28/10/2024, 17:00 — 18:00 — Amphitheatre Pa1, Mathematics Building
Alfio Quarteroni, Politecnico di Milano, Milan, Italy & École Polytechnique Fédérale de Lausanne, Switzerland
Revisiting the Role of Mathematicians in the Era of Artificial Intelligence
In applied mathematics, effective problem solving begins with precise problem formulation, highlighting the importance of the initial problem-setting phase. Without a clearly defined problem, identifying suitable tools and techniques for resolution becomes arduous and often futile. This transition from problem setting to problem solving is pivotal within the broader framework of knowledge advancement. Despite the remarkable progress of AI tools, they remain reliant on the groundwork laid by human intelligence. Mathematicians, leveraging their adeptness in discerning patterns and relationships within data and variables, play a crucial role during this phase. This lecture will introduce fundamental mathematical concepts encompassing both traditional machine learning and scientific machine learning. The latter offers an optimal platform for the harmonious fusion of problem setting and problem solving, bolstered by profound domain expertise.
See also
Poster
06/06/2023, 11:00 — 12:00 — Abreu Faro Amphitheatre Online
Gilles Brassard, Université de Montréal
Quantum Cryptography from dream to reality (and beyond)
Although practised as an art and science for ages, cryptography had to wait until the mid-twentieth century before Claude Shannon gave it a strong mathematical foundation. However, Shannon's approach was rooted in his own information theory, itself inspired by the classical physics of Newton and Einstein. When quantum theory is taken into account, new vistas open up both for codemakers and codebreakers. Is this a blessing or a curse for the protection of privacy? We shall discuss quantum cryptography, from its humble origins more than a half-century ago to its current blooming, and speculate about prospects for the future. No prior knowledge in cryptography or quantum theory will be assumed.
Biography
Brassard is renowned for his groundbreaking contributions to the field of quantum cryptography, including pioneering work in quantum teleportation, quantum entanglement distillation, quantum pseudo-telepathy, and the classical simulation of quantum entanglement. In collaboration with Charles H. Bennett, Brassard invented the BB84 protocol for quantum cryptography in 1984, which he later extended to include the Cascade error correction protocol. This protocol efficiently detects and corrects noise caused by eavesdropping on quantum cryptographic signals.
In recognition of his outstanding achievements, Brassard has received numerous prestigious awards. He was awarded the Prix Marie-Victorin, the highest scientific award of the government of Quebec, in 2000, and was elected as a Fellow of the International Association for Cryptologic Research in 2006. In 2010, he was awarded the Gerhard Herzberg Canada Gold Medal, Canada's highest scientific honour. Brassard was also elected a Fellow of the Royal Society of Canada and the Royal Society of London in 2013. In December of that year, he was named an Officer in the Order of Canada by the Governor-General of Canada, the Right Honourable David Johnston. In 2018, he received the Wolf Prize in Physics, and in 2019 he was awarded both the BBVA Foundation Frontiers of Knowledge Award in Basic Science and the Micius Quantum Prize.
Most recently, in September 2022, Brassard was awarded the Breakthrough Prize in Fundamental Physics, the world's largest science prize.
30/01/2018, 17:00 — 18:00 — Abreu Faro Amphitheatre
Charles Fefferman, Princeton University
Interpolation and Approximation in Several Variables
Let $X$ be our favorite Banach space of continuous functions on $\mathbb{R}^n$ (e.g. $C^m$ , $C^{m,α}$ , $W^{m,p}$). Given a real-valued function $f$ defined on an (arbitrary) given set $E$ in $\mathbb{R}^n$ , we ask: How can we decide whether $f$ extends to a function $F$ in $X$? If such an $F$ exists, then how small can we take its norm? What can we say about the derivatives of $F$? Can we take $F$ to depend linearly on $f$?
What if the set $E$ is finite? Can we compute an $F$ whose norm in $X$ has the smallest possible order of magnitude? How many computer operations does it take? What if we ask only that $F$ agree approximately with $f$ on $E$? What if we are allowed to discard a few points of $E$ as “outliers”; which points should we discard?
A fundamental starting point for the above is the classical Whitney extension theorem.
The results are joint work with Arie Israel, Bo’az Klartag, Garving (Kevin) Luli, and Pavel Shvartsman.
See also
Poster
