Following a conference in Hatfield.
The idea that parts of mathematics manifest themselves in different ways in different places.
Many cohomologies give the same result when applied to a variety. Grothendieck’s idea of a universal entity of which these are avatars.
See Mathematical Robustness and very interesting comment by Denis-Charles Cisinski.
The kind of things I see as ‘robust’ are for example some particular objects from geometry. For instance, the projective spaces (or more generally, the Grassmanians) are robust in the sense they make sense in very different contexts, namely different geometries: it is almost sufficient to make sense of ${\mathrm{GL}}_{n}$. This makes some other complicated objects robust as well: singular cohomology, de Rham cohomology, K-theory and cobordism are robust in the sense they make sense in very different contexts. This might mean that what we call geometry is quite robust as well: it can be seen in very wide contexts (algebraic geometry, differential geometry, analytic geometry, Faltings almost algebraic geometry, but also tropical geometry, Toën and Vaquié geometry under $\mathrm{Spec}(\mathbb{Z})$, Durov geometry over commutative monads, and I don’t speak of their derived versions…
Another example is the theory of categories (say 1-categories): this is a very robust theory in the sense that the theory of $(\mathrm{\infty},1)$-categories is just some kind of reincarnation of it: most of the basic statements from 1-category theory extend in a straightforward way to statements in $(\mathrm{\infty},1)$-category theory.
This kind of way of being robust is certainly related to mathematical reality (in the sense of Lautman, say). This is so real (I mean in the practice of the mathematician) that this is also the kind of things we might even hope to axiomatize: for instance, an axiomatic approach to geometries (e.g. J. Lurie already started to do so). This is where the expressive power of category theory becomes wonderful.
But isn’t this different? Avatars of cohomology are new ways to calculate a quantity given a space, whereas above the idea is that a specific cohomology makes sense in different settings.
Like avatars which fail to materialise. Example of simple sporadic groups (‘sporadic groups are avatars’). Need to check out work on pseudofinite groups (models of theory of finite groups). This makes sense of Cisinki’s point about ${\mathrm{GL}}_{n}$, here applied to pseudofinite fields, i.e., ultraproducts.
Borovik: “Re avatars: Grothendieck meant some real hard cases. At a more elmentary level, well-known objects of “everyday” mathematics are frequently manifestations of more general concepts.
My favourite example is cross product of vectors (apparently, formaly introduced by Gibbs in 19th century) which is the Lie algebra of the group of rotations of three-dimensional Euclidean space.
Or Euclid, who never mentioned that angles and distances can be measured by the same kind of numbers.
To be fair, angles and distances in the real world (which Euclid was attempting to model) are different kinds of numbers; that is, they have different physical dimensions. Did Euclid ever mention that angles and ratios of distances can be measured by the same kind of numbers? Perhaps not; this could have led him to measure angles in radians, which he did not do. Perhaps this could not have been understood rigorously before Archimedes? —Toby
Or planar crystallographic groups present in Islamic mosaics.
A mathematical object or structure starts to influence the shape of mathematical theories centuries before it is discovered. This is one of the most worrying problems in history of mathematics.”
Groups as natural kinds blog
Friends of Harvard Mathematics talk
Ideas, principles and notions are very closely related, e.g., each applies to duality.
Philosophy
Undermining justification/discovery distinction, or is it better to say that what we do is widen the scope of justification to include the value of a mathematical theory?
Understanding the operation of constraints in a non-empirical field. Assessment via narrative, Alasdair Macintyre, etc.
Both above present opportunities to link up with other branches of philosophy, e.g., aesthetics and ethics.
Communication of research mathematics to wider audience, etc. as below.
Better management of mathematical research, because we will have helped mathematicians to articulate their activities in a way that research administrators can understand. Plainly, mathematicians don’t need our help in talking to each other. But we all have to explain our work to managers, administrators, etc..
More effective popularisation and integration of mathematics into the wider culture. There is a steady market for popular mathematics–this stuff can enrich it by articulating the research experience.
Helping to resist the flight from maths in schools. STEM and all that.