Topological quantum computation with non-Abelian anyons offers a promising path toward fault-tolerant universal quantum computation. However, the practical realization of such a system remains challenging due to the difficulty of finding suitable topological materials. In this work, we provide a comprehensive blueprint for constructing a large-scale quantum computer based on the quantum double model $\mathcal{D}(S_3)$, a specific non-Abelian topological order. We implement logical computations using quantum circuits on qubits and qutrits, including a single non-Clifford gate, compatible with near-term quantum devices. This work bridges the gap between abstract mathematical frameworks and noise-resilient quantum computation on near-term devices. Our proposal offers a promising path to realize an anyon-based quantum computer.

In recent years there was a significant progress in the theory of pseudo-differential operators on filtered manifolds. In my talk I will introduce a wider class of manifolds which I call manifolds with a tangent Lie structure. I will explain a coarse approach to pseudo-differential theory which gives a simplified pseudo-differential calculus containing only operators of order 0 and negative order. This calculus easily leads to the Atiyah-Singer type index theorem for operators of order 0 on manifolds with a tangent Lie structure. For filtered manifolds this calculus agrees with the known Hormander and van Erp - Yuncken calculi, which allows to extend the index theorem to operators of any order.

In the last decade the observer moduli space of Riemannian metrics with positive curvature conditions have become more and more popular. So far, results in this direction only work if the dimension of the underlying manifold is bigger than 5 or if the manifold is spin. I will present a different construction that works in dimension 4 and does not rely on spin geometry.

The coarse Novikov conjecture claims that a certain assembly map from the K-homology of a metric space to the K-theory of its Roe algebra is injective. It was verified for a large class of metric spaces including the space admitting a fibred coarse embedding into Hilbert space. In this talk, I will talk about the notion of fibred coarse embedding introduced by X. Chen, Q. Wang and G. Yu, and its generalization, so called FCE-by-FCE structure. I will also talk about the coarse Novikov conjecture for a space with an FCE-by-FCE structure. This is based a joint work with L. Guo, Q. Wang and G. Yu.

In this talk, I will give a gentle overview of various incarnations of the Hodge Theory: abelian and nonabelian, complex, p-adic and mod p. I will then explain some recent advances in the mod p nonabelian setting, emphasizing how they fit into the grand picture. I try to make this talk accessible to all kinds of geometers.