SPP 2026 · Geometry at Infinity
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SPP Main Conference and Farewell
The SPP Main Conference 2025, which took place in the first week of April 2025 at the MPI in Leipzig, marks the end of the Priority Programme SPP2026: Geometry at Infinity and therefore also …
Basic Science Lifetime Award in Mathematics 2025
After the Abel Prize and the Breakthrough Prizes the next ones in line are the Basic Science Lifetime Awards. In 2025 in the category mathematics they were awarded to Shigefumi Mori for his …
Breakthrough Prizes 2025
Recently the Breakthrough Prizes were awarded (press release). The Breakthrough Prize in Mathematics 2025 goes to Dennis Gaitsgory for his central role in the proof of the geometric …
Abel Prize 2025
The Abel Prize 2025 is awarded to Masaki Kashiwara for his fundamental contributions to algebraic analysis and representation theory, in particular the development of the theory of D-modules …
Immersed but not embedded homology classes
Let \(X\) be a finite complex and \(z \in H_m(X)\) an integral homology class. One can ask whether there exists a closed, oriented, \(m\)-dimensional manifold \(M\) with fundamental class …
W-Triviality of 9-Fold Suspensions
Here is a result that I randomly stumbled upon. Let us call a space W-trivial if every real vector bundle over it has trivial total Stiefel-Whitney class. Atiyah and Hirzebruch now proved in …
The \(S^1\)-Stability Conjecture in Dimension 4
Last December I blogged about the \(S^1\)-stability conjecture for psc-metrics (link). Unfortunately, the result I explained there turned out to be wrong: Counter-examples can be found in …
Optimality of Gerver’s Sofa
The moving sofa problem is one of those deceptively simple yet incredibly difficult “real-world” math problems. Despite its straightforward formulation, it has remained unsolved …
The \(S^1\)-Stability Conjecture for psc-Metrics
Jonathan Rosenberg introduced the following conjecture: A closed manifold \(M\) admits a Riemannian metric of positive scalar curvature if and only if the product \(M \times S^1\) admits …
K-homology class of the Euler characteristic operator
When studying the Atiyah-Singer index theorem one usually sees four main examples. Atiyah-Singer operator: Its topological index is the \(\hat{A}\)-genus and its analytical index can be …
Polynomial and metallic manifolds
I recently came across the notion of a polynomial, resp. metallic manifold. Since I have never heard of it before and I assume that the same applies to most of you, I wanted to share its …
Hausdorff medal 2024
The Hausdorff medal is awarded biennially by the European Set Theory Society for the most influential work in set theory in the last five years. This years recipient is Gabriel Goldberg.
The Probability of the Law of Excluded Middle
This is basically a repost of this blog post by John Carlos Baez: link. I wanted to share it with you since I find the result unexpectedly interesting. The Law of Excluded Middle states that …
AI achieves silver-medal at IMO
Google developed AI models that achieved on this year’s IMO questions a silver-medal (with only one point away from a gold-medal). You can read more about it in this blog post of …
All triangulations have a common stellar subdivision
Already in April a paper was put on the arXiv (2404.05930) by Karim Adiprasito and Igor Pak proving the following result: Every two triangulations of a geometric complex in Euclidean space …
Cantor-Medaille 2024 für Felix Otto
Felix Otto bekommt die Cantor-Medaille der Deutschen Mathematiker-Vereinigung (DMV) für herausragende wissenschaftliche Leistungen […] von außergewöhnlicher Tiefe und höchstem Einfluss …
Sergei Novikov 1938 – 2024
A week ago Sergei Novikov passed away. He was a notable mathematician working in topology. The conjecture he posed in 1965 about the homotopy invariance of higher signatures of manifolds is …
Kervaira invariant one problem
Last month a solution to the Kervaire invariant one problem in the final remaining dimension 126 was announced (link to a mathstodon post). I have already mentioned this problem in an …
Shaw Prize 2024
The Shaw Prize in Mathematical Sciences 2024 is awarded to Peter Sarnak for his development of the arithmetic theory of thin groups and the affine sieve, by bringing together number theory, …
Jim Simons 1938 – 2024
Three days ago Jim Simons passed away. He was a great mathematician (the Chern-Simons form is named after him together with Shiing-Shen Chern) and the most successful hedge fund manager of …
Blaschke’s Inclusion Theorem
An extremely classical theorem from planar geometry is the following one: If \(B_1\) and \(B_2\) are two strictly convex, compact planar domains with smooth boundaries touching at some point …
A Mathematical Conventions Survey
I wanted to share the results of this funny survey with you: 100 Questions About Mathematical Conventions. But beware, if you click on the link you will loose an hour of your life time. 😉
Rationalized stable homotopy groups of spheres
Computing the stable homotopy groups of spheres \(\pi_*^s\) is a major problem in stable homotopy theory. In chromatic homotopy theory one studies, for every prime \(p\), a tower \[\cdots …
Ars legendi-Fakultätenpreis Mathematik 2024
Der diesjährige Ars legendi-Fakultätenpreis für exzellente Hochschullehre in der Mathematik geht an Prof. Anselm Knebusch. Prof. Anselm Knebusch von der Hochschule für Technik Stuttgart …
Abel Prize 2024
The Abel Prize 2024 goes to Michel Talagrand for his groundbreaking contributions to probability theory and functional analysis, with outstanding applications in mathematical physics and …
Metric spaces and C*-algebras III
This is a continuation of the previous two posts I and II. In the first one I defined the uniform Roe algebra \(C_u^*(X)\) of a metric space \(X\) of bounded geometry and mentioned the …
Isoclinism of groups
Do you know when two groups \(G\) and \(H\) are called isoclinic to each other? I have never heard of this notion before, but there was a paper on the arXiv listing today using it in its …
Analysis, Geometry and Topology of Positive Scalar Curvature Metrics
I wasn’t traveling much (actually, almost not at all) to conferences or workshops in the last few years. First because of the COVID-19 pandemic and afterwards because I got a child and …
Fachwissen von Lehrkräften
Über die Jahre hinweg habe ich mich regelmäßig mit Kolleg*innen darüber unterhalten, wie relevant die Menge und Tiefe an fachmathematischer Ausbildung von angehenden Gymnasiallehrkräften …
Crafoord Prize 2024
The Crafoord Prize is awarded since 1982 by the Royal Swedish Academy of Sciences and the Crafoord Foundation. It is awarded in the categories mathematics and astronomy, geosciences, …
An Olympiad-level AI system for geometry
The International Mathematical Olympiad (IMO) announced a few months ago the AIMO Prize: 5 million USD to the first AI that can win a gold medal! At least in geometry, this seems to be …
A new lower bound for sphere packing
Recall that the classical sphere packing problem is the following: What is the maximum proportion of \(\mathbb{R}^d\) that can be covered by a collection of non-overlapping spheres of volume …
Lotterien in Antragsverfahren
Heute kam die Ankündigung, dass die Stiftung Innovation in der Hochschullehre im Februar wieder eine Freiraum-Förderung ausschreiben wird (hier ein Link zu dem Blogbeitrag über die …
Non-trivial units of complex group rings
One of most important mathematical results from 2021 was Gardam’s refutation of the unit conjecture (blog post): He constructed the first non-trivial unit in a group ring. The …
Pseudo-isotopies of 4-manifolds
One of the landmark results about the topology of 4-manifolds is Freedman’s proof of the 4-dimensional Poincaré conjecture. The problem with this result was that after some time people …
Formalizations in LEAN
I already blogged a few times about computer assisted verification of proofs, for example here: link. It seems to me that this topic is accelerating fast right now, especially formalizations …
Freiraum 2023
Letztens wurden die Projekt verkündet, welche bei Freiraum 2023 gefördert werden. Im Rahmen von Freiraum können Mittel für die Verwirklichung von Projekten in der Lehre beantragt werden, und …
Uncomplemented subspaces in Banach spaces
In any Hilbert space \(H\), every closed subspace \(M\) therein is complemented, i.e. there exists a closed subspace \(N\) with \(M \oplus N \cong H\). One possible choice for \(N\) is …
2024 Breakthrough Prize in Mathematics
The 2024 Breakthrough Prize in Mathematics goes to Simon Brendle for transformative contributions to differential geometry, including sharp geometric inequalities, many results on Ricci flow …
Metric spaces and C*-algebras II
In the previous post about metric spaces and C*-algebras we saw the definition of the uniform Roe algebra \(C^*_u(X)\) of a metric space \(X\) and discussed its remarkable property that two …
The Kervaire Conjecture
Think about your favourite non-trivial group. Now add a generator to it and then add any relation. Is the resulting group still non-trivial? The above question is known as the Kervaire …
Can you hear your location on a manifold?
Can one hear the shape of a drum? is a famous paper by Mark Kac from 1966. The precise mathematical question stated therein is the following: If the Laplace operators of the two Riemannian …
A chiral aperiodic monotile
Two months ago I blogged about the resolution of the famous problem whether a single shape can tile the entire plane, but only aperiodically (link). There was just one drawback to the …
Shaw Prize 2023 in Mathematical Sciences
The Shaw Prize this year is awarded to Vladimir Drinfeld and Shing-Tung Yau for their contributions related to mathematical physics, to arithmetic geometry, to differential geometry and to …
The invariant subspace problem in Hilbert spaces
I was a bit surprised today to see a preprint by Per Enflo on the arXiv (arXiv:2305.15442). He is already 79 years old (Wikipedia)! Per Enflo is famous for several fundamental results in …
Metric spaces and C*-algebras
Let \(X\) be a metric space of bounded geometry. The latter means that for all \(r > 0\) exists an \(n_r \in \mathbb{N}\) such that every ball in \(X\) of radius \(r\) has at most \(n_r\) …
Positive vs nonnegative sectional curvature
The sectional curvature is one of the main invariants of Riemannian manifolds. But despite its importance, there is actually little known about questions like: What is the distinction (for …
Hilbert spaces and C*-algebras without Choice
Today a paper was posted on the arXiv by Blackadar, Farah and Karagila (arXiv:2304.09602) that examines Hilbert spaces and C*-algebras in ZF set theory without the assumption of any Axiom of …
Uniform and coarse embeddings of Banach spaces
Let \(X\) and \(E\) be Banach spaces. The metrics on these spaces induce both uniform and coarse structures, and we can ask whether the following two statements are equivalent to each other: …
An aperiodic monotile
It is finally done – the long standing question whether there exists a single tile that can cover the entire plane, but only aperiodically, is answered! Here is the answer …
Ars legendi-Fakultätenpreis Mathematik 2023
Der Ars legendi-Fakultätenpreis Mathematik ist dieses Jahr an Prof. Dr. Claudia Kirch verliehen worden. Der Preis wird für herausragende, innovative und beispielgebende Leistungen in Lehre, …
Abel Prize 2023
The Able Prize 2023 is awarded to Luis A. Caffarelli for his seminal contributions to regularity theory for nonlinear partial differential equations including free-boundary problems and the …
Coherence of one-relator groups
The recent series of results about one-relator groups (see, e.g. the previous two blog posts) culminates now in the final breakthrough: The conjecture of Baumslag from the 70s (that all …
Homological coherence of one-relator groups
Seems to be an interesting time right now to be working on coherent groups (previous blog posts: link and link). Recall that coherent groups are those whose finitely generated subgroups are …
Virtually free-by-cyclic groups
Last month I blogged about coherent groups (i.e. groups whose finitely generated subgroups are finitely presented). There I also referred to an article of Daniel Wise about coherent groups …
Differentials in the Adams spectral sequence
In this post we consider the Adams spectral sequence which computes the stable homotopy groups of spheres at the prime 2. The classes \(\{h_j\}_{j\ge 0}\) on the 1-line of the second page …
Coherent groups
Recall that a group G is called coherent if every finitely generated subgroup of G is finitely presented. Main examples are 3-manifold groups and (virtually) polycyclic groups. Instead of …
Regularity of minimizing hypersurfaces
Let us consider the following classical problem from geometry (the case \(n=3\) is basically Plateau’s problem): Let \(\Gamma\) be a smooth, closed, oriented, \((n−1\))-dimensional …
Illustrating the Impact of the Mathematical Sciences
A series of posters and some other related media were produced by the National Academy of Sciences of the USA to showcase mathematics of the twenty-first century and its applications in the …
Double Soul Conjecture
Recall the Soul Theorem of Cheeger and Gromoll: If \((M,g)\) is a complete and connected Riemannian manifold of non-negative sectional curvature, then there exists a closed, totally convex …
Progress on the union-closed sets conjecture
The union-closed sets conjecture is the following extremely easy to state conjecture about subsets of finite sets: Assume that \(\mathcal{F}\) is a family of subsets of \(\{1, 2, \ldots, …
Status-Bias im Peer-Review
In der aktuellen Forschung & Lehre ist ein Beitrag mit dem Titel Status-Bias im Peer-Review-Verfahren, den ich sehr interessant fand. Der Beitrag beginnt mit folgenden Worten: …
Conjectures about polynomials
Recently I stumbled upon the following two conjectures about polynomials: The first one is the Jacobian conjecture: We have polynomials \(f_1, \ldots, f_n\) in the variables \(x_1, \ldots, …
Non-positive immersions
In a blog post about hyperbolicity of one-relator groups I mentioned the following property that a 2-dimensional complex \(X\) might have: \(X\) is said to have non-positive immersions if …
Announcing a result on the arXiv
Today Rachel Greenfeld and Terence Tao announced via the arXiv that they can disprove the periodic tiling conjecture (arXiv:2209.08451). I do not want to discuss here in detail the contents …
Predicting the future of arbitrary functions
Let \(S\) be any non-empty set. If you have a function \(f\colon \mathbb{R} \to S\) and you tell me its values on an interval \((-\infty,t)\), can I predict which value it will have at the …
Fields medalists 2022
Recently the Fields medalists 2022 (and all the other prizes given out by the IMU) were announced: official page. You can read about some of the achievements of these people on the blog of …
Shaw Prize 2022
The Shaw Prize in Mathematical Sciences 2022 (link) is awarded to Noga Alon and to Ehud Hrushovski for their remarkable contributions to discrete mathematics and model theory with …
Finite subgroups of homeomorphism groups
If \(M\) is a compact topological manifold, can one say something about its group of homeomorphisms \(\mathrm{Homeo}(M)\)? Of course one can, there is quite a lot to say about it, and there …
A quantitative coarse obstruction to psc-metrics
Recently, Guo and Yu pushed the following result to the arXiv (math.KT/2203.15003): For any \(R > 0\) and positive integer \(m\), there exists a constant \(k(R,m)\) such that the following …
Abel Prize 2022
The Abel Prize 2022 was awarded to Dennis Sullivan “for his groundbreaking contributions to topology in its broadest sense, and in particular its algebraic, geometric and dynamical aspects”. …
Hyperbolicity of one-relator groups
Recall that a one-relator group is a group \(G\) admitting a presentation of the form \(G = \langle x_1, \ldots, x_n \ | \ r = 1\rangle\), where \(r\) is a (cyclically reduced) word in the …
ICM 2022 takes place as a fully virtual event
The International Mathematical Union announced that the International Congress of Mathematicians 2022 will not be held in Saint Petersburg but will take place as a fully virtual event …
Mathematikleistungen von Schüler*innen der gymnasialen Oberstufe
Im Journal für Mathematik-Didaktik ist letztes Jahr ein Artikel von Rolfes-Lindmeier-Heinze erschienen (DOI:10.1007/s13138-020-00180-1), welcher die seit 1995 durchgeführten …
Lehmer’s conjecture and the Fuglede-Kadison determinant
Lehmer’s conjecture is one of the most striking open problems in number theory. It roughly postulates that the complex roots \(a\) with \(|a| > 1\) of a polynomial with integral …
Wolf Prize 2022
The Wolf Prize in Mathematics 2022 is awarded to George Lusztig for “groundbreaking contributions to representation theory and related areas.” The Wolf Prize is an international …
Positive scalar curvature and the conjugate radius
A classical result in Riemannian geometry is the theorem of P. O. Bonnet and S. B. Myers stating that a complete Riemannian \(n\)-manifold \(M\) with Ricci curvature bounded from below by …
Alon-Jaeger-Tarsi conjecture
The Alon-Jaeger-Tarsi conjecture states the following: For any (finite) field F with at least four elements and any non-singular matrix M over F there is a vector x such that both x and Mx …
Properly positive scalar curvature
An interesting (at least to me) research theme in the geometry of manifolds is the question about the existence of positive scalar curvature metrics on closed manifolds. Since I also like to …
An implication of the Farrell-Jones conjecture
A ‘well-known’ implication of the Farrell-Jones conjecture (for a given group G) is that the map \[\widetilde{K_0(\mathbb{Z}G)} \to \widetilde{K_0(\mathbb{Q}G)}\] in reduced …
Kaplansky’s direct finiteness conjecture
Not too long ago I blogged about the first counter-example to Kaplansky’s unit conjecture (link) stating that there are no non-trivial units in the group ring K[G] for K a field and G …
Message from the EMS president
In the last EMS Magazine (2021/No. 121) Volker Mehrmann reflected in his editorial (link) on the bygone (virtual) European Congress 8ECM. At the end he asked to write to him our opinions …
Lie groups acting on countable sets
Does every connected Lie group act faithfully on a countable set? In other words: is every Lie group a subgroup of \(\mathrm{Sym}(\mathbb{N})\)? This question is sometimes called …
Topological CAT(0)-manifolds
It is an interesting and important fact that a contractible manifold (without boundary) is not necessarily homeomorphic to Euclidean space. This makes the classical Cartan-Hadamard theorem, …
PSC obstructions via infinite width and index theory
In a recent preprint (arXiv:2108.08506), Yosuke Kubota proved an intriguing new result on the relation of largeness properties of spin manifolds and index-theoretic obstructions to positive …
(Non-)Vanishing results for Lp-cohomology of semisimple Lie groups
For a locally compact, second countable group \(G\) one can define the continuous \(L^p\)-cohomology \(H^*_{ct}(G,L^p(G))\) of \(G\) and the reduced version \(\overline{H}^*_{ct}(G,L^p(G))\) …
New book about Freedman’s proof
Today I learnt from an article in the QuantaMagazine (link to article) that there is finally a new book trying to explain Freedman’s proof of the 4-dimensional Poincaré conjecture …
Loewner’s “forgotten” theorem
Yesterday I discovered an instructive paper on “Loewner’s forgotten theorem” by Peter Albers and Serge Tabachnikov on the arXiv. I don’t know in how far the result …
Rigidity implication of the Novikov conjecture
Part of my own research is related to the Novikov conjecture, and hence I am always interested to see applications of it. When reading the preprint Essentiality and simplicial volume of …
Space of psc-metrics on non-spin manifolds
Recently I was having a look at the preprint Essentiality and simplicial volume of manifolds fibered over spheres by Thorben Kastenholz and Jens Reinhold (arXiv:2107.05892). In Theorem C …
A question about the first \(L^2\)-Betti number
In a recent arXiv preprint (arxiv:2106.15750) J. A. Hillman discusses a new homological approach towards some old results on 3-manifold groups due to Elkalla. In his article he runs into an …
Computer assisted verification of contemporary mathematics
Half a year ago Steffen Kionke wrote a blog post on condensed mathematics wherein he mentioned that Peter Scholze put up a challenge to formally verify a key fundamental result of a joint …
A hyperbolic 5-manifold which fibres over the circle and subgroups of hyperbolic groups
Recently appeared a highly interesting paper by Italiano, Martelli, Migliorini on the arXiv (2105.14795) (Okay, this was already three weeks ago – I don’t follow the …
GDM-Monat 2021 – Vortrag zur Doppelten Diskontinuität
Auf dem GDM-Monat im März 2021 (voriger Beitrag dazu: link) hörte ich auch den Vortrag von Viktor Isaev zu dem Thema: Entwicklungsverläufe von Studierenden bezüglich ihrer Wahrnehmung zur …
Singmaster’s conjecture
Singmaster’s conjecture is an easy to state conjecture about Pascal’s triangle. It is easy to see that every natural number in Pascal’s triangle occurs only a finite number …
GDM-Monat 2021 – Hauptvortrag zum Mathematischen Beweisen
Die Gesellschaft für Didaktik der Mathematik (GDM) hat wegen der Corona-Pandemie anstatt einer Präsenztagung im Jahre 2021 im März einen Monat lang Vorträge und Veranstaltungen …
Shaw Prize 2021
The Shaw Prize 2021 in the Mathematical Sciences goes to Jean-Michel Bismut and Jeff Cheeger for their remarkable insights that have transformed, and continue to transform, modern geometry. …
Characterizing Euclidean 3-space
In my previous post about the recent preprint of Jiang Wang we saw the following characterization of Euclidean 3-space among all contractible 3-manifolds: It is the only one that admits a …
Online Fernklausuren in Mathematik
Im letzten Semester mussten aufgrund der Pandemie viele Prüfungen in anderen (oft digitalen) Formaten abgehalten werden. Für das laufende Sommersemester – auch wenn sich die Lage …
Contractible 3-manifolds and positive scalar curvature, III
I have already blogged several times about complete Riemannian metrics of positive scalar curvature on 3-manifolds: here, here and here. Now it seems that finally Jian Wang proved the result …
A relative index theorem too good to be true?
A few days ago I was reading with Rudolf Zeidler the recent preprint by Zhizhang Xie (arXiv:2103.14498). We were especially focusing on the proof of the relative index theorem therein and …
Topological complexity
Rummaging in the arXiv I ran across this article and the notion of topological complexity which is really appealing. The idea of topological complexity isn’t quite new, it was …
Extensions, coarse embeddability and the coarse Baum-Connes conjecture
One of the pinnacle results so far about the (strong) Novikov conjecture is Guoliang Yu’s proof that it holds for groups which are coarsely embeddable into a Hilbert space. In fact, he …
Multiplying matrices
Two years ago I blogged about recent developments about multiplying integers. The next most important operation in (applied) mathematics is multiplying matrices. The usual way of doing this …
Abel Prize 2021
The Abel Prize Laureates 2021 were announced today. They are László Lovász and Avi Wigderson for … their foundational contributions to theoretical computer science and discrete …
Hantzsche-Wendt manifolds
Recently a preprint was put on the arXiv:2103.01051 about Hantzsche-Wendt manifolds. I did not know what these manifolds are, got interested and started reading the introduction. By …
Chern assembly maps
The Baum-Connes conjecture asserts that for a group G the analytic assembly map, which is nowadays usually defined using KK-theory, \[\mu_*^{K\!K}\colon RK_*^G(\underline{EG}) \to …
Unit conjecture disproved!
There are three conjectures about group rings of torsion-free groups that are attributed to Kaplansky. To state them, let \(K\) be a field, \(G\) be a torsion-free group and denote by …
Database of online seminar talks
Due to the current situation there are numerous online seminar talks organized all over the world. I was quite happy to discover this week that the website researchseminars.org maintains a …
Tetrahedra
Consider the following three basic questions about tetrahedra: Does a given tetrahedron tile space?Which tetrahedra are scissors-congruent to a cube?Can one describe the tetrahedra all of …
The commutator subgroups of surface groups
Every subgroup of infinite index in a surface group is a free group. There are many ways to see this, for instance, using a bit of topology. An infinite index subgroup corresponds to a …
Condensed mathematics
This post is inspired by Alex Engel’s post “Validity of results II” on incorrect results and computer proof checking. Before I get there, let me start with some background …
Gegenseitige Korrektur in Lehrveranstaltungen
Dieses Semester halte ich eine Vorlesung für Studierende im dritten Semester (ich habe hier über den mathematischen Inhalt gebloggt). Zu dieser Vorlesung gibt es offiziell keinen …
Validity of results, II
In July I wrote a post about Validity of results. It was about a recent preprint pointing out an erroneous proof in an influential paper. By chance I stumbled yesterday upon this question on …
How to hear the corners of a drum
A short while ago I was blogging about the classical isospectrality problem which goes back to Mark Kac’ famous question: Can one hear the shape of a drum? Some basic information …
The group of germs at infinity of line homeomorphisms
Recently, I learned that the group G∞ of germs at +∞ of orientation preserving homeomorphisms of the real line has two remarkable properites: it is simple and left-orderable. I thought it is …
Elementarisierung der Fachinhalte für die Schule
In der Zeitschrift für Didaktik der Naturwissenschaften ist letztens ein Artikel erschienen über das Thema Quantenphysik in der Schule mit dem Ziel Leitlinien für Lehrerfortbildungen …
Ergebnisse der Wissenschaftsbefragung
Gestern sind die Ergebnisse der DZHW-Wissenschaftsbefragung veröffentlicht worden. Die Wissenschaftsbefragung ist eine bundesweite repräsentative Trendstudie. [… Sie] versteht sich als …
Triangulating real projective n-space
How many vertices do you need to triangulate the real projective n-space? From this blog post of Gil Kalai I learned about a recent preprint (arXiv:2009.02703) by …
Resolution of Keller’s conjecture
Keller’s conjecture states that in any tiling of Euclidean space by identical hypercubes there are two cubes that meet face to face.
Catastrophe theory
This term I am teaching a course on catastrophe theory for 2nd year students. When you look up this topic on Wikipedia, you will see words like chaos and singularities mentioned, and you …
Immersions of manifolds into Euclidean space
Recall the well-known result of Whitney that any (compact) smooth \(n\)-manifold admits an immersion into \(\mathbb{R}^{2n-1}\). Today there was a preprint posted on the arXiv …
Blockseminar on Dirac operators and scalar curvature
In mid-October we gathered for one week in Bollmannsruh (somewhat west of Berlin) to work our way through the seminal paper Positive scalar curvature and the Dirac operator on complete …
Gehaltsverhandlungen als PostDoc
Gehaltsverhandlungen als PostDoc? Das gibt’s doch nur für Professoren … Das dachte ich mir zumindest bis jetzt und ging davon aus, dass man in Deutschland solange auf E13 (bzw. …
Seminar on soap bubbles and positive scalar curvature
Together with Rudolf Zeidler I am organizing a reading seminar this winter on generalized soap bubbles and positive scalar curvature. The goal of it is to read the corresponding preprint by …
Nobel Prize for Sir Roger Penrose
Sir Roger Penrose won the Nobel Prize in Physics for the discovery that black hole formation is a robust prediction of the general theory of relativity. I have to admit that I connected him …
Positive scalar curvature metrics on manifolds with boundary
Usually when I blog here about positive scalar curvature, the manifolds I consider are assumed to have no boundary. In this post I want to explain the basics of what happens when the …
Akademisches und schulbezogenes Fachwissen
Das braucht man doch nie in der Schule! ist ein oft gehörtes Argument, wenn sich Studierende des gymnasialen Lehramts der Mathematik über die Inhalte ihrer Fachvorlesungen beschweren. Sie …
Preise auf der DMV-Jahrestagung 2020
Die Jahrestagung 2020 der DMV fand vor zwei Wochen statt. Hier könnt ihr einen Kurzbericht dazu lesen: link. An dieser Stelle möchte ich kurz von den zwei auf dieser Jahrestagung verliehenen …
Can one hear orientability?
Mark Kac asked in a paper from 1966 the following question: Can one hear the shape of a drum? The mathematically precise question is the following: Assume that \((M,g)\) and \((N,h)\) are …
Breakthrough Prizes 2021
Three days ago the Breakthrough Prize Foundation announced the recipients of the 2021 Breakthrough Prizes. I focus in this post only on the prizes in mathematics (there are also prizes in …
Contractible 3-manifolds and positive scalar curvature, II
Let \((M,g)\) be a complete, contractible Riemannian \(3\)-manifold (without boundary). Chang-Weinberger-Yu (link) proved that if \((M,g)\) has uniformly positive scalar curvature, then …
Isometry groups of hyperbolic surfaces
A month ago Aougab, Patel and Vlamis posted a preprint on the arXiv (arXiv:2007.01982) about the question which groups, for a fixed orientable surface of infinite genus, can be realized as …
Spaces of positively curved Riemannian metrics
It is by now a classical topic in index theory to study on a (closed) Riemannian (spin) manifold the space of all Riemannian metrics of positive scalar curvature. We have several results …
Quasi-local operators
In the first post of this series we asked at the end two questions – in this post we start working towards the answers in the general setup of the third post of this series. Our setup …
The unplanned impact of mathematics
In 2011 Nature published a short article about The unplanned impact of mathematics which I stumbled upon just now. Since it is just 4 pages long, I can recommend reading it just for fun …
Computer assistance and pure mathematics
Today I want to tell you a story of a preprint in pure mathematics that came into existence only by crucial help of precise computer computations. To explain the results, let us first define …
Validity of results
Can you vouch for the validity of the results in papers that you cite and use in your own articles? In April this year Blagojević, Cohen, Crabb, Lück and Ziegler posted a preprint on the …
Conjugation Curvature
Recently I saw some papers on the arXiv on conjugation curvature of finitely generated groups (also called medium-scale curvature, transportation curvature, metric Ricci curvature or …
Collatz conjecture
Given a natural number n, the Collatz sequence it generates is the following: if n is even, then divide it by 2, if n is odd, then multiply it by 3 and add 1; and now iterate this procedure. …
Large scale properties of 3-manifold groups
A week ago there was a preprint posted on the arXiv by Peter Haïssinsky and Cyril Lecuire about Quasi-isometric rigidity of three manifold groups (arXiv:2005.06813). Building on work by many …
Banach conjecture
There was a paper today in the arXiv mailing list (arXiv:2006.00336) proving yet another case of the Banach conjecture. I never heard of this conjecture before, but it is easy to state and …
Shaw Prize 2020
The Shaw Prize 2020 in the Mathematical Sciences goes to Alexander Beilinson and David Kazhdan “for their huge influence on and profound contributions to representation theory, as well …
MINTchallenge
Der Club MINT, eine Initiative des Stifterverbands, ruft regelmäßig die MINTchallenges aus in denen “schlaue Ideen zur Lösung aktueller Herausforderungen der MINT-Bildung an …
Instability of Anti-de Sitter Space-Time
A recent article in the QuantaMagazine (link) discusses a paper of Georgios Moschidis (arXiv:1812.04268) who proved instability of Anti-de Sitter space-time for a certain Einstein-matter …
EMS Prizes 2020
Though the 8th European Congress of Mathematics was postponed to 2021, the recipients of the EMS Prizes 2020 were already announced: https://8ecm.si/news/69.
Local-to-global principles for the topology of boundaries of hyperbolic groups
Two weeks ago a paper was posted (by Benjamin Barrett) on the arXiv (arXiv:2004.11650) proving the following theorem about Gromov boundaries of word hyperbolic groups: Let \(G\) be a …
Remote teaching for the working mathematician
Ich bekam als Reaktion auf meinen Blogbeitrag zu Remote teaching einige eMails, in denen nach konkretem Rat und praktischen Tips gefragt wurde für das kommende Semester. Vorlesungen …
Aspherical manifolds and positive scalar curvature
Recall the following conjecture about aspherical manifolds (i.e., manifolds whose universal cover is contractible): If M is a closed, aspherical manifold, then M does not admit any …
Remote teaching
The Corona Crisis is throwing off our lives. I was also not spared, though this is now probably complaining on a high comfort level: As some of you know, I was on a sabbatical since last …
A Prime Breakthrough
It seems that at the beginning of this year a major breakthrough on prime numbers was achieved. I learned about it from this blog: link. Let me summarize the result quickly for you if you …
Laplace operator and covering spaces
Let M be a closed Riemannian manifold and denote by X its universal covering space equipped with the pulled back Riemannian metric. There is an intimate relation between the Laplace operator …
Lindelöf hypothesis
Today I stumbled across a news article ( link ) written about someone (actually, Athanassios Fokas – a known mathematician) having announced progress on the Lindelöf hypothesis ( …
Girls and boys performing in STEM fields
This is actually already old news (it is from September 2018), but since then it was on my list of things to blog about and only now I found the time to actually do it.
Shaw Prize 2019
This year’s Shaw Prize in Mathematical Sciences goes to Michel Talagrand (link to the press release, link to his Wikipedia page).
Compact hyperbolic non-spin manifolds
Recently a preprint was posted on the arXiv ( arXiv:1904.12720 ) claiming to have constructed the first examples of compact orientable hyperbolic non-spin manifolds (in every dimension at …
Impressions from the SPP Conference
Some pictures from the SPP Conference in April 2019 in Münster. Let me start with a picture of the speaker of the SPP, Bernhard Hanke, explaining at the beginning of the conference the next …
Reports on Impact of Mathematics
A week ago the EMS posted links to reports on the impact of mathematical research on society and economy. You can access these reports here: link. Though probably for many too long to read …
Multiplying integers
The classical algorithm, that everyone knows from elementary school, for multiplying two n-digit integers runs in \(O(n^2)\)-time. Recently, there was a preprint posted on HAL (link) in …
Two quite different prizes
In the last few weeks were two quite different prizes awarded, on which I want to report. In March 14th, 2019, the L’Oréal-UNESCO For Women in Science International Awards (homepage, UNESCO …
AMS Special Session on the Mathematics of John Roe
There was a Special Session on Coarse Geometry, Index Theory, and Operator Algebras: Around the Mathematics of John Roe at the Spring Central and Western Joint Sectional Meeting of the AMS …
2019 Abel Prize
Karen Uhlenbeck (Wikipedia) is the 2019 Abel Prize Laureate (Wikipedia, Homepage) … for her pioneering achievements in geometric partial differential equations, gauge theory and …
2019 Christopher Heyde Medal
Three days ago the Australian Academy of Sciences announced the recipients of its 2019 Honorific Awards (Twitter, Webpage). Among them is Professor Geordie Williamson FAA FRS (Wikipedia), …
Global existence for non-linear wave equations
Non-linear wave equations (in 3+1 dimensions) are ubiquitous in physics, playing a vital role in many subfields of it. There are many methods available to show short-time existence of …
Ten Years of Polymath
Ten years ago the first Polymath project was launched (link to the original proposal of Timothy Gowers). Polymath can be described as massively collaborative mathematics, i.e., a large …
Wolf Prize in Mathematics 2019
The Wolf Prize in one of the most prestigious prizes in mathematics (Wikipedia entry). This year’s laureates are Prof. Gregory Lawler (Wikipedia, Homepage) and Prof. Jean Francois le …
The Whitehead manifold and positive scalar curvature
Recall that the Whitehead manifold is a contractible 3-manifold which is not homeomorphic to Euclidean 3-space (Wikipedia entry). It was proven by Chang-Weinberger-Yu (link) that the …
Sir Michael Atiyah 1929 – 2019
Sir Michael Atiyah died three days ago on January 11th, 2019 (news of The Royal Society). A nice obituary (together with one for Jean Bourgain) may be found on this blog: link. Here is …
Coarse embeddings and non-positive curvature
Let \((X,d)\) be a complete, geodesic metric space. \((X,d)\) is called an Alexandrov space of global non-positive curvature if for every quadruple of points \(x,y,z,w\) such that \(w\) is a …
Cantor medal awarded to Hélène Esnault
The DMV awarded the Cantor medal to Hélène Esnault from the FU Berlin (press release). The Cantor medal is the most important prize the DMV awards.
Non-negative scalar curvature and mean convex boundaries
In a recent preprint ( arXiv:1811.08519 ) E. Barbosa and F. Conrado derive for manifolds with boundary topological obstructions to the existence of non-negative scalar curvature metrics with …
MSJ Geometry Prize 2018
The Geometry Prize 2018 of the Mathematical Society of Japan was awarded to Shouhei Honda for his work on Geometric analysis on convergence of Riemannian manifolds and to Yuji Odaka for his …
Selberg’s lemma and negatively curved Hadamard manifolds
Selberg’s lemma is a fundamental result about linear groups. It states that every finitely generated subgroup of \(\mathrm{GL}(n,K)\), where \(K\) is a field of characteristic zero, is …
Breakthrough Prize 2019
Ten days ago the winners of the Breakthrough Prize 2019 were announced (press release, AMS Blog). In mathematics the award goes to Vincent Lafforgue for “ground-breaking contributions …
Classification of static vacuum black holes
In a series of two papers ( arXiv:1806.00818 and arXiv:1806.00819 ) Martin Reiris Ithurralde classifies all metrically complete solutions of the static vacuum Einstein equations with compact …
Vier Exzellenzcluster in der Mathematik
Vor drei Tagen ( Pressemitteilung der DFG ) wurde die Entscheidung über die zukünftigen Exzellenzcluster veröffentlicht. Vier davon wird es in der Mathematik geben: in Bonn, in Münster, in …
Hotness and student evaluations
Are student evaluations biased by gender? No, say the authors of this article on slate.com, but instead they are biased by hotness: link.
Shaw Prize 2018
Luis Caffarelli (wikipedia, homepage) from the University of Texas at Austin will receive the Shaw Prize 2018 in Mathematics (press release) for “his groundbreaking work on partial …
Thomas Friedrich 1949 – 2018
Thomas Friedrich was a German mathematian working in differential geometry and global analysis. He passed away in February 2018. Thomas Friedrich contributed substantially to the development …
Operators of finite propagation
In the starting post of this series we considered infinite band matrices (with uniformly bounded entries) acting on infinite vectors and asked at the end the question how to determine …
Strong cosmic censorship conjecture
The cosmic censorship conjectures concern the singularities arising in general relativity. In May the QuantaMagazine published an article (link) about a potential disproof of a strong …
News around the ICM 2018
The ICM 2018 is currently taking place in Rio de Janeiro. Here are the prize winners of the IMU prizes and of the K-theory foundation, and the new IMU Executive Committee. The Fields …
Contractible 3-manifolds and positive scalar curvature
It is known that \(\mathbb{R}^3\) admits a complete metric of uniformly positive scalar curvature. In fact, for any closed manifold \(X\) and any \(k \ge 3\) the manifold \(X \times …
Prizes, prizes, prizes
Several prizes have been awarded in the past few weeks to mathematicians. Kyoto Prize The Kyoto Prize 2018 in the category Basic Sciences was awarded to Masaki Kashiwara from the RIMS at …
Peter Scholze a new director at MPI Bonn
Peter Scholze is a new director at the Max Planck Institute for Mathematics in Bonn. You can read a bit more about this news here: link.
Equivariant band matrices and Fourier series
Recall that in the first post of this series we claimed that there exists an infinite matrix \(T\) which is in the closure (in operator norm) of the band matrices with uniformly bounded …
Two new presidents
Professor Volker Mehrmann from the TU Berlin was elected as the new president of the European Mathematical Society (EMS). His four-year term will start January 1st, 2019. More information …
Stable minimal surfaces in 3-manifolds
Meeks-Pérez-Ros conjectured in their article “Stable constant mean curvature surfaces” (2008) the following: if a closed, connected Riemannian 3-manifold N does not admit any …
Manfredo do Carmo 1928-2018
Manfredo do Carmo was a Brazilian mathematician working in differential geometry. 1978 he was an invited speaker at the ICM in Helsinki, and 1971-1973 he was president of the Brazilian …
Mathematik in der Blütezeit des Islam
Die Wikipedia veranstaltet jährlich einen Schreibwettbewerb (link). Es gibt jeweils einen Jurypreis und einen Publikumspreis, und dieses Jahr ging der Publikumspreis an einen mathematischen …
Chromatic Number of the Plane is at least 5
The problem From the usual Euclidean plane we form the following graph: the points of the plane are the vertices of our graph, and two vertices are connected by an edge if they are exactly …
Norms of infinite matrices
This is the first post of a series of posts in which we will eventually venture deep into the realm of coarse geometry. But we will always be motivated by questions which are related to the …
John Roe 1959-2018
John Roe, the founder of coarse index theory, passed away last month after a long fight against cancer. His web page is still online ( http://sites.psu.edu/johnroe/ ) and can be visited to …
Nemmers Prize 2018
The Frederic Esser Nemmers Prize in Mathematics goes this year to Assaf Naor “for his profound work on the geometry of metric spaces, which has led to breakthroughs in the theory of …
Snowflakes at infinity
The countless shapes of snowflakes have long raised the curiosity of many scientists, among others the famous Kepler. They have by now been classified by empirical observation into 80 …
Ricci flow and diffeomorphism groups of 3-manifolds
A new paper proves the contractibility of the space of constant curvature metrics on all 3-manifolds except possibly real projective space. Bamler, Kleiner: Ricci flow and diffeomorphism …
A locally hyperbolic 3-manifold that is not hyperbolic
A preprint with a new example shows that the understanding of infinitely generated Kleinian groups will be more complicated than for the finitely generated ones. Cremaschi: A locally …
What’s hot at MathOverflow 23/2017
The two ways Feynman diagrams appear in mathematics
Bavarian Geometry/Topology Meeting
These days, there was the 2nd Bavarian Geometry/Topology Meeting, organized by Fabian Hebestreit and Markus Land, and hopefully becoming a tradition as the NRW topology meeting which by now …
Breakthrough Prize for higher-dimensional geometry
The award ceremony is certainly not what mathematicians are used to, and there are certainly many things that one can say for and against such monstrous awards and the ambience around. In …
Quasi-isometric groups with no common model geometry
Do quasi-isometries between groups always arise from actions on a common model space? Previous counterexamples invoked central extensions of lattices, e.g., of surface groups. A new …
S-matrices and the big unification
A week ago, the long-awaited preprint Scattering Forms and the Positive Geometry of Kinematics, Color and the Worldsheet by Arkani-Hamed, Bai, He, and Yan, appeared on the ArXiv. Michael …
What’s hot at math overflow 22/2017
Concepts in topology successfully transferred to graph theory and combinatorics with non-trivial applications?
A deformation of instanton homology for webs
The four color theorem from graph theory is certainly the most famous problem for which so far only a brute force computational proof exists. A new preprint of Kronheimer-Mrowka supports an …
What’s hot at mathoverflow 21/2017
Concepts in topology successfully transferred to graph theory and combinatorics with non-trivial applications?
Slides and Pictures from the Kickoff Meeting
Listening to 33 math talks within 2 times 6 hours is certainly a unique experience. If you missed the event, you may enjoy the slides of the talks: All Slides for Download
Gauß in Regensburg
The German Mathematical Society (DMV) offers twice a year the „Gauß Lecture“, an overview lecture with a well-known mathematician. The lecture is intended to show current developments in …
What in the world is topological quantum matter?
Always wondered what the work of the 2016 Nobel prize laureates might have to do with topology? This recent video from Fan Zhang may give a first idea:
What’s hot at mathoverflow 20/17
Why is the definition of higher homotopy groups the “right one”? Why is there no symplectic version of spectral geometry? Pullback and homology Is this generalization of …
Séminare Bourbaki: Convex-cocompactness in higher rank
Today, there was the Séminaire Bourbaki, which takes place every four months in Paris. The list of the talks can be found here. Interesting from the point of view „Geometry at Infinity“ was …
Zimmer’s conjecture for actions of SL(m,Z)
Zimmer’s conjecture aims to extend Margulis’ superrigidity theorem to a nonlinear setting. Except for actions on the circle there has not been much progress in the last 35 years. …
What’s hot at mathoverflow 19/17
Is there a geometric interpretation for Reidemeister torsion?
Vladimir Voevodsky 1966-2017
Vladimir Voevodsky – one of the most influential mathematicians of the last decades – deceased yesterday at the age of only 51 years.
What’s hot at mathoverflow 18/17
Open subsets of Euclidean space in dimension 5 and higher admitting exotic smooth structures
Non-positively curved groups and spaces
This week there was the workshop “Non-positively curved groups and spaces” in Regensburg.
Future features at Zentralblatt?
This week there was the common meeting of the Austrian and German mathematical societies at the University of Salzburg and besides many interesting math talks there was also a session about …
What’s hot at mathoverflow 17/17
3D Billiards problem inside a torus
What’s hot at mathoverflow 16/17
Finite-order self-homeomorphisms of \({\mathbb R}^n\)
Quasiconvexity and Dehn filling
One of the most important theorems in 3-manifolds topology is Agol’s Theorem asserting that every closed 3-manifold is virtually fibered and in particular virtually Haken. Agol proved …
Kuhkubus
Spiegel Online reports today (Escher in 3D) on work of Alexander Gürten from the contest Math Creations. It is about bulls tesselating Euclidean space. Have a look at the linked video:
What’s hot at mathoverflow 15/17
Intuition for symplectic groups
Snowflakes and unicorns
The New York Times has an article about the untimely death of Marina Ratner and Maryam Mirzakhani: With Snowflakes and Unicorns, Marina Ratner and Maryam Mirzakhani Explored a Universe in …
A rank-one CAT(0) group is determined by its Morse boundary
In this series we will introduce and discuss new preprints that are somehow related to the topics of the programme “Geometry at Infinity”. By incidence, this first article fits …
Curvature and global shape 2017
Some impressions from the conference “Curvature and global shape” that took place July 24-28 in Münster.
First blog post
This blog will eventually be integrated into the website of the DFG priority programme “Geometry at Infinity“. It will discuss all kind of things that might be of potential interest to …