Ana Caraiani is standing on the bridge near the Imperial College London campus and she does work in mathematics that bridges distant areas of the field.
Philipp Ammon is a writer.
The mathematician Andrew Wiles gave Ana Caraiani a challenging problem for her senior thesis. Caraiani had less luck with her assigned problem than he did for his 1994 proof of Fermat's Last Theorem. She remained undaunted despite the fact that she didn't make much headway.
She said that the exercise wasn't meant to solve the problem. I think Wiles was telling me that I shouldn't be working on things that I know how to do. It is worth taking on problems that are hard.
She learned how to do mathematical research. You can't always follow a linear path. If you get stuck on one part of the problem, you should move on to another part. Caraiani's experience has helped her in her work on a broad collaborative effort to connect disparate areas of mathematics. The Langlands program is one of the biggest, most ambitious and most challenging undertakings in mathematics today and was built on work by Robert Langlands in the 1960s.
Caraiani is a professor at Imperial College London and a Royal Society university research fellow. She faced obstacles that had nothing to do with her ability when she was a child. She won two gold medals in two years and a silver medal in 2001, becoming the first woman in a decade to qualify for the International Mathematical Olympiad as a high school student. Despite her success, she did not feel welcome or encouraged.
She said that some people, including math teachers, told her not to get her hopes up. I wanted to prove them wrong.
Caraiani talked with Quanta about her experiences in mathematics and her work on the Langlands program, which has been described as a way to create a grand unified theory of mathematics. The interview was edited for clarity.
The Langlands program is one of the most ambitious projects in mathematics today. She often meets with her colleagues at the Dalby Court at Imperial College.
Philipp Ammon is a writer.
I faced explicit discouragement in high school, whereas today there is a lot of explicit encouragement. People around me have experienced subtle forms of discrimination. It is harder to build a research program and establish long-term relationships if others view you differently. It is harder to get taken seriously if you are always having to prove yourself.
I have made my name and have been luckier than most. I know that mathematics is not inclusive for all people, not just women. There is a huge barrier to entry in my field, the Langlands program, which requires so much specialized knowledge.
I am doing what I can to help others in this field, but I feel it is not enough. I try to make space for women at conferences and within my own research group. I am happy that my group has a higher percentage of women.
What drew you to that field?
Wiles encouraged me to go to Harvard to study with Richard Taylor, who helped work out a key piece of the Fermat proof. I knew I would do Langlands because he did it.
There was more to it for me. Forging connections between different areas of mathematics is what the Langlands program is about. I realized that if I chose this area, I wouldn't have to limit myself to just one thing in math. We can throw everything we have at them in the hopes of making progress when we don't know how to prove.
What is happening in this enterprise?
A lot of what my colleagues and I do involves building bridges between two different sectors, which are called Galois groups and representations on one side and modular forms and their generalizations on the other.
Let's start with Galois groups. We can use x2 - 3 - 0 as an example. There is an obvious symmetry between the two numbers, as they are reflections of each other across the y. The group of roots to the equations is not the group of symmetries that are shared by the roots of the equation.
The equations get very complicated when you start looking at degree 5 with terms like x5 or y5. A galois representation is a way of simplification. Instead of looking at the whole group, you look at the parts. It is similar to converting a 3D object into a 2D picture, but the picture doesn't have all the information the original object had, but it still contains enough for some purposes.
These connections are intellectually satisfying. There is a practical value too.
The other side of the divide?
A modular form is a highly symmetric function on the upper half of the complex plane, in which the x- axis represents real numbers and the y- axis represents imaginary numbers. We consider this a nice function because it is smooth and doesn't have jagged edges. That is another way of saying it is not the same as before.
The upper plane can be divided into little regions. If you know the value the function takes in one tiling, you can predict the value it will assume in another. We can glue some of the infinitely many tilings to other tilings to create a modular curve.
The Langlands program shows that they are similar.
The bridge from modular forms to Galois representations was first constructed in the 1970s and has been strengthened ever since.
You can get a string of numbers from the modular-form side and the same string of numbers from the Galois side using completely different procedures. The modular forms are highly symmetric functions and you have to rewrite them in order to get the same results. You can get a string of numbers from the coefficients. You can get the same string of numbers from the Galois side.
It is still stunning to me that this can happen in practice because of the amount of mathematics you have to go through.
A lot of what my colleagues and I do is building bridges.
Philipp Ammon is a writer.
That is correct. The first bridge only takes one way. The Taylor-Wiles method was invented to complete the proof of Fermat's Last Theorem and is used to go from Galois representations to modular forms. We can travel in both directions now.
Why do you go to this trouble? What do these bridges allow you to do?
It is satisfying to show a similarity between parts of math. There is a practical value too. If the problem is mathematical, it can be easier to solve it on one side. Sometimes a difficult problem can be solved by going to the other side and doing more things. You have to be able to move freely in both directions if you want to prove something.
One of the biggest goals in the field is to figure out how to build our bridges in more general settings. We can keep expanding the Langlands program by doing that.
How did you help build these bridges?
The Taylor-Wiles method worked well in two dimensions but not in three. Frank Calegari and David Geraghty came up with a plan to boost it so it could be used in a 3D setting. They said that this strategy would work if they could solve three of the theories they had formulated.
In order to accommodate new phenomena that occurred in the 3D setting, Peter and I made the first bridge bigger and wider than the original 2D version.
In late 2015, we realized that we could use the work we had just done to solve the second question about where the bridge would land. We came up with a strategy that we thought could work. Taylor suggested that we organize a workshop at the Institute for Advanced Study to solve the second conjecture.
Caraiani doesn't think the Langlands program will explain everything in mathematics, but she thinks it may connect to every area of the field someday.
Philipp Ammon is a writer.
The proof goes from a very geometric one to a very number-theoretic one. We decided we weren't the best people to do the number-theoretic part, so we ended up doing the geometric part. If we kept this to ourselves, progress would be quicker.
Was it?
We found a way to get around the third conjecture after we solved the second one. The bridge was built in the opposite direction from the Galois side. The roadblock that had hindered the Taylor-Wiles method was successfully carried past. The bridge worked in different dimensions. The paper was posted online on Christmas Day and is currently being reviewed.
Where does that leave you now?
Two special cases were proved by our treatment of Calegari and Geraghty. I am working with James to solve the most general version of the conjecture.
Even though we bypassed it before, I am still interested in the third conjecture because it predicts things about related objects called Shimura varieties, which I want to understand better.
There are settings where we don't know how to build a bridge. One of the biggest goals in the field is to figure out how to construct our bridges in more general settings using arbitrary number systems and arbitrary polynomials. We can keep expanding the Langlands program by doing that.
How far can unification go?
Langlands theory will not evolve in a way that it can explain everything in mathematics, though I think it may touch on every area.
Robert Langlands had a grand vision. The field is broadening its scope because of the network of conjectures he formulated decades ago. The more bridges we cross, the more we think of new places to go. When we make progress, it gives us a better idea of how much more we can do. I don't think anyone expects this program to be finished soon.
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