### Spring 2022 Mathematics Colloquium

###### Time and Location:

Minard 112 at 3:00 PM (No refreshments this semester.)

Special Colloquia or Tri-College Colloquia venues and times may vary, please consult the individual listing.

#### Thursday, January 13 Peder Thompson (Niagara University)

Vanishing of Serre’s intersection multiplicity for triangulated categories

**TIME CHANGE: This talk will be from 1:00 pm - 2:00 pm.**

Abstract: This talk will tell a story about how we can use homology to measure geometric invariants, such as the number of points of intersection between two curves. A classic geometric description of how many points of intersection are possible goes back to Bézout in the 1700s, and these ideas led to the development of some beautiful mathematics. In the 1950s, Serre introduced intersection multiplicity to commutative and homological algebra, providing a fascinating link between geometry and algebra. The goal of this talk will be to survey some of my recent results in intersection theory, especially in the contexts of regular local rings, hypersurface rings, and graded complete intersection rings. Finally, I will discuss an ongoing project to extend these ideas into the realm of triangulated categories.

#### Tuesday, January 18 Katherine Raoux (MPIM Bonn)

Knots, surfaces and 4-manifolds

Abstract: Manifolds are smooth spaces that nearly everyone encounters and they are the central objects studied by topologists. One- and two-dimensional manifolds, curves and surfaces, are the subjects of pinnacle 19th century achievements. The 20th century brought a lot of progress in understanding higher dimensional manifolds, and, in particular, the tools developed to study 3- and 4-manifolds have a distinct geometric flavor.

In this talk, I'll describe my research on knots, surfaces and 4-manifolds. We'll start by looking at how knots bound surfaces and discuss how the possible surfaces depend on the ambient space. In particular, I will present a lower bound for the genus of such surfaces, derived from a modern tool called Heegaard Floer theory. I will also discuss connections to geometric structures on manifolds. Some of this is joint work with Matthew Hedden.

#### Thursday, January 20 Patrick Orson (MPIM Bonn)

Embedding spheres in 4-manifolds

Abstract: An embedded sphere in a manifold determines a homotopy class. But, switching this round, which homotopy classes in a manifold are determined by embedded spheres? Answering this question is particularly challenging inside 4-manifolds, where successful embedding techniques from other dimensions often fail. Understanding 2-spheres in 4-manifolds is also central to the classification of 4-manifolds themselves.

In this talk I will discuss the 2-sphere embedding question for topological 4-manifolds. After working through some examples, I will focus on a strategic approach to the problem called surgery theory. I will describe joint work where I have used surgery theory to answer the sphere embedding question, and other related questions, in infinite families of 4-manifolds.

#### Tuesday, January 25 Saeed Nasseh (Georgia Southern University)

INTERACTIONS BETWEEN COMMUTATIVE ALGEBRA AND OTHER AREAS: SOLUTIONS TO LONG-STANDING CONJECTURES

Abstract: Algebraic geometry, number theory, and invariant theory are the primary fields of applications of commutative algebra. In its modern shape, commutative algebra originated in the works of Kummer, Dedekind, Kronecker, Krull, Noether, and especially Hilbert during the nineteenth century and was vastly developed later in the outstanding work of Grothendieck on modern algebraic geometry. In the 1950’s, homological algebra (that originates from algebraic topology) made its way into the commutative algebra toolbox in the works of Auslander-Buchbaum and Serre. Tight connections between commutative algebra, representation theory, combinatorics, and deformation theory appeared in the past century in numerous places (including the work of Auslander) as well.

Another tool from algebraic topology that has recently sparked interest among commutative algebraists is differential graded (DG) homological algebra. The use of techniques from DG homological algebra was established by Avramov, Buchsbaum, Eisenbud, Foxby, Halperin, Kustin, and Miller (to name a few) in commutative algebra, for instance, via DG algebra structures on Koszul complexes and free resolutions. It has been shown recently that these techniques can be applied to solve non-trivial problems in commutative algebra.

In this talk, I introduce two long-standing major conjectures in commutative algebra - namely, Vasconcelos’ Conjecture (1974) and the Auslander-Reiten Conjecture (1975) - that have been solved (completely and partially, respectively) in my recent works using DG homological algebra and techniques from the above mentioned areas of mathematics. I will describe, in down-to-earth terms, the cast of characters that play fundamental roles in the solutions of these conjectures. Finally, I will sketch a new method (that "partially" includes differential equations) and I will show evidence that it might be used in solving the Auslander-Reiten Conjecture in its full generality (this is a part of an in-progress joint work with Maiko Ono and Yuji Yoshino).

#### Thursday, January 27 Janet Page (University of Michigan)

Extremal Varieties in Positive Characteristic

**TIME CHANGE: This talk will be from 1:00 pm - 2:00 pm.**

Abstract: Algebraic geometry aims to understand the shapes defined by polynomial equations, called algebraic varieties, and commutative algebra often provides the tools with which to do so. For centuries mathematicians have studied which varieties contain lines (or other linear spaces) as subsets and how many lines they contain. Perhaps one of the most famous classically known facts about lines on algebraic varieties is one proved by Cayley and Salmon in 1849--that every (smooth) cubic surface contains 27 lines. A hundred years later, Segre showed that a smooth surface of degree d > 2 over the complex numbers has at most (d-2)(11d-6) lines; when d = 3, we see that smooth cubic surfaces actually attain this upper bound. One might ask whether Segre's theorem holds when we look at surfaces over other fields. In this talk, I'll introduce a class of surfaces that have more lines than Segre's upper bound, which are defined over a field of positive characteristic. These surfaces fit into a class of varieties that we recently defined and studied using commutative algebraic tools, and I will discuss some of their surprising--and extremal--algebraic and geometric properties.

#### Tuesday, March 8 Daoji Huang (University of Minnesota)

The “Lifting Dream” of Schubert calculus

Abstract: : In the 19th century, Hermann Schubert was interested in questions in enumerative geometry, such as “given four arbitrary lines in 3-space, how many lines intersect all four lines?” The modern treatment of such questions centers around the study of the cohomology ring of the complete flag variety, with a basis given by the classes of Schubert varieties. While this topic has been extensively studied, a central question which seeks a combinatorial description of the structure constants, remains open. However, the same question has been fully solved using many different methods in the special case when the classes are pullbacks from the Schubert classes in Grassmannians, in which case the Schubert structure constants are known as the Littlewood-Richardson coefficients. The “lifting dream” of Schubert calculus hopes to find ways to lift the Littlewood-Richardson rules to the general case. In this talk, I will discuss potential venues to lift the tableaux combinatorics for the Grassmannian case using bumpless pipe dreams, and some success stories so far.