Mathematical Institute, University of Freiburg, Teaching

Javier Fresan:
Multiple zeta values: from classical to motivic

Time and place

Thursday, 14.7.16, 10:00-11:00, Raum 125, Eckerstr. 1

Abstract

Multiple zeta values are real numbers of the form \n\nζ(s₁,...,sᵣ)= Σ (1 / n₁ˢ¹...nᵣˢʳ)\n\nwhere the sum is over all integers n₁ > n₂ > ... > nᵣ ≥ 1, the sᵢ ≥ 1 are integers and s₁ ≥ 2. When r = 1, these are nothing else than the special values of the Riemann zeta function. The case r = 2 was also considered by Euler, back in 1775. After more than two centuries of oblivion, multiple zeta values were rediscovered and popularized in recent years, thanks to the work of mathematicians like Brown, Deligne, Goncharov and Zagier, as well as physicists who discovered that many Feynman amplitudes in quantum field theory can be expressed in terms of these numbers.\n\nThe goal of this series of four lectures will be to sketch a proof of a recent theorem of Brown saying that any multiple zeta value can be written as a linear combination with rational coefficients of ζ(s₁,...,sᵣ) with sᵢ ∈ {2,3}. This addresses the "algebraic part" of conjectures by Hoffman and Zagier. Despite the elementary nature of the statement, the only known proof so far uses quite sophisticated techniques, based on the representation of multiple zeta values as periods of mixed Tate motives over the integers. Brown’s theorem will be our excuse to present some of these beautiful mathematics.\n\nThe first lecture of the series will be "colloquium style". A rough plan for the others is as follows:\n\nLecture 2: MZV and the fundamental group (iterated integrals, Chen’s the- orem, mixed Hodge structureson the pro-unipotent completion)\n\nLecture 3: MZV as periods of mixed Tate motives (Tannakian category over a number field, ramification, the motivic fundamental group of P¹ \b {0, 1, ∞})\n\nLecture 4: Proof of Brown’s theorem\nBesides the original references, I will be mainly following the survey "Classical and motivic multiple zeta values" which I wrote with José Ignacio Burgos Gil and is available at https://people.math.ethz.ch/~jfresan/mzv.pdf.

Javier Fresan:
Multiple zeta values: from classical to motivic

Time and place

Friday, 15.7.16, 10:00-11:00, Raum 318, Eckerstr. 1

Abstract

Multiple zeta values are real numbers of the form \n\nζ(s₁,...,sᵣ)= ∑ (1 / n₁ˢ¹...nᵣˢʳ)\n\nwhere the sum is over all integers n₁ > n₂ > ... > nᵣ ≥ 1, the sᵢ ≥ 1 are integers and s₁ ≥ 2. When r = 1, these are nothing else than the special values of the Riemann zeta function. The case r = 2 was also considered by Euler, back in 1775. After more than two centuries of oblivion, multiple zeta values were rediscovered and popularized in recent years, thanks to the work of mathematicians like Brown, Deligne, Goncharov and Zagier, as well as physicists who discovered that many Feynman amplitudes in quantum field theory can be expressed in terms of these numbers.\n\nThe goal of this series of four lectures will be to sketch a proof of a recent theorem of Brown saying that any multiple zeta value can be written as a linear combination with rational coefficients of ζ(s1,...,sᵣ) with sᵢ ∈ {2,3}. This addresses the "algebraic part" of conjectures by Hoffman and Zagier. Despite the elementary nature of the statement, the only known proof so far uses quite sophisticated techniques, based on the representation of multiple zeta values as periods of mixed Tate motives over the integers. Brown’s theorem will be our excuse to present some of these beautiful mathematics.\n\nThe first lecture of the series will be "colloquium style". A rough plan for the others is as follows:\n\nLecture 2: MZV and the fundamental group (iterated integrals, Chen’s the- orem, mixed Hodge structureson the pro-unipotent completion)\n\nLecture 3: MZV as periods of mixed Tate motives (Tannakian category over a number field, ramification, the motivic fundamental group of P¹ \b {0, 1, ∞})\n\nLecture 4: Proof of Brown’s theorem\nBesides the original references, I will be mainly following the survey "Classical and motivic multiple zeta values" which I wrote with José Ignacio Burgos Gil and is available at https://people.math.ethz.ch/~jfresan/mzv.pdf.

Javier Fresan:
Multiple zeta values: from classical to motivic

Time and place

Monday, 18.7.16, 10:00-11:00, Raum 414, Eckerstr. 1

Abstract

Multiple zeta values are real numbers of the form \n\nζ(s₁,...,sᵣ)= ∑ (1 / n₁ˢ¹...nᵣˢʳ)\n\nwhere the sum is over all integers n₁ > n₂ > ... > nᵣ ≥ 1, the sᵢ ≥ 1 are integers and s₁ ≥ 2. When r = 1, these are nothing else than the special values of the Riemann zeta function. The case r = 2 was also considered by Euler, back in 1775. After more than two centuries of oblivion, multiple zeta values were rediscovered and popularized in recent years, thanks to the work of mathematicians like Brown, Deligne, Goncharov and Zagier, as well as physicists who discovered that many Feynman amplitudes in quantum field theory can be expressed in terms of these numbers.\n\nThe goal of this series of four lectures will be to sketch a proof of a recent theorem of Brown saying that any multiple zeta value can be written as a linear combination with rational coefficients of ζ(s1,...,sᵣ) with sᵢ ∈ {2,3}. This addresses the "algebraic part" of conjectures by Hoffman and Zagier. Despite the elementary nature of the statement, the only known proof so far uses quite sophisticated techniques, based on the representation of multiple zeta values as periods of mixed Tate motives over the integers. Brown’s theorem will be our excuse to present some of these beautiful mathematics.\n\nThe first lecture of the series will be "colloquium style". A rough plan for the others is as follows:\n\nLecture 2: MZV and the fundamental group (iterated integrals, Chen’s the- orem, mixed Hodge structureson the pro-unipotent completion)\n\nLecture 3: MZV as periods of mixed Tate motives (Tannakian category over a number field, ramification, the motivic fundamental group of P¹ \b {0, 1, ∞})\n\nLecture 4: Proof of Brown’s theorem\nBesides the original references, I will be mainly following the survey "Classical and motivic multiple zeta values" which I wrote with José Ignacio Burgos Gil and is available at https://people.math.ethz.ch/~jfresan/mzv.pdf.

Javier Fresan:
Multiple zeta values: from classical to motivic

Time and place

Tuesday, 19.7.16, 10:00-11:00, Raum 414, Eckerstr. 1

Abstract

Multiple zeta values are real numbers of the form \n\nζ(s₁,...,sᵣ)= ∑ (1 / n₁ˢ¹...nᵣˢʳ)\n\nwhere the sum is over all integers n₁ > n₂ > ... > nᵣ ≥ 1, the sᵢ ≥ 1 are integers and s₁ ≥ 2. When r = 1, these are nothing else than the special values of the Riemann zeta function. The case r = 2 was also considered by Euler, back in 1775. After more than two centuries of oblivion, multiple zeta values were rediscovered and popularized in recent years, thanks to the work of mathematicians like Brown, Deligne, Goncharov and Zagier, as well as physicists who discovered that many Feynman amplitudes in quantum field theory can be expressed in terms of these numbers.\n\nThe goal of this series of four lectures will be to sketch a proof of a recent theorem of Brown saying that any multiple zeta value can be written as a linear combination with rational coefficients of ζ(s1,...,sᵣ) with sᵢ ∈ {2,3}. This addresses the "algebraic part" of conjectures by Hoffman and Zagier. Despite the elementary nature of the statement, the only known proof so far uses quite sophisticated techniques, based on the representation of multiple zeta values as periods of mixed Tate motives over the integers. Brown’s theorem will be our excuse to present some of these beautiful mathematics.\n\nThe first lecture of the series will be "colloquium style". A rough plan for the others is as follows:\n\nLecture 2: MZV and the fundamental group (iterated integrals, Chen’s the- orem, mixed Hodge structureson the pro-unipotent completion)\n\nLecture 3: MZV as periods of mixed Tate motives (Tannakian category over a number field, ramification, the motivic fundamental group of P¹ \b {0, 1, ∞})\n\nLecture 4: Proof of Brown’s theorem\nBesides the original references, I will be mainly following the survey "Classical and motivic multiple zeta values" which I wrote with José Ignacio Burgos Gil and is available at https://people.math.ethz.ch/~jfresan/mzv.pdf.

Talks at the mathematical institute

Veranstaltungen und Vorträge

Summer term 2016