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Thinking Inside the Box: A combinatorial approach to Schubert Calculus (Sami H. Assaf, USC)
October 4, 2023 @ 4:15 pm - 5:30 pm
Title: Thinking Inside the Box: A combinatorial approach to Schubert Calculus
Speaker: Sami H. Assaf, Department of Mathematics, University of Southern California
Abstract: Given 2 lines in the plane, how many points lie on both? If we rule out the case where the two lines are the same, and we work in projective space so that parallel lines share the point at infinity, then the answer is always the same: 1 point of intersection. Suppose instead we’re given 4 lines in space. How many other lines meet all four of those? Schubert asked, and in some cases answered, such problems in enumerative geometry in his celebrated treatise in 1879. He argued that when the answer is finite, it does not depend on the choice of the linear spaces. Attempts to formalize Schubert’s “Principle of Conversation of Number” have led us to modern Schubert calculus and intersection theory, which has ramifications in geometry, topology, combinatorics, and even string theory. I will survey the history of this field from Schubert to now, highlighting a partial solution that involves enumerating ways of putting numbers in boxes.
Sami Assaf is a Professor of Mathematics and a Gabilan Distinguished Professor of Science and Engineering at the University of Southern California. Her fundamental research in combinatorics, geometry and probability is supported by grants from the National Science Foundation and the Simons Foundation. She has delivered plenary addresses for AMS meetings as well as for the annual Formal Power Series and Algebraic Combinatorics conference. Professor Assaf is recipient of a USC Mentoring Award for Faculty Mentoring Undergraduates, directs a local Math Circle for elementary school students, and currently serves as Director of Graduate Studies for the Department of Mathematics where she works to foster a culture of inclusivity, diversity, and excellence for all students.