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

Geometric Invariant Theory is a central tool in the construction of moduli spaces, and shares the property ubiquitous among such tools that certain so-called 'unstable' objects must be excluded if the moduli space is to be well behaved. However, instability in GIT is a structured phenomenon: after making a choice of a certain invariant inner product, one has the HKKN stratification of the parameter space which, morally, sorts the objects according to how unstable they are. I will explain how one can use recent results of Berczi-Doran-Hawes-Kirwan in Non-Reductive GIT to perform quotients of these unstable strata as well, extending the classifications given by classical moduli spaces. This can be carried out, at least in principle, for any moduli problem that can be posed using GIT, and I will discuss two examples in particular: unstable (i.e. singular) curves, and coherent sheaves of fixed Harder-Narasimhan type. The latter of these is joint work with Gergely Berczi, Victoria Hoskins and Frances Kirwan.
 

I will explain how the definition of Bridgeland stability condition on a triangulated category C can be generalised to allow for massless objects. This allows one to construct a partial compactification of the stability space Stab(C) in which each `boundary stratum' is related to Stab(C/N) for a thick subcategory N of C, and has a neighbourhood which fibres over (an open subset of) Stab(N). This is joint work with Nathan Broomhead, David Pauksztello, and David Ploog.
 

Enumerative  invariants since 1995 are defined as integrals of cohomology classes over a particular homology class, called the virtual fundamental class. When there is a torus action, the virtual fundamental class is localized to the fixed points and this turned out to be the most effective technique for computation of the virtual integrals so far. About 10 years ago, Jun Li and I discovered that when there is a cosection of the obstruction sheaf, the virtual fundamental class is localized to the zero locus of the cosection. This also turned out to be quite useful for computation of Gromov-Witten invariants and more. In this talk, I will discuss a generalization of the cosection localization to real classes which provides us with a purely topological theory of Fan-Jarvis-Ruan-Witten invariants (quantum singularity theory) as well as some GLSM invariants. Based on a joint work with Jun Li at arXiv:1806.00116.