Roadmap Algorithms for Semi-Algebraic Sets: A Survey

Abstract: Designing singly exponential, rather than doubly exponential, algorithms has been a central theme in algorithmic real algebraic geometry. One of the first major advances beyond emptiness testing was a singly exponential algorithm for computing the number of connected components of a semi-algebraic set given by a quantifier-free formula. The key object in this algorithm is a roadmap: a semi-algebraic subset of dimension at most one that captures the connectivity of the original set.

Roadmaps have since become a fundamental tool for connectivity problems in semi-algebraic geometry. In fact, all known singly exponential algorithms for deciding connectivity proceed by computing a roadmap. The complexity of roadmap algorithms has improved steadily over time, through new geometric, algebraic, and algorithmic ideas, while remaining in the singly exponential regime.

In this talk, I will survey the history of roadmap algorithms, describe recent improvements in their complexity, and discuss applications beyond connectivity, including the computation of Betti numbers and higher-dimensional variants of sampling algorithms. This talk is based on joint work with several collaborators, whom I will mention during the talk.

How to Prove It: Computation, Proofs, and Mathematical Rigor

Abstract: Computers have dramatically expanded the scale of experimentation and visualization in mathematics, becoming indispensable tools for formulating conjectures and discovering proofs. This phenomenon largely predates the rise of AI, which nonetheless vastly amplifies its scope and impact. Yet computer-aided mathematics remains a striking blind spot in the rigorous peer-review standards enforced by the most prestigious mathematical journals. This talk examines the role of computational tools in contemporary mathematics, with a particular focus on interactive theorem proving and its potential to finally bridge this gap.

Fast Matrix Multiplication - Theory and Practice

Abstract: Matrix multiplication is everywhere: In AI, algo-trading, simulations, graphics and more. AI / deep neural network applications, spend up to 95% of the time on matrix multiplication. Matrix multiplication is expected to consume over 1% of the electricity worldwide this year. In recent years, the tech giants responded with high-performance math software (for GPU and CPU) and with hardware accelerators. Yet, all state-of-the-art solutions are based on the wasteful cubic-time classic matrix multiplication algorithm, despite more than five decades of research on sub-cubic time algorithms. Why is that?

In this talk I will present a brief history of the ongoing race (of humans and machines) for faster matrix multiplication. How to multiply large matrices and small ones. Do and don't when precision and reproducibility matter. And what of these algorithmic advances can be used in practice.

Various aspects of creative telescoping: a personal WZ story

Abstract: I will review some lesser known WZ-infected directions including WZ seeds, reduction of holonomic rank, matrix-valued telescoping and modern applications to number theory. As one may guess from this abstract, I happen to be a collector of unusual applications of creative telescoping, so that any further examples of this sort from the participants of ISSAC -- before, during and after the symposium -- will be highly appreciated.