Aufblasung affiner Varietäten
Montag, 4.7.16, 16:15-17:15, Raum 404, Eckerstr. 1
Mechanisms and Modelling of Dislocation Patterns
Dienstag, 5.7.16, 14:00-15:00, Raum 226, Hermann-Herder-Str. 10
A Möbius invariant decomposition of the Möbius energy
Dienstag, 5.7.16, 16:15-17:15, Raum 404, Eckerstr. 1
Abstract:\nWe consider the Möbius energy for closed curves in R^n\nso-called since it is invariant under Möbius transformations.\nWe can decompose the energy into three parts, each of which is\nMöoius invariant.\nThe decomposition gives is easy-to-analyze components, e.g., for\nderiving the first and second variational formulas and estimates, and\nfor giving information concerning the minimizers of the energiers.\nThis is a joint work with Dr. Aya Ishizeki (Saitama University)
tba
Mittwoch, 6.7.16, 16:30-17:30, Raum 404, Eckerstr. 1
Donnerstag, 7.7.16, 17:00-18:00, Hörsaal II, Albertstr. 23b
reserviert
Vorstellungsvortrag: Descent
Donnerstag, 7.7.16, 17:00-18:00, Hörsaal II, Albertstr. 23b
The notion of descent is ubiquitous in mathematics. An object satisfies descent\nwhenever its nature is determined by local conditions, for instance:\n\n1. (sheaf condition) A function on a topological space can be given locally on an open\ncovering. The local functions can be glued to a global function if they agree on\noverlaps.\n\n2. (descent) A vector bundle on a topological space can - by definition - be given\nlocally on an open covering. The local bundles are glued to a global one by means of\nglueing data on overlaps that satisfy a compatibility condition on "overlaps of\noverlaps".\nThis comes in many flavors such as descent for modules over rings, families of\nvarieties, etc.\n\n3. (cohomological descent) Any type of cohomology of a topological space or\nalgebraic variety can be recovered (in a certain sense) from the cohomology of an\nopen cover. The "glueing data" in this case is much more complicated and carries the\nessential information.\nWe will explain in this talk how all these instances of descent (and many more) are\nunified by adopting a higher-categorical point of view, the examples above becoming\ndescent for set-like objects, 1-category like objects, or infinity-categorical objects. As\nmodel for "infinity-categorical" questions of descent, we present the theory of fibered\nderivators, the topic of the habilitation thesis of the speaker. Our main motivation has\nbeen descent for Grothendieck six-functor formalisms (encoding Serre duality,\nVerdier duality, etc.).
Algebraic part of motivic cohomology
Freitag, 8.7.16, 10:15-11:15, Raum 404, Eckerstr. 1
Alumni-Tag 2016
Freitag, 8.7.16, 15:00-16:00, Hörsaal Rundbau, Albertstr. 21a
Higher homotopies
Montag, 11.7.16, 16:15-17:15, Raum 404, Eckerstr. 1
Starting with the relation of infinite loop spaces and generalized cohomology theories, we use some simple examples to illustrate some special homotopy invariant properties of infinite loop spaces. Then we go on and introduce various delooping machines. In the end of the talk, a description of infinite loop space by Gamma-space will be given.
Convergent semi-Lagrangian methods for the Monge-Ampère equation on unstructured grids
Dienstag, 12.7.16, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
Min-Max theory and the Willmore conjecture, Fernando C. Marques, André Neves
Dienstag, 12.7.16, 16:15-17:15, Raum 404, Eckerstr. 1
Wie viele Ramsey Ultrafilter gibt es?
Mittwoch, 13.7.16, 16:30-17:30, Raum 404, Eckerstr. 1
Multiple zeta values: from classical to motivic
Donnerstag, 14.7.16, 10:00-11:00, Raum 125, Eckerstr. 1
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.
The mathematics behind LIGO Experiment's first ever detection of gravitational waves
Donnerstag, 14.7.16, 17:00-18:00, Hörsaal II, Albertstr. 23b
On 11 February 2016 the LIGO and Vigo Collaborations announced\nthe detection of gravitational waves. These gravitational waves were produced\nabout 1.3 billion years ago from the inspiral and merger of a pair of black holes\nof 29 and 36 solar masses into a single one of 62 solar masses. The difference\nin their masses was transformed in gravitational radiation, which propagated\nthrough the spacetime as gravitational waves, to reach the Earth on 14\nSeptember 2015. These observations demonstrate the existence of binary\nstellar-mass black hole systems. They also provide the first direct detection of\ngravitational waves and the first observation of a binary black hole merger.\n\nIn this talk I will present the mathematics behind this recent detection of\ngravitational waves, whose existence was predicted by Einstein in 1916, one\nyear after he formulated his equation for General Relativity.
The mathematics behind LIGO Experiment's first ever detection of gravitational waves
Donnerstag, 14.7.16, 17:00-18:00, Hörsaal II, Albertstr. 23b
On 11 February 2016 the LIGO and Vigo Collaborations announced\nthe detection of gravitational waves. These gravitational waves were produced\nabout 1.3 billion years ago from the inspiral and merger of a pair of black holes\nof 29 and 36 solar masses into a single one of 62 solar masses. The difference\nin their masses was transformed in gravitational radiation, which propagated\nthrough the spacetime as gravitational waves, to reach the Earth on 14\nSeptember 2015. These observations demonstrate the existence of binary\nstellar-mass black hole systems. They also provide the first direct detection of\ngravitational waves and the first observation of a binary black hole merger.\n\nIn this talk I will present the mathematics behind this recent detection of\ngravitational waves, whose existence was predicted by Einstein in 1916, one\nyear after he formulated his equation for General Relativity.\n\nDownload the invitation as PDF.
Multiple zeta values: from classical to motivic
Freitag, 15.7.16, 10:00-11:00, Raum 318, Eckerstr. 1
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.
Reconstructing branching lineages in single cell genomics
Freitag, 15.7.16, 12:00-13:00, Raum 404, Eckerstr. 1
Single-cell technologies have recently gained popularity in developmental biology because they allow resolving potential heterogeneities due to asynchronicity of differentiating cells. Popular multivariate approaches for analyzing such data are based on data normalization, followed by dimension reduction and clustering to identify subgroups. However, in the case of cellular differentiation, we cannot expect clear clusters to be present - instead cells tend to follow continuous branching lineages.\n\nWe show that modeling the high-dimensional state space as a diffusion process, where cells move to close-by cells with a distance-dependent probability well reflects the differentiating characteristics. Based on the underlying diffusion map transition kernel, we then propose to order cells according to a diffusion pseudo time, which measures transitions between cells using random walks of arbitrary length. This allows for a robust identification of branching decisions and corresponding trajectories of single cells. We demonstrate the method on single-cell qPCR data of differentiating mouse haematopoietic stem cells as well as on RNA sequencing profiles of embryonic stem cells.\n\nAs outlook if time permits, I will outline how to use this pseudotime in combination with dynamic models to construct a mechanistic understanding of the regulatory process, based on recent work regarding ODE-constrained mixture modeling.
Multiple zeta values: from classical to motivic
Montag, 18.7.16, 10:00-11:00, Raum 414, Eckerstr. 1
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.
TBA
Montag, 18.7.16, 16:15-17:15, Raum 404, Eckerstr. 1
A Local Index Formula for the Intersection Euler Characteristic of an Infinite Cone
Montag, 18.7.16, 16:15-17:15, Raum 404, Eckerstr. 1
The study of global analysis of spaces with (isolated) cone-like singularities has started with work of Cheeger in the 80s and has seen a rich development since. One important result is the generalisation of the Chern-Gauss-Bonnet theorem for these spaces, which is due to Cheeger. It establishes a relation between the \(L^2\)-Euler characteristic of the space, the integral over the Euler form and a local contribution \(\bgamma\) of the singularities. The ``Cheeger invariant'' \(\bgamma\) is a spectral invariant of the link manifold. \n\nThe aim of this talk is to establish a local index formula for the intersection Euler characteristic of a cone. This is done by studying local index techniques as well as the spectral properties of the model Witten Laplacian on the infinite cone. As a result we express the absolute and relative intersection Euler characteristic of the cone as a sum of two terms, one of which is Cheeger's invariant \(\bgamma\).
Multiple zeta values: from classical to motivic
Dienstag, 19.7.16, 10:00-11:00, Raum 414, Eckerstr. 1
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.
Localization on generalized flag manifolds
Dienstag, 19.7.16, 10:15-11:15, Raum 318, Eckerstr. 1
The orbits of the adjoint action of a compact, connected Lie\ngroup on its Lie algebra are called generalized flag manifolds. Under\nsuitable hypotheses, they constitute an ideal ground for the ABBV\nlocalization theorem: we will introduce the necessary notions, and apply\nthis result to quickly compute the volume of such GFMs.
Das Monge-Kantorovich-Problem
Dienstag, 19.7.16, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
Gaspard Monge beschäftigte sich als erster mit dem optimalen Transportproblem. Wir werden in diesem Vortrag seine Formulierung des Problems betrachten und sie nach Leonid Kantorovich abschwächen.\nWir werden die Kantorovich-Dualität sehen, die uns erlaubt, die Suche nach einem optimalen Maß im Kantorovich-Problem auf die Suche nach einem optimalen Funktionenpaar zurückzuführen. Anschließend wollen wir die Existenz einer Lösung des Kantorovich-Problems für eine quadratische Kostenfunktion zeigen.\n
Variational formulae for the sigma_r energy
Dienstag, 19.7.16, 16:15-17:15, Raum 404, Eckerstr. 1
Random reals and infinite time Turing machines
Mittwoch, 20.7.16, 16:30-17:30, Raum 404, Eckerstr. 1
Donnerstag, 21.7.16, 17:00-18:00, Hörsaal II, Albertstr. 23b
reserviert
Smoothing theory
Montag, 25.7.16, 16:15-17:15, Raum 404, Eckerstr. 1
Since Milnor discovered exotic 7-spheres in 1956, it is known that a\ntopological manifold can have several non-diffeomorphic smooth\nstructures. The aim of smoothing theory is to calculate the set S(X) of\nsmoothings of a given topological manifold X in terms of a homotopy\ntheoretical property:\nS(X) turns out to be in bijection with the set of lifts of a certain\nclassifying map.\n\nIn my talk I will introduce all necessary concepts such as mircobundles\nand piecewise linear manifolds and try to illustrate their properties.\nThen the fundamental theorem will be stated and important parts of the\nproof will be sketched. In the end I hope to give some practical\nadvices on how to calculate structure sets.\n
Constrained BV functions on covering spaces and a solution to Plateau's type problems
Dienstag, 26.7.16, 09:00-10:00, Raum 404, Eckerstr. 1
Numerical approximation of positive power curvature flow via deterministic games
Dienstag, 26.7.16, 10:30-11:30, Raum 404, Eckerstr. 1
Diffuse Interfaces and Topology: A Phase-Field Model for Willmore's Energy
Dienstag, 26.7.16, 11:30-12:30, Raum 404, Eckerstr. 1
tba
Dienstag, 26.7.16, 14:00-15:00, Raum 404, Eckerstr. 1
wird noch bekanntgegeben
Dienstag, 26.7.16, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
Some Möbius invariant geometric evolution equations
Dienstag, 26.7.16, 15:30-16:30, Raum 404, Eckerstr. 1
Some singular perturbation problems involving curvature
Dienstag, 26.7.16, 16:30-17:30, Raum 404, Eckerstr. 1
Evolution of planar networks
Mittwoch, 27.7.16, 09:00-10:00, Raum 404, Eckerstr. 1
Convergence of thresholding schemes for multi-phase mean-curvature flow
Mittwoch, 27.7.16, 10:30-11:30, Raum 404, Eckerstr. 1
A frame energy for immersed tori
Mittwoch, 27.7.16, 11:30-12:30, Raum 404, Eckerstr. 1
The Ricci Flow on manifolds with almost non-negative curvature operator
Mittwoch, 27.7.16, 14:00-15:00, Raum 404, Eckerstr. 1
Mean curvature flow without singularities
Mittwoch, 27.7.16, 15:30-16:30, Raum 404, Eckerstr. 1
Variational analysis of a mesoscale model for bilayer membranes
Mittwoch, 27.7.16, 16:30-17:30, Raum 404, Eckerstr. 1
Mathematical 2-dimensional conformal field theory
Donnerstag, 28.7.16, 14:00-15:00, Raum 125, Eckerstr. 1
A conformal field theory (CFT) is a quantum field theory (QFT)\ninvariant under conformal transformations. Unlike general QFTs, CFTs\ncan be defined rigorously. In this talk we will concentrate on two\nsets of axioms suitable for considering 2D CFT: Wightman axioms and\nvertex algebras.\n\nWightman axioms were created by Wightman in 1950s to consider general\nQFTs. After a slight well-known modification, Wightman axioms can\ndescribe 2D genus 0 CFTs. Mathematically, Wightman axioms are a part\nof functional analysis.\n\nVertex algebras were introduced by Borcherds in 1980s and they\ndescribe a chiral half of 2D CFT. For vertex algebras and related\nwork, most notably for the proof of Conway--Norton monstrous moonshine\nconjecture, Borcherds received the Fields Medal. Mathematically,\nvertex algebras are a part of algebra.\n\nThe plan for this talk is to carefully define both sets of axioms,\nprovide the steps of the proof of Kac's Theorem that a 2D Wightman\n(Möbius) CFT gives rise to two commuting (Möbius) conformal vertex\nalgebras, and finally give details of how one can combine two unitary\n(quasi-)vertex operator algebras to get a 2D Wightman (Möbius) CFT.\n\nNo prior knowledge of physics is required.