Kashiwara’s theory of crystal bases provides a powerful tool for studying representations of quantum groups. Crystal bases retain much of the structural information of their corresponding representations, whilst being far more straightforward and ‘stripped-back’ objects (coloured digraphs). Their combinatorial description often enables us to obtain concrete realizations which shed light on the representations, and moreover turn challenging questions in representation theory into far more tractable problems.

After reviewing the construction and basic theory regarding quantum groups, I will introduce and motivate crystal bases as ‘nice q=0 bases’ for their representations. I shall then present (in both finite and affine types) the construction of Young wall models in the important case of highest weight representations. Time permitting, I will finish by discussing some applications across algebra and geometry.

The quadratic Euler characteristic of an algebraic variety is a (virtual) symmetric bilinear form which refines the topological Euler characteristic and contains interesting arithmetic information when the base field is not algebraically closed. For smooth projective varieties, it has a quite concrete expression in terms of the cup product and Serre duality for Hodge cohomology. However, for singular varieties, it is defined abstractly (using either cut and paste relations or motivic homotopy theory) and is still rather mysterious. I will first introduce this invariant and place it in the broader context of quadratic enumerative geometry. I will then explain some progress on concrete computations, first for symmetric powers (joint with Lenny Taelman) and second for conductor formulas for hypersurface singularities (older results with Marc Levine and Vasudevan Srinivas on the one hand, and joint work in progress with Ran Azouri, Niels Feld, Yonathan Harpaz and Tasos Moulinos on the other).