At swissQuant, we are committed to harnessing the transformative power of mathematics in finance and technology. As we approach International Day of Mathematics on 14 March, we are excited to highlight the profound impact of mathematical concepts through the insights of Professor Alessio Figalli, a distinguished member of our Scientific Advisory Board and Fields Medal Laureate.
Professor Figalli’s expertise in optimal transport not only highlights the historical significance of mathematical concepts, but also their modern applications that continue to revolutionise various industries. Read more about Prof. Figalli’s background and work at ETH Zurich below.
Understanding optimal transport
“Optimal transport is a very old topic; it is about moving material from one place to another in the most efficient way possible,” explains Professor Figalli.
Key takeaways:
- Historical significance: Optimal transport dates back to the French Revolution, pioneered by mathematician Gaspard Monge, who sought efficient ways to move materials for military fortifications.
- Economic and mathematical development: Initially viewed as an economic problem, optimal transport has evolved into a complex mathematical challenge, attracting interest for its depth and the variety of problems it can solve.
- Practical applications in modern technology: Optimal transport finds practical applications in Artificial Intelligence and machine learning, especially in image comparison and optimisation problems.
- Interdisciplinary connections: The principle of optimal transport extends beyond mathematics, influencing fields such as fluid dynamics, probability theory, and atmospheric phenomena.
- Future potential: Its adaptability in machine learning algorithms promises more stable and efficient outcomes in optimisation problems.
The journey of optimal transport
From historical roots to modern applications
Optimal transport, initially a logistical strategy during the French Revolution, has transcended its military origins to become a cornerstone of modern computer science.
“It started during the French Revolution and then during the Napoleonic campaign, when Gaspard Monge, a French mathematician, wanted to understand the most efficient way to take materials, extract them, and then move them to places where they were needed to build fortifications,” reflects Professor Figalli.
This evolution from a tactical military tool to a multifaceted mathematical concept exemplifies the adaptive nature of mathematical theories.
A quantum leap: from logistics to economics
The 20th century ushered in a new chapter for optimal transport as it transitioned from the battlefield to the economic arena.
The visionary Russian mathematician Kantorovich reconceived optimal transport as an economic model, demonstrating how it could streamline the distribution of goods, thereby reducing costs.
His pioneering work not only earned him the Nobel Prize in Economics but also expanded the horizons of optimal transport into new domains.
Exploring the diverse applications of optimal transport
Optimal transport, a field that has intrigued mathematicians for decades, is recognised for its broad applicability in various fields of mathematics and science. Its profound impact is evident in areas ranging from fluid dynamics to machine learning, showcasing the versatility of this mathematical concept.
- Fluid dynamics: The relationship between optimal transport and fluid dynamics proved to be a pivotal discovery. Techniques from optimal transport have been instrumental in studying the complex equations that govern fluid movement.
- Probability and differential equations: Subsequent explorations revealed the relevance of optimal transport in probability theory and the study of partial differential equations, further expanding its utility across mathematical disciplines.
Revolutionising AI and machine learning
In today’s digital world, optimal transport plays an important role in Artificial Intelligence and machine learning, especially in image comparison.
Professor Figalli elaborates on this application:
“This means that I can think of optimal transport as a way to compare pictures because you give me two pictures made of pixels, and then I ask myself how expensive it is for me to transport all the pixels present in the first image to the pixels in the second image.”
This approach not only enriches our understanding of data comparison but also advances the ability of AI technologies to recognise and process complex patterns, underscoring their utility in the rapidly evolving technological landscape.
Optimisation: applications in Natural Sciences and Engineering
In meteorology, the movement of clouds and atmospheric fronts is described by complex equations. Optimal transport theory underlies this process, as clouds, composed of tiny water particles, strive for efficiency in their movement through the air.
Clouds transition from one configuration to another over time, with particles being optimally transported from one point to another within the Cloud. This hidden optimisation process reflects nature’s tendency to seek optimal solutions in the movement of clouds.
In the technological realm, optimal transport has found a novel application as a loss function in machine learning algorithms. By facilitating the comparison of data distributions, it offers a fresh perspective on optimisation challenges, promising improved stability and convergence of algorithmic solutions.
Envisioning the future: the infinite potential of Mathematics
As we anticipate new discoveries, the journey of optimal transport continues to inspire innovation. Its evolution from a practical solution to a profound mathematical concept and a tool for modern technology reflects the transformative power of mathematics, promising endless possibilities in quantitative finance and beyond.
The journey of optimal transport from its origins to its role in cutting-edge technologies is a powerful reminder of the endless possibilities of mathematical exploration.
At swissQuant, we are continually inspired by the ways in which mathematical principles can be harnessed to innovate and solve real-world challenges.
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