Counting Curves | www.caltech.edu

Whereas counting is the primary ability a baby learns in arithmetic, it is also one thing studied on the highest ranges of the self-discipline, albeit in a extra thrilling manner. New professor of arithmetic Tony Yue Yu analysis entails counting curves in a geometrical house, which locations his work within the subject of enumerative geometry. One of many earliest examples of enumerative geometry is the Drawback of Apollonius, named after a mathematician in historic Greece. On this downside, one counts the variety of circles which can be tangent to 3 given circles in a airplane (black within the illustration). There are generally eight such tangent circles; one is proven in pink.

Such questions will not be solely intuitively interesting, but additionally virtually necessary as a result of counting geometric objects with sure constraints is similar sort of downside as counting the variety of options to a system of equations.

Yu is growing a brand new idea of curve counting through what known as non-Archimedean geometry. Usually, whenever you take two numbers reminiscent of 1 and 100, through which one quantity is lower than the opposite, you possibly can add the lesser quantity to itself repeatedly to finally surpass the better quantity (100). This idea stems from work by Archimedes, an historic Greek mathematician. Nevertheless, in non-Archimedean geometry, you possibly can maintain including up smaller numbers however you’ll by no means surpass the bigger quantity. Yu explains that these “unique, non-intuitive” numbers are on the coronary heart of his work.

Yu, who grew up in Ningbo, China, accomplished his undergraduate research in arithmetic at Peking College in 2010, and later accomplished his graduate research on the Ecole normale supérieure in Paris. He was a everlasting researcher within the French Nationwide Centre for Scientific Analysis till becoming a member of the Caltech school in 2021.

We met with Yu over Zoom to be taught extra about his analysis and the way it pertains to an idea acquainted to physicists often called mirror symmetry.

When did you first develop into occupied with math?

I’ve been fascinated by math and science since childhood. Rising up in Ningbo, there have been loads of science actions and competitions for teenagers, and I loved all of them. I went to Peking College, majoring in arithmetic, as a result of in highschool I used to be capable of learn college textbooks on many different science topics however could not perceive a lot from the mathematics textbooks. I grew to become very interested in trendy arithmetic.

Then I went to Ecole normale supérieure in Paris for graduate faculty. Paris is the birthplace of recent algebraic geometry, based by Alexander Grothendieck within the Sixties. Individuals wish to say that Paris is the middle of the style world, however it’s also a middle of mathematical analysis in lots of areas. As soon as in Paris, surrounded by so many mathematicians, it was pure for me to pursue mathematical analysis.

Are you able to inform us extra about non-Archimedean geometry?

The Archimedean property says that given any two optimistic numbers A < B, if we add A to itself sufficiently many instances, the sum A+ ⋯ +A will finally exceed B.

You’ll say that that is apparent as a result of that is how size behaves in our every day life. Nevertheless, in trendy arithmetic, there’s a nice curiosity to check portions and geometric areas the place the Archimedean property fails. We name them non-Archimedean numbers and non-Archimedean areas. On this realm, numbers don’t fall on a quantity line nor symbolize any notion of distance. They’re unique and do not match our instinct.

Non-Archimedean geometry is a department of algebraic geometry, the place we research geometric shapes outlined over non-Archimedean numbers. Since we don’t dwell in a non-Archimedean world, it’s arduous to check non-Archimedean areas, and plenty of analysis mathematicians thought of it to be a troublesome and summary subject.

I took an interest within the subject whereas I used to be a graduate pupil in Paris. At some point I requested my advisor what a non-Archimedean house is. He replied that it’s some very “furry” house. I used to consider mathematical objects as austere and solemn, and I could not imagine how a geometrical house might be furry like an animal! I grew to become very fascinated by the topic afterward.

Are you able to inform us extra about enumerative geometry?

Enumerative geometry is about counting geometric objects, such because the Drawback of Apollonius, through which one counts circles in a airplane. Whereas it’s enjoyable and necessary to rely and compute the exact numbers, the joys of the sphere is to find deeper structural relations behind these numbers. One of the mysterious relations is described by so-called mirror symmetry, which is a duality of shapes first found by theoretical physicists learning string idea, a mathematical idea that goals to explain the elemental particles and forces in nature.

No matter whether or not string idea might be confirmed by experiments, it has made an excellent affect on mathematical analysis. In mirror symmetry, the numbers of curves in a single house might be associated to options of differential equations on the mirror house. The complete extent of this phenomenon, in addition to its underlying mathematical mechanism, are nonetheless largely unknown.

What issues are you engaged on?

I initiated the research of enumerative geometry utilizing non-Archimedean strategies, specifically with the purpose of fixing conjectures within the subject of mirror symmetry. Actually, non-Archimedean areas seem naturally within the research of mirror symmetry through a course of often called degeneration of areas. One can consider degeneration as crashing an enormous difficult house into smaller, easier damaged items. The parameter for the degeneration turns into a non-Archimedean quantity, and the degeneration course of offers rise to a non-Archimedean house. Nevertheless, most researchers weren’t eager on making use of non-Archimedean strategies to curve counting issues as a result of non-Archimedean geometry was thought of to be unique and troublesome. My analysis has been exploring this route in the previous couple of years, and it has turned out to be a rewarding expertise.

I purpose to additional develop this non-Archimedean strategy and hopefully make new contributions to the mathematical basis of mirror symmetry. It’s also necessary to match this work with strategies studied by different researchers. Whereas mirror symmetry fully revolutionized the sphere of enumerative geometry, I additionally stay up for exploring purposes of mirror symmetry to broader areas of algebraic geometry, reminiscent of moduli idea and birational geometry. Each concern the classification of areas, and classification has at all times been a central theme throughout totally different areas of arithmetic.

Is there anything you wish to add?

Along with being the indispensable software for science and expertise, math can be an artwork. Generally you hear a bit of music and also you prefer it. Math has plenty of surprises however requires years of coaching to completely admire.