# Teaching Math on the Global Stage

When he isn’t teaching algebra at Columbia or abroad, Amadou Bah likes to soak up the cultural offerings on campus.

Amadou Bah, a Ritt Assistant Professor of mathematics, was raised in Senegal and educated in Paris. Months after finishing his dissertation in 2021, he moved to New York to join Columbia. This spring, Bah visited Gabon to take part in the Algebraic Days of Gabon, a math event designed to inspire young researchers, an experience he described as a special opportunity to connect with researchers he may not otherwise meet.* Columbia News* caught up with Bah to discuss that experience, and what brought him to Columbia.

**What was your recent talk at the conference in Gabon about?**

It was a series of four lectures on the Weil conjectures for curves over finite fields. The hardest part of these conjectures is also known as the Riemann hypothesis over finite fields because it is a geometric analogue of the famous (and still unsolved) Riemann hypothesis related to the distribution of prime numbers. I was focusing on describing a proof André Weil gave, in the late 40s, of his conjectures in the dimension one case (curves), which he also formulated in higher dimension.

Vaguely speaking, the conjectures relate, in a very striking and nontrivial way, two different worlds: on the one hand, the arithmetic of multivariable polynomial equations (namely the counting of their solutions over finite fields of cardinality the power of a fixed prime number), and on the other the topology of the geometric space they define; the latter is captured, in the case of curves, by the invariant called the genus, which is the number of holes in the space when it is viewed as a chain of donuts (Riemann surface).

It was really one of the high points of twentieth century mathematics when its proof was completed in the 70s by Pierre Deligne (building on foundations laid by Grothendieck).

What I was hoping to do was introduce tools from algebraic geometry that enter into the proof of this theorem to get the graduate students and recent PhDs who were attending the conference interested in exploring them. A lot of the mathematics that we use today were born out of the attempt to prove this theorem.

It was great to connect with students from the region. The conference I presented at is annual, and if there’s an opportunity, I’d definitely want to go back.

**What brought you to Columbia?**

I studied in Senegal until high school and then moved to Paris for a program that prepared me to study in college and graduate school, which I completed at the Institut des Hautes Études Scientifiques. It’s actually the institution where a lot of the math solving the Weil conjectures (and the final step) was discovered.

I moved to New York right after defending my PhD thesis, in 2021. I like the diversity of the city, and all the opportunities I have to explore its parks, museums, and restaurants.

**How common is it for students from Senegal to choose the path you did, moving to France for higher education?**

It’s very common. Senegal was a French colony for some time, so there’s a history of students moving to Paris that goes back around 100 years. The president of Senegal was educated in France almost a century ago. The more unusual part of my journey might be studying math, or moving to the U.S.

**What does your schedule look like as a Ritt assistant professor?**

This year I taught three algebra classes, and co-organized a seminar called Automorphic Forms and Arithmetic. We’d have outside speakers come every Friday, often visiting from universities around the U.S. and international ones, too. It was a busy year.

**How would you describe your work to a family member or friend with no background in math?**

That’s hard, actually. It’s one of the tragedies of mathematics: We’re doing all these beautiful things, but it’s very hard to share it with other people. Maybe if I had an hour I could give them a good idea. What I generally tell people is that I’m working on arithmetic geometry. A bit more specifically: What I’m most interested in is ramification theory for Galois representations, and connecting recent developments there to some aspects of the Langlands Program, which Columbia has strong experts in, including my postdoctoral advisor, Michael Harris.

Now the beauty in all this is somewhat subjective and what I mean by it has changed over time. At first, what attracted in mathematics were the logical consistency and the sense of order (each piece falling into place) that a proof exudes. Nowadays, I am more excited by the surprising: an unexpected tool makes an appearance in a proof; often, I later on realize that my surprise was a reflection of my ignorance at that time: knowledge leading to indifference.

**What do you like to do in your free time?**

I like museums, restaurants, and parks. I love the Met, especially its 19th century painters. The University community feels more integrated than French universities do, and I really enjoy getting to attend events around Columbia. I live right around the corner from the Lenfest Center for the Arts, where the Master of Fine Arts students staged their end of semester plays, and I really enjoyed going to see those this spring.