Students who want to pass, need to present short talks on each module. Topics are listed under module description. Each chosen topic should be covered by 1-2 students, each person giving 30-45 minutes talk on a topic. (i.e. each person talks several times, once for each module, but you can work in pairs)

module 1: complex analysis. literature: Hassani *Mathematical Physics* (chapter), Stein&Shakarchi *Complex Analysis*. **Outline**:

- recalling complex calculus
- holomorphic functions, Cauchy-Riemann relations
- power series, Hadamard convergence lemma, analytic functions
- contour integration, basic properties
- Cauchy-Goursat theorem, with consequences: existence of primitive, computation of some integrals (e.g. Fresnel),
- Cauchy integral formula with consequences:
- derivatives as integrals and analyticity of holomorphic functions
- Liouville lemma and fundamental theorem of algebra
- mean value property, maximum modulus principle (and short excursion into harmonic functions)
- Morera theorem
- analytic continuation

- analytic continuation
- Laurent expansion, properties of Laurent series, examples of exapansions
- residue formula (via Laurent expansion)
- classification of singularities (briefly, example of essential singularity), formula for residues for poles,
- computation of integrals via residues
- meromorphic functions, singularities and infinity, Riemann sphere
- argument principle, winding number
- branch of complex logarithm (Riemann surfaces left out)

Topics for student’s talks (in case the given literature is not enough for the length of your talk and for for appetite, I recommend looking into B. Simon: *Comprehensive course in analysis 2A*: (+), (++) denote how broad is a topic (but sometimes it is easier to present a broader topic, because one does not have to enter many details).

-Riemann surfaces (+)

-gamma and beta functions (++)

-Riemann’s zeta function (++)

-Schwarz reflection principle (+)

-dispersion relations (+)

-elliptic functions (++)

-conformal maps (++)

complex analysis talks:

-Friday 24.06, 1415-1545 April&Tom zeta function, Gisela&Crina conformal maps

-Thursday 30.06: 915-1145: Ilay&Weihao gamma and beta functions, Matthias&co elliptic functions

Thursday 7.07: 9:15-1145: Kaylani Riemann surfaces,

Thursday 14.07/Friday 15.07: Tamjeed&Rafiqus Schwarz Reflection Principle, Dan&Andrii dispersive relations, Hector Jacobi’s elliptic function

module 2: functional analysis&PDEs,

literature: Hassani *Mathematical Physics* (selected chapters), Rudin *Real and Complex Analysis* Chapters 3-5, Simon *Comprehensive course in Analysis*

(selected chapters), Reed&Simon *Methods of Modern Mathematical Physics *(selected chapters)

**outline**:

- vector spaces, completeness and separability, linearly dense sets
- a crash course in Lebesgue integral
- examples: spaces of sequences l^p, function spaces L^p, C^0, proof of completeness and separabililty of L^2 and of C^0 (Bernstein polynomials), Stone’s theorem
- inner product, Schwarz inequality, Hilbert spaces, best approximation theorem
- orthonormal sets, Bessel inequality, complete o.n. sets and equivalent conditions, examples: Legendre polynomials, trigonometric polynomials, classical Fourier series
- functionals, linear continuous functionals, norm of a functional, dual space and its completeness
- Riesz representation theorem and application to elliptic PDEs, generalisation, i..e. Lax-Milgram mentioned
- weak convergence, sin example
- strong boundedness vs weak convergence in separable spaces/spaces with predual separable space, lwsc of norm
- application to evolutionary PDEs, Navier-Stokes weak solutions mentioned
- application: direct method in Calculus of Variations: basic example of L^p energy, mentioned more general case of coercive, lwsc functional
- recalling Schwarz class S and Fourier transform from math3, metrisability of S
- tempered distributions S’, their density in S
- Schwarz’s extension of linear, continuous maps S->S to linear, continuous maps S’->S’
- examples of such extensions: distributional derivative, delta, convolution, FT
- application: fundamental solution, mentioned Malgrange-Ehrenpreis theorem

topics for talks:

The idea of talks is to explore several further directions of functional analysis. There are two ‘chapters’, which constitute consistent intrinsic ‘narrative’. Each of you should, individually or in pairs, choose single topic of one of the chapters. I want however the group dealing with each chapters 1 to be present at the entire chapter presentation. (of course I recommend staying at all presentations)

reading for both chapters is: Reed-Simon I (Functional Analysis) [RS], Rudin, Chapter 5 [R], and additional materials in moodle. Even if none/only one of the books is below mentioned for detailed coordinates, I recommend looking at both books: some proofs are nicer in the one, some in the other.

Chapter 1.

Topics: Baire theorem with applications

1.1. Baire theorem with proof [RS III.5, pp.79-81]

1.2. Banach-Steinhaus theorem (as a result of Baire) with two corollaries: joint continuity [RS III.5, pp.81-82] and boundedness of weakly converging sequences (google or see material [M1] in moodle)

1.3. open mapping/inverse mapping theorem/closed graph theorem (as a result of Baire) [RS III.5, pp.82-83]

1.4. application of 1.4: Hellinger-Toeplitz theorem (I would appreciate explaining me also the QM context) [RS III.5, pp.84]

Chapter 2. Hahn-Banach theorem with applications. Reading: moodle material [M2], consult Reed-Simon I (Functional Analysis) [RS] Chapter III, Rudin, Chapter 5 [R] if needed

2.1 Hahn-Banach theorem with proof [M2 p.7-9]

2.2. Hahn-Banach corollaries/versions: ‘separating functionals’ [M2 p.9-10] and mentioning geometric H-B [M2. p. 11-13]

2.3. applications of Hahn-Banach: (L^\infty)^* is not L^1, uniqueness of weak limits [M2 p.10-11 and p.16]

2.4 Ergodic theory [RS II pp.54-60], this can be take by up to 3 people and admittedly has nothing to to with H-B, I added it for balance and out of interest.

Talks remarks and plan:

Talks will be held in A314, neues augusteum.

The first round of talks (complex analysis) was of very diverse quality, one of them quite unrelated to the intended topic. Please follow the suggested literature (as a minimum, I am glad if you dig deeper) and cover the themes listed in the above descriptions of your talk. I will contact individually those of you who must pay particular attention, otherwise they don’t pass the subject (i.e. those, whose first talk was unsatisfactory).

In case you want to discuss the content in advance, I am available on 1.08 and on 2.08 afternoon (4-6 pm) at A307, neues augusteum.

The talks should serve two purposes: the talking person convincing me that they follow basics of what we did and what they are talking about, and the entire group of us learning something more. Therefore I encourage everyone to come and listen. In case I am not convinced, I will (more actively than in the first round) ask further questions/basics of covered by us functional analysis.

I have split up talks into several groups. Members of each group have their talks closely related to each other, so every group should be present at all of this groups’ talks. Thanks to a finer splitting, each of you can have a talk up to 30 mins (and not shorter than 20 mins).

group I Thursday, 4.08 13-14:30

2.4 Abril, Crina, Tom

group II, Thursday, 4.08, 15-18

1.1 Rafiqus, Tamjeed 15-16

1.3. Weihao 16-16:30

1.4 Kaylani, Ilay 16:30-17:30

group III, Thursday, 4.08, 18-19

1.2 Matthias, Anupam

group IV Friday 5.08 9-10

2.1 9:30-10 Gisela

2.3 10-10:30 Hector

group V Friday 11-12

1.15 Andrii, Dan (materials via email)

all the in-person talks (i.e. all except for Matthias&Anupam) take place in Felix-Klein Horsaal, Paulinum. It is on the 5th floor of the main building, in its ‘church’ corner (i.e. Paulinum), the large seminar room.

the online talk takes place at

https://conf.fmi.uni-leipzig.de/b/jan-jkc-jq7-083