convex spaces
      4) Krein-Milman and Caratheodory
      5) Riesz Representation theorem

Part II: II_p-summing maps and applications
      1) Grothendieck-Pietsch factorization theorem
        2) Grohendieck inequality
        3) Basis and unconditional basis
        4) Uniqueness of unconditional basis in l_1
        5) Shur mutipliers

Part III:  Tensor norms (f=finite dimensional theory, i=infinite dimensional theory)
     f 1) Lewis Lemma and applications
     f 2) Trace duality for II_p and N_q
      f 3) Banach Mazur compactum and John's theorem
      i 1) smalles ant biggest tensor norm
      i 2) Approximation Problem
      i 3) injective and surjective hull of  pi-tensor product
      i 4) gamma-norms

Part IV: Special topics-consent of instructor ans students
   

Books: 
I will use selected topics from different resources
Defant, Andreas(D-OLD); Floret, Klaus(D-OLD)
Tensor norms and operator ideals.
North-Holland Mathematics Studies, 176.
North-Holland Publishing Co., Amsterdam, 1993. xii+566 pp. ISBN 0-444-89091-2

Lindenstrauss, Joram; Tzafriri, Lior
Classical Banach spaces. I.
Sequence spaces. Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 92.
Springer-Verlag, Berlin-New York, 1977. xiii+188 pp. ISBN 3-540-08072-4

Lindenstrauss, Joram; Tzafriri, Lior
Classical Banach spaces. II.
Function spaces. Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], 97.
Springer-Verlag, Berlin-New York, 1979. x+243 pp. ISBN 3-540-08888-1

Pisier, Gilles
Factorization of linear operators and geometry of Banach spaces.
CBMS Regional Conference Series in Mathematics, 60.
Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1986. x+154 pp. ISBN 0-8218-0710-2

Pisier, Gilles(1-TXAM)
The volume of convex bodies and Banach space geometry.
Cambridge Tracts in Mathematics, 94.
Cambridge University Press, Cambridge, 1989. xvi+250 pp. ISBN 0-521-36465-5; 0-521-66635-X


Pisier, Gilles(F-PARIS6-E)
Similarity problems and completely bounded maps.
Second, expanded edition. Includes the solution to "The Halmos problem". Lecture Notes in Mathematics, 1618.
Springer-Verlag, Berlin, 2001. viii+198 pp. ISBN 3-540-41524-6

Pisier, Gilles(1-TXAM)
The operator Hilbert space ${\rm OH}$, complex interpolation and tensor norms. (English. English summary)
Mem. Amer. Math. Soc. 122 (1996), no. 585, viii+103 pp.


Wojtaszczyk, P.(PL-PAN)
Banach spaces for analysts.
Cambridge Studies in Advanced Mathematics, 25.
Cambridge University Press, Cambridge, 1991. xiv+382 pp. ISBN 0-521-35618-0


Grading:
There will be homework

Homework




MATH 347 X3H  Fundamental mathematics-Fall2005 


 
 

Time:  MWF 12:00-12:50

Location:  Altgeld Hall  447

Instructor: Marius Junge      Course email   Office hours:  tba
 

Course description


Part I: Elementary Concepts
  key words: introduction (here is the text), numbers, sets, functions, relations, groups, bijection, cardinality
  key goals: Discuss what we should do and what we will do, introduction to mathematical language, first proofs, induction, a feel for inequalities.


Part II: Properties of numbers
  key words: Congruence relation, some finite rings, Fermat's little Theorem, rational numbers (the geometric and the formal viewpoint), irrational                      numbers (a beginning)
  key goals: Understand the arithmetic axioms for Q (R) (also by comparison to finite rings Zn).

Part III: 
Discrete Mathematics (if we find time)

  key words: finite probability space, conditional probability, Bayes formula, some expectations, pigeonhole principle and applications.  
  key goals: counting the right way may be trickier than you think - discrete probability has many down to earth applications.

Part IV: Continuous Mathematics
   key words: completeness axioms, least upper bound, archimedean principle, monotone and convergent sequences, Cauchy sequences,  important                       series, limits and continuity
  key goals: understanding how to prove and disprove  `for all ... exists...' statements, the art of 'epsilon-delta'-arguments, see the continuum from                       the mathematical point of view.

  

Books:
John P. d'Angelo and Douglas West: problem-solving and proofs


Grading:
Homework (weekly) and projects  (1/3)
Midterms (2)  (1/3)
Final exam  (1/3)
Projects should be prepared by individual students or in groups of two students. Every student has to prepare at least one project. There are several ways to find projects. 1)  Some projects will be posted in class and I will ask for volunteers.  2) The text book offers an abounded reservoir of problems, for example four problems at the beginning of each section, many exercises marked with (!) are very instructive and we will not find time to discuss most of the interesting examples. Any of those problems can be used for a project and you should notify me when finding something which interests you (reading a little ahead is a good recipe here). Some applications of the pigeonhole principle are really cool and great for projects. The solution 3) is a project of my choice (and closer to what I find interesting ?).
Your creativity and involvement in projects will be pivotal if I have to decide between grades at the end of the term.

Comments: 

This is not an ordinary course! Here you may learn what mathematics is all about and you may decide to hate or love it-I hope you will love it. Proofs, ideas, and concepts will be practiced and discussed  here. But this cannot be done in a content free environment. We have to consider  interesting  problems  and questions. You will learn that the question `What are real numbers?' has an answer, but not necessarily an easy answer.
In this course you will be asked to act and learn by yourself in the form of  solving problems, writing proofs and most importantly think. I understand (and I do remember) that the first steps towards own proofs are time consuming  and sometimes confusing. Well, confusion or milder forms 'that doesn't sound write' are good  first steps towards true understanding - nothing to worry about. To the contrary, if you have never been puzzled or surprised in a math course before, then something went wrong. 
I am looking forward to share some of the many interesting questions in mathematics with you this term.
 

Incoming test  Homework1   
                       

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