The project has uncovered a rich tradition of constructivist thinking in classical geometry. Anchoring geometry in construction devices that generalize Euclid’s ruler and compass was a way of ensuring rigor, not dissimilar to the motivations that led to a revival of operationalism, positivism, constructivism, and intuitionism in the 20th century. With this lens we have uncovered new layers of meaning in mathematical works of antiquity and the 17th century. With our reconstruction of the underlying research program, the specific technical choices of these works take on a broader significance and are seen to be part of a coherent dialog.

The interplay between mathematics and (theoretical) physics has inspired a large amount of research. This proposal focuses on a particular topic in this area called the Landau- Ginzburg/conformal field theory correspondence. This correspondence dates from the late 80s and early 90s in the theoretical physics literature and it relates very different algebraic structures describing on the one hand boundary conditions and defects for Landau–Ginzburg models and N=2 superconformal field theories on the other. Thanks to very recent developments in both the theory of Landau-Ginzburg models and mathematical descriptions of conformal field theories (leading to some first examples of this correspondence), we have all the necessary tools for pushing forward our mathematical understanding of this correspondence. The aim of this project is to provide a mathematical statement for the Landau–Ginzburg/ conformal field theory correspondence. For this, first I will complete a classification of examples and explore their properties, which will reveal how a conjecture will look like. This research will involve techniques of representation theory, quantum algebra and category theory. Once this task is achieved, I will embed all these results into a precise conjecture using higher categorical methods. The final stage of this project is to prove this conjecture, and explore further extensions of these results beyond the N=2 superconformal field theories.

Operads are a tool blending algebra and geometry together: they describe intricate algebraic structures, in which elements can be multiplied in many different ways and deformed into each other. They admit applications in many different parts of mathematics, such as differential topology, number theory, and mathematical physics. In this research we investigate operads by finding ways of breaking them up into simpler pieces and then reconstructing them. We will in particular apply our results to the study of embedding spaces between manifolds.

Statistics has taken a more and more prominent role in our world. Arithmetic statistics concerns the average or limiting behavior of arithmetic objects. With the help of computers, we have produced vast amounts of data for these arithmetic objects, which exhibit a lot of random behavior. It is however often already a non-trivial task to develop heuristics for the random behavior of arithmetic objects, let alone actually prove randomness. The researcher will rigorously prove some of the available heuristics in several important cases.

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