**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

**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.

**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.

**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

**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.

**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)

**Exams: Presentation**

**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.

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%)

In this course you have to learn proofs, proofs and real analysis.

**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**

Projects

**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

**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.

**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.

**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.

**Time:** MWF 3-4 pm

**Location: ** Altgeld
Hall 347

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

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 HallGrading:

Remark: the notes in Part IV are based on P. Loeb's lecture notes for real analysis 2003

Homework (1/3-individual submission (always Mondays),

2 Midterm exams (1/3)

Final exam (1/3)

**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.Grading:

Charles W. Curtis: Linear Algebra-An Introductory approach,

Springer 1984, Sections 2 and 7.

Homework (30%)-individual submission (always Mondays), projects (10%)

2 Midterm exams (30%)

Final exam (30%)

**
Math 595 --Introduction to Banach spaces-Spring 2005**

**Time:** MWF
12-12.50pm

**Location: ** Altgeld
Hall 441

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

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)Grading:

Tensor norms and operator ideals.

North-Holland Mathematics Studies, 176.

North-Holland Publishing Co., Amsterdam,1993. xii+566 pp.ISBN0-444-89091-2

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.ISBN3-540-08072-4

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.ISBN3-540-08888-1

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.ISBN0-8218-0710-2

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.ISBN0-521-36465-5; 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.ISBN3-540-41524-6

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.ISBN0-521-35618-0

There will be homework

MATH 347 X3H Fundamental mathematics-Fall2005

**Time:** MWF
12:00-12:50

**Location: ** Altgeld
Hall 447

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.

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 proofsGrading:

Homework (weekly) and projects (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 ?).

Midterms (2) (1/3)

Final exam (1/3)

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

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