Stefano Filipazzi Finds Order in Limitless Complexity

Stefano Filipazzi studies the geometry of shapes you can’t draw. 

The new assistant professor of Mathematics is an algebraic geometer. “What I like to do is geometry — study shapes and try to understand their properties. The algebra part of being an algebraic geometer comes from focusing on shapes that can be described by a polynomial equation,” Filipazzi explained. 

For those whose memories of pre-calculus are a bit dusty, polynomial equations are built by adding up simple ingredients, such as numbers, variables and powers. Like Lego bricks, they can be little building blocks (such as y = x+2, which describes a straight line) but they can be combined into unimageable shapes — literally. 

“Not all shapes can be built this way, so it's a bit restrictive,” Filipazzi said, “but it also gives you power, because not only can you use geometry, and sometimes calculus, to study these shapes, but you can also use tools from more advanced algebra.” 

Filipazzi’s specialty is birational geometry, a subfield that looks for the simplest building blocks of these algebraic shapes, called “varieties,” and the ways they can be classified. It’s a bit like classifying pottery. Many pots can be gently reshaped — squeezed a little here, stretched a little there —without changing what they fundamentally are. Birational geometry formalizes these small “surgeries” and asks: What are the fundamental pot-types, and how can we use them to construct more complicated vessels? 

A central thread in his research is a long-running quest for finiteness: Is there a finite number of basic types of shapes? A straight line can slide up or tip and still be a line; a clay pot can be bent slightly and remain a pot. Filipazzi studies when such “deformations” stay within the same fundamental type, and when no amount of gentle reshaping can turn one kind of object into another — like trying to turn a handle-less bowl into a mug. 

Before coming to Duke, Filipazzi earned his Ph.D. at the University of Utah, held a postdoc at the University of California, Los Angeles, and then continued his research at the Federal Technology Institute (EPFL) in Lausanne, Switzerland. His love for algebraic geometry started earlier than that, though, at the University of Pavia, in Italy, where he completed his bachelor’s and master’s degrees. 

“I liked both algebra and geometry, and I liked a professor who taught a bit of both. It was too early for me to even know that what I was learning was algebraic geometry, but that’s how I got into it,” he said. “I guess I was lucky, because there were three or four students who liked algebraic geometry in my cohort, and we would study together and motivate each other. It was nice, you know, going through the growing pains together.” 

Hoping to give Duke students the same inspiration, this fall Filipazzi is teaching a one-month graduate mini-course that introduces core ideas from his research, aimed at giving students a feel for the big picture. 

What keeps him motivated is the hope that a carefully crafted toolkit — those sanctioned “surgery” moves and structural insights — can convert sprawling, high-dimensional possibilities into a finite, understandable map. It’s the algebraic geometer’s version of finding order in what looks like limitless complexity.