Spherical Designs are very special points on the sphere with the property that the average of a low-degree polynomial over the points is the same as the global average of the polynomial on the sphere. As it turns out, the definition can be suitably interpreted to make sense on a finite combinatorial Graph as well.  The arising structures are breathtakingly pretty (many pictures will be shown). They can be interpreted as the analogue of Platonic bodies in graphs. Graphs can have many more symmetries than Euclidean space and, correspondingly, some of these point structures are remarkably symmetric. This is also naturally related to Extremal Combinatorics where classical Theorems (the Erdos-Ko-Rado Theorem or the Deza-Frankl theorem)  suddenly turn into beautiful special cases.   If we only consider the hypercube graph {0,1}^d, we naturally encounter problems from coding theory.   A probabilistic interpretation tells us new things about the speed with which random walks on the graph become random.   There will be pictures, a survey of recent results by C. Babecki, K. Golubev, D. Shiroma, R. Thomas and many, many open problems.

Given a natural number n and a graph H, the extremal number ex(n, H) is the largest number of edges in an H-free graph on n vertices. One of the outstanding questions concerning extremal numbers, originally posed by Erdos and Simonovits, is the rational exponents conjecture, which asks whether for every rational number r between 1 and 2, there is a graph H such that c n^r < ex(n, H) < C n^r for some positive constants c and C. We will discuss some of the recent progress on this conjecture.

The stochastic block model has been one of the most fruitful research topics in community detection and clustering. Recently, community detection on hypergraphs has become an important topic in higher-order network analysis. We consider the detection problem in a sparse random tensor model called the hypergraph stochastic block model (HSBM). We prove that a spectral method based on the non-backtracking operator for hypergraphs works with high probability down to the generalized Kesten-Stigum detection threshold conjectured by Angelini et al (2015). We characterize the spectrum of the non-backtracking operator for sparse random hypergraphs and provide an efficient dimension reduction procedure using the Ihara-Bass formula for hypergraphs. As a result, the community detection problem can be reduced to an eigenvector problem of a non-normal matrix constructed from the adjacency matrix and the degree matrix of the hypergraph. Based on joint work with Ludovic Stephan (EPFL). https://arxiv.org/abs/2203.07346

Hamilton-Jacobi PDEs and optimal control problems are widely used in many practical problems in control engineering, physics, financial mathematics, and machine learning. For instance, controlling an autonomous system is important in everyday modern life, and it requires a scalable, robust, efficient, and data-driven algorithm for solving optimal control problems. Traditional grid-based numerical methods cannot solve these high-dimensional problems, because they are not scalable and may suffer from the curse of dimensionality. To overcome the curse of dimensionality, we developed several neural network methods for solving high-dimensional Hamilton-Jacobi PDEs and optimal control problems. This talk will contain two parts. In the first part, I will talk about SympOCNet method for solving multi-agent path planning problems, which solves a 512-dimensional path planning problem with training time of less than 1.5 hours. In the second part, I will show several neural network architectures with solid theoretical guarantees for solving certain classes of high-dimensional Hamilton-Jacobi PDEs. By leveraging dedicated efficient hardware designed for neural networks, these methods have the potential for real-time applications in the future. These are joint works with Jerome Darbon, George Em Karniadakis, Peter M. Dower, Gabriel P. Langlois, and Zhen Zhang.

Community detection in the stochastic block model is one of the central problems of graph clustering. In this setup, spectral algorithms have been one of the most widely used frameworks for the design of clustering algorithms. However, despite the long history of study, there are still unsolved challenges. One of the main open problems is the design and analysis of ``simple'' spectral algorithms, especially when the number of communities is large. 

In this talk, I will discuss two algorithms. The first one is based on the power-iteration method. Our algorithm performs optimally (up to logarithmic factors) compared to the best known bounds in the dense graph regime by Van Vu (Combinatorics Probability and Computing, 2018). 

Then based on a connection between the powered adjacency matrix and eigenvectors, we provide a ``vanilla'' spectral algorithm for large number of communities in the balanced case. Our spectral algorithm is as simple as PCA (principal component analysis).

This talk is based on joint works with Chandra Sekhar Mukherjee. (https://arxiv.org/abs/2211.03939)