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New Trends in contemporarty mathematics: Multipliers

Informacje ogólne

Kod przedmiotu: 360-MS2-2NTa
Kod Erasmus / ISCED: (brak danych) / (brak danych)
Nazwa przedmiotu: New Trends in contemporarty mathematics: Multipliers
Jednostka: Wydział Matematyki
Grupy:
Punkty ECTS i inne: 1.00 Podstawowe informacje o zasadach przyporządkowania punktów ECTS:
  • roczny wymiar godzinowy nakładu pracy studenta konieczny do osiągnięcia zakładanych efektów uczenia się dla danego etapu studiów wynosi 1500-1800 h, co odpowiada 60 ECTS;
  • tygodniowy wymiar godzinowy nakładu pracy studenta wynosi 45 h;
  • 1 punkt ECTS odpowiada 25-30 godzinom pracy studenta potrzebnej do osiągnięcia zakładanych efektów uczenia się;
  • tygodniowy nakład pracy studenta konieczny do osiągnięcia zakładanych efektów uczenia się pozwala uzyskać 1,5 ECTS;
  • nakład pracy potrzebny do zaliczenia przedmiotu, któremu przypisano 3 ECTS, stanowi 10% semestralnego obciążenia studenta.

zobacz reguły punktacji
Język prowadzenia: angielski
Rodzaj przedmiotu:

obowiązkowe

Założenia (opisowo):

(tylko po angielsku) Analysis; Functional analysis (not essential); basic group theory.

Tryb prowadzenia przedmiotu:

mieszany: w sali i zdalnie

Skrócony opis: (tylko po angielsku)

These lectures introduce the theory arising from the entrywise product of matrices. We will explore the motivation for studying this product, and prove several important characterisations and properties. Finally, we will investigate other situations where this entrywise product occurs, and understand how this can be unified with our study of matrices.

Pełny opis: (tylko po angielsku)

Mathematics, cycle 2

Discipline: Theoretical Mathematics

Year of study 2; semester 4

Lecture: 15 hours

ECTS credits: 1

Type: elective course

Learning activities: 15 hours (100%) lectures.

Outside class: 2 hours homework for grading

The study of multipliers began with Schur's results on the entrywise product of matrices (with infinite index sets). The theory was further developed and used by Grothendieck, in his investigation of Banach spaces. Parallel to this is the theory of Fourier multipliers -- functions which multiply the terms of a Fourier series to give a new Fourier series. In abstract harmonic analysis this notion is generalised to arbitrary locally compact groups, where Fourier multipliers are known as Herz-Schur multipliers. The connections between these two parallel notions were put in their final form by Bozejko and Fendler. It was also realised that this can be unified using groupoids, and multipliers of groupoids opened more areas of investigation.

The purpose of this course is to give a tour of the theory of multipliers, beginning with the entrywise product of matrices, and generalising to operators on Hilbert space. We will then look at Herz-Schur multipliers, developing the parallel theory of multipliers on groups, and giving the connections between these two notions of multipliers. In the short final section we introduce groupoids and indicate how they can be used to unify and generalise these results.

Literatura: (tylko po angielsku)

Topics in matrix analysis; Horn and Johnson; Cambridge University Press; 2009.

Herz-Schur multipliers; Todorov; lectures given at Fields Institute; 2014. Available at https://www.fields.utoronto.ca/programs/scientific/13-14/harmonicanalysis/operatoralg/slides/Todorovfieldslec15.pdf.

Efekty uczenia się: (tylko po angielsku)

KA7_UW09 (Can use mathematical analysis, including Banach and Hilbert spaces)

KA7_UK05 (English)

KA7_UU02 (Can independently search literature)

Metody i kryteria oceniania: (tylko po angielsku)

Attendance.

Grading.

Zajęcia w cyklu "Rok akademicki 2022/23" (zakończony)

Okres: 2022-10-01 - 2023-06-30
Wybrany podział planu:
Przejdź do planu
Typ zajęć:
Wykład, 15 godzin więcej informacji
Koordynatorzy: Andrew McKee
Prowadzący grup: Andrew McKee
Lista studentów: (nie masz dostępu)
Zaliczenie: Zaliczenie na ocenę
Rodzaj przedmiotu:

fakultatywne

Tryb prowadzenia przedmiotu:

mieszany: w sali i zdalnie

Skrócony opis: (tylko po angielsku)

These lectures introduce the theory arising from the entrywise product of matrices. We will explore the motivation for studying this product, and prove several important characterisations and properties. Finally, we will investigate other situations where this entrywise product occurs, and understand how this can be unified with our study of matrices.

Pełny opis: (tylko po angielsku)

Mathematics, cycle 2

Discipline: Theoretical Mathematics

Year of study 2; semester 4

Lecture: 15 hours

ECTS credits: 1

Type: elective course

Learning activities: 15 hours (100%) lectures.

Outside class: 2 hours homework for grading

The study of multipliers began with Schur's results on the entrywise product of matrices (with infinite index sets). The theory was further developed and used by Grothendieck, in his investigation of Banach spaces. Parallel to this is the theory of Fourier multipliers -- functions which multiply the terms of a Fourier series to give a new Fourier series. In abstract harmonic analysis this notion is generalised to arbitrary locally compact groups, where Fourier multipliers are known as Herz-Schur multipliers. The connections between these two parallel notions were put in their final form by Bozejko and Fendler. It was also realised that this can be unified using groupoids, and multipliers of groupoids opened more areas of investigation.

The purpose of this course is to give a tour of the theory of multipliers, beginning with the entrywise product of matrices, and generalising to operators on Hilbert space. We will then look at Herz-Schur multipliers, developing the parallel theory of multipliers on groups, and giving the connections between these two notions of multipliers. In the short final section we introduce groupoids and indicate how they can be used to unify and generalise these results.

Literatura: (tylko po angielsku)

Topics in matrix analysis; Horn and Johnson; Cambridge University Press; 2009.

Herz-Schur multipliers; Todorov; lectures given at Fields Institute; 2014. Available at https://www.fields.utoronto.ca/programs/scientific/13-14/harmonicanalysis/operatoralg/slides/Todorovfieldslec15.pdf.

Opisy przedmiotów w USOS i USOSweb są chronione prawem autorskim.
Właścicielem praw autorskich jest Uniwersytet w Białymstoku.
ul. Świerkowa 20B, 15-328 Białystok tel: +48 85 745 70 00 (Centrala) https://uwb.edu.pl kontakt deklaracja dostępności USOSweb 7.0.3.0 (2024-03-22)