Fall 2022
Math 662: Schubert calculus and flag manifolds

• 12:00—12:30 C.J. Bott
• 12:30—13:00 Thomas Yahl
• 13:00—13:30 Weixun Deng
• 13:30—14:00 Jordy Lopez
• 14:00—14:30 Break (and catch up)
• 14:30—15:00 Joshua Crouch
• 15:00—15:30 Tiansu Zhang
• 15:30—16:00 Derek Wu

• Tiansu Zhang: Toric matrix Schubert Variety
  Abstract: Given a matrix Schubert variety X_π we can decompose it as Y_π×Cⁿ. We will charactize when Y_π is toric. We then relate acylic root polytopes with moment polytope Φ(P(Y_π)) of the projectivization of Y_π.
• Jordy García: Introduction to A¹-Enumerative Geometry
  Abstract: A¹-Enumerative Geometry applies tools from homotopy theory to the study of classical enumerative geometry. We survey papers from Brazelton, Pauli and Wickelgren to present results in the area, such as enriched local degrees of morphisms of smooth schemes, A¹-Milnor numbers, as well as using Macaulay2 to compute A¹-Euler numbers.
• Derek Wu: More combinatorial aspects of Schubert polynomials
  Abstract: We have already seen Schubert polynomials defined in terms of Rothe diagrams and pipe dreams. In this presentation, we continue with more combinatorial objects that give rise to Schubert polynomials, such as reduced expressions, balanced tableaux, and insertion algorithms. We also show the bijections between these various objects, and give some ideas as to why these are the correct objects to study.
• C.J. Bott: Generalized Flag Varieties
  Abstract: In addition to the Grassmannians and complete flag varieties we've learned about in class, there are many other flavors of flag varieties! I will present the topic of generalized flag varieties: their dimensions and global coordinates, local (Stiefel) coordinates, generalization of the Bruhat order poset and permutations, their corresponding Schubert cells and varieties, and representing such objects and their intersection theory on a computer.
• Joshua Crouch: Probabilistic Schubert Calculus
  Abstract: The machinery of Schubert calculus is extraordinarily powerful when working over the complex numbers, but it does not work over the real numbers. An alternative approach is probabilistic Schubert calculus where one is interested in the expected number of points of intersection of real Schubert varieties in random position. We will discuss the codimension-one case of special Schubert varieties.
• Thomas Yahl: Galois groups of Schubert Problems
  Abstract: To each Schubert problem with finitely many solutions, there is a Galois group that acts on these solutions and controls the complexity of computing them explicitly. For Grassmanians of small dimension, these groups have been extensively studied, leading to interesting examples and conjectures as to what these groups are more generally. We explore what is currently known about these groups and present some examples.
• Matthew Faust: Quasisymmetric Schur Polynomials
  Abstract: Quasisymmetric Schur polynomials are the quasisymmetric counterpart to Schur polynomials. Similar to Schur polynomials they can be obtained from a tableaux called composition tableau, and they form a basis for the quasisymmetric functions. We will examine how these polynomials are constructed and present the many parallel properties that this weaker collection of polynomials exhibit.
• Weixun Deng: Schubert Calculus and Representation Theory
  Abstract: It has been known that the intersection index of the Schubert varieties in the Grassmannian coincides with the dimension of the space of invariant vectors in a suitable tensor product of finite-dimensional irreducible representations of the general linear group. We will describe a direct connection between the representation theory of GL_n and classical Schubert calculus on the Grassmannian, which goes via the Chern-Weil theory of characteristic classes.

Last modified: Mon Nov 28 12:30:17 CST 2022 by sottile