L6

In an article published in 2009, Dave Benson described, for a finite group $G$, the mod $p$ homology of the space $\Omega(BG^\wedge_p)$ --- the loop space of the $p$-completion of $BG$ --- in purely algebraic terms. In joint work with Carles Broto and Ran Levi, we have tried to better understand Benson's result by generalizing it. We showed that when $\mathcal{C}$ is a small category, $|\mathcal{C}|$ is its geometric realization, $R$ is a commutative ring, and $|\mathcal{C}|^+_R$ is a plus construction of $|\mathcal{C}|$ with respect to homology with coefficients in $R$, then $H_*(\Omega(|\mathcal{C}|^+_R);R)$ is the homology any chain complex of projective $R\mathcal{C}$-modules that satisfies certain conditions. Benson's theorem is then the special case where $\mathcal{C}$ is the category associated to a finite group $G$ and $R=F_p$, so that $p$-completion is a special case of the plus construction.
 

The extremal number ${\rm ex}(n,F)$ of a graph $F$ is the maximum number of edges in an $n$-vertex graph not containing $F$ as a subgraph. A real number $r \in [0,2]$ is realisable if there exists a graph $F$ with ${\rm ex}(n , F) = \Theta(n^r)$. Several decades ago, Erdős and Simonovits conjectured that every rational number in $[1,2]$ is realisable. Despite decades of effort, the only known realisable numbers are $0,1, \frac{7}{5}, 2$, and the numbers of the form $1+\frac{1}{m}$, $2-\frac{1}{m}$, $2-\frac{2}{m}$ for integers $m \geq 1$. In particular, it is not even known whether the set of all realisable numbers contains a single limit point other than two numbers $1$ and $2$.

We discuss some progress on the conjecture of Erdős and Simonovits. First, we show that $2 - \frac{a}{b}$ is realisable for any integers $a,b \geq 1$ with $b>a$ and $b \equiv \pm 1 ~({\rm mod}\:a)$. This includes all previously known ones, and gives infinitely many limit points $2-\frac{1}{m}$ in the set of all realisable numbers as a consequence. Secondly, we propose a conjecture on subdivisions of bipartite graphs. Apart from being interesting on its own, we show that, somewhat surprisingly, this subdivision conjecture in fact implies that every rational number between 1 and 2 is realisable.

This is joint work with Jaehoon Kim and Hong Liu.

The Oberwolfach Research Institute for Mathematics (Mathematisches Forschungsinstitut Oberwolfach/MFO) was founded in late 1944 by the Freiburg mathematician Wilhelm Süss (1895-1958) as the „National Institute for Mathematics“. In the 1950s and 1960s the MFO developed into an increasingly international conference centre.

The aim of my project is to analyse the history of the MFO as it institutionally changed from the National Institute for Mathematics with a wide, but standard range of responsibilities, to an international social infrastructure for research completely new in the framework of German academia. The project focusses on the evolvement of the institutional identity of the MFO between 1944 and the early 1960s, namely the development and importance of the MFO’s scientific programme (workshops, team work, Bourbaki) and the instruments of research employed (library, workshops) as well as the corresponding strategies to safeguard the MFO’s existence (for instance under the wings of the Max-Planck-Society). In particular, three aspects are key to the project, namely the analyses of the historical processes of (1) the development and shaping of the MFO’s workshop activities, (2) the (complex) institutional safeguarding of the MFO, and (3) the role the MFO played for the re-internationalisation of mathematics in Germany. Thus the project opens a window on topics of more general relevance in the history of science such as the complexity of science funding and the re-internationalisation of the sciences in the early years of the Federal Republic of Germany.

We consider the behaviors of global solutions to the initial value problems for the multi-dimensional Navier-Stokes(Euler)-Fokker-Planck equations. It is shown that due to the micro-macro coupling effects of relaxation damping type, the sound wave type propagation of this NSFP or EFP system for two-phase fluids is observed with the wave speed determined by the two-phase fluids. This phenomena can not be observed for the pure Fokker-Planck equation and the Navier-Stokes(Euler) equation with frictional damping.

We discuss shock reflection problem for compressible gas dynamics, von Neumann conjectures on transition between regular and Mach reflections. Then we describe recent results on existence and uniqueness of regular reflection solutions for potential flow equation, and discuss some techniques involved in the proof. The approach is to reduce the shock reflection problem to a free boundary problem, and prove existence and uniqueness by a version of method of continuity. This involves apriori estimates of solutions in the elliptic region of the equation of mixed type, with ellipticity degenerating on some part of the boundary. For the proof of uniqueness, an important property of solutions is convexity of the free boundary. We will also discuss some open problems.

This talk is based on joint works with G.-Q. Chen and W. Xiang.

In the 6th century, the phenomena of irregularity of the solar motion and parallax of the moon were found by Chinese astronomers. This made the calculation of solar eclipse much more complex than before. The strategy that Chinese calendar-makers dealt with was different from the geometrical model system like Greek astronomers taken as. What Chinese astronomers chose is a numerical algorithm system which was widely taken as a thinking mode to construct the theory of mathematical astronomy in old China. 

Relatively little is known about the correspondence of William Burnside, a pioneer of group theory in the UK. There are only a few dozen extant letters from or to him, though they are not without interest. However, one of the most noteworthy letters to or at least about him, in that it had a special mention in his obituary in the Proceedings of the Royal Society, has not been positively identified. It's not clear who it was from or when it was sent. We'll look at some possibilities.

The International Congresses of Mathematicians (ICMs) have taken place at (reasonably) regular intervals since 1897, and although their participants may have wanted to confine these events purely to mathematics, they could not help but be affected by wider world events.  This is particularly true of the 1936 ICM, held in Oslo.  In this talk, I will give a whistle-stop tour of the early ICMs, before discussing the circumstances of the Oslo meeting, with a particular focus on the activities of the Nazi-led German delegation.

The emergence of analytic methods in the 17th century opened a new way in order to tackle the elucidation of certain quantities. The strong presence of the circle-squaring problem, focused mainly the attention on π, on which besides the serious doubts about its rationality, it arises an awareness---boosted by the new algebraic approach---of the difficulty of framing it inside algebraic boundaries. The term ``transcendence'' emerges in this context but with a very ambiguous meaning.

The first great step towards its comprehension, took place in the 18th century and came from Johann Heinrich Lambert's hand, who using a new analytical machinery---continued fractions---gave the first proof of irrationality of π. The problem of keeping this number inside the algebraic limits, also receives an especial attention at the end of his Mémoires sur quelques propriétés remarquables des quantités transcendantes, circulaires et logarithmiques, published by the Berlin Academy of Science in 1768. In this work, Lambert after giving to the term ``transcendence'' its modern meaning, conjectures the transcendence of π and therefore the impossibility of squaring the circle.

TBC