Together with many other people and after a long development I could prove that a Poincaré duality group of cohomological dimension 2 is the group of a Riemann surface. That was actually a conjecture of Jean-Pierre Serre. "You have to prove it!" he had always insisted.

The duality theorems for a manifold will be derived from the corresponding ones for nets and co-nets of group systems ... Of the two types of duality theorems — the and s — the first could be obtained more easily from the corresponding one for a group system and its character system ... But when applied to the Alexander type this method fails. Also in using the net and co-net theory for both types, the less complicated Poincaré theorem serves as an introduction for the Alexander theorem.

Poincaré clearly posed these two problems and... seven years later... independently... Einstein and Poincaré... solved... the problem of defining definition at spatially separated points...

... Formal groups. This topic is by far the deepest and most imaginative creation of Dieudonné, realized when Dieudonné was nearing 50, supposedly the term for an active mathematical life. It can be seen as the creation of a differential calculus for groups over a field of characteristic p > 0 (possibly finite). The methods of calculus do not work, and one has to resort to pure algebra. There were a number of forerunners: a version of Taylor’s formula in characteristic p > 0 due to Dieudonn ́e himself, the ideas of Delsarte about convolution operators (as explained in Book IV, chapter 6 of Bourbaki’s Éléments), a definition of the Lie algebra of a Lie group and its enveloping algebra in terms of distributions on the group (by L. Schwartz). But the impetus came from the book by Chevalley, in 1951, about algebraic groups. Chevalley had developed a purely algebraic version of Lie theory, but restricted to fields of characteristic 0. The case of characteristic p > 0 was “terra incognito”.
In a long series of papers, published between 1954 and 1958, later on collected into a book ... Dieudonné explored in depth this new world.

The quantum dualities, which are known as S-duality or U-duality, extend the classical T-duality and lead to a beautiful and coherent picture of stringy dualities. These exchange highly quantum situations with semiclassical backgrounds, exchange different branes, etc. As in the classical dualities, among all dual descriptions there is at most one description which is natural because it is semiclassical. All other dual descriptions are very quantum mechanical.

Duality Theorems. In: 35, 1948, 188–202.

Poincaré analyses how the reality of three dimensional Euclidean (or non-Euclidean) space, has been constructed from our daily experiences as a human being with the objects that are most important for us (rigid bodies), and closely around. This does not mean that this three dimensional space is an ‘invention’ of humanity. It exists, but the way we have ordered, and later on formalized it, by means of specific mathematical models, does make part of it. In other words, what we call the three dimensional reality of space partly exists in its own and partly exists by the structures that we have constructed, relying on our specific human experience with it.

In the development of the theory of algebraic functions of one variable the introduction by Riemann of the surfaces that bear his name has played a well-known part. Owing to the partial failure of space intuition with the increase in dimensionality, the introduction of similar ideas into the field of algebraic functions of several variables has been of necessity slow. It was first done by Emile Picard, whose work along this line will remain a classic. A little later came the capital writings of Poincaré in which he laid down the foundations of Analysis Situs, thus providing the needed tools to obviate the failure of space intuition.

[I]n 1898... Henri Poincaré wrote... "On the Measure of Time" and he said... there are two fundamental problems to do with time. One... with the definition of duration... What does it mean to say that a second today is the same... [H]e said there's another issue... [not] so widely recognized. ...[H]ow do you define simultaneity at spatially separated points?

All existence seemed to be based on duality, on contrast. Either one was a man or one was a woman, either a wanderer or sedentary burgher, either a thinking person or a feeling person-no one could breathe in at the same time as he breathed out, be a man as well as a woman, experience freedom as well as order, combine instinct and mind. One always had to pay for one with the loss of the other, and one thing was always just as important and desirable as the other.

Although group theory is certainly relevant for nineteenth-century physics, it really started to play an important role with the work of Lorentz and Poincaré, and became essential with quantum mechanics. Heisenberg opened up an entirely new world with his vision of an internal symmetry, the exploration of which continues to this day in one form or another. Beginning in the 1950s, group theory has come to play a central role in several areas of physics, perhaps none more so than in what I call fundamental physics ...

fear is born of duality —

The and the are duals of each other—the vertices of one match the face midpoints of the other and vice versa.

Not all the geometrical structures are "equal". It would seem that the riemannian and complex structures, with their contacts with other fields of mathematics and with their richness in results, should occupy a central position in differential geometry. A unifying idea is the notion of a G-structure, which is the modern version of an equivalence problem first emphasized and exploited in its various special cases by Elie Cartan.