Fall 2022 Mathematics Colloquium

Time and Location:

Minard 112 at 3:00 PM (Refreshments at 2:30 in Minard 404)

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

Tuesday, October 11             Oliver Pechenik (University of Waterloo)

Cell decompositions and rings-with-bases

Abstract: When I first saw linear algebra, every vector space was $\mathbb{R}^n$, it came with a "standard coordinate basis", and linear transforms were the same as matrices. Later, I learned I was supposed to treat abstract vector spaces abstractly without specifying an arbitrarily-chosen basis or working in coordinates. Even later, I learned that sometimes there is a "canonical" basis and it's powerful to work in those coordinates after all. In this talk, we'll look at the cohomology and K-theory of some moduli spaces of flags and loops and see that they have canonical bases, with applications to the combinatorics of symmetric and quasisymmetric functions.

Thursday, October 13             Joshua Swanson (University of Southern California)

"Spinny pictures": connecting Catalan combinatorics, alternating sign matrices, and plabic graphs

Abstract: "Rotating" trees is a key step in balancing data structures like B-trees and AVL-trees. Modern databases and digital life are built upon this surprisingly crucial operation. Mathematically, a particularly beautiful instance of tree rotation arises by bijectively encoding complete binary trees as non-crossing matchings on a circle and literally spinning the circular matching. Certain non-obvious properties on the tree level become obvious when translated to the context of such a "spinny picture".

Far from being an isolated curiosity, we will show that such "spinny pictures" serve as a bridge between a remarkably diverse array of topics spanning enumerative combinatorics, representation theory, and algebraic geometry. We will in particular discuss our recent resolution of a nearly 30-year-old problem related to work of Kuperberg and Khovanov involving promotion on 4-row rectangular tableaux. The background assumed will be low, and pictures will be plentiful.

Joint work with Christian Gaetz, Oliver Pechenik, Stephan Pfannerer, and Jessica Striker.

Tuesday, November 1          Tri-College Colloquium
Doug Anderson                at NDSU   

The effects of discretization on Hyers-Ulam stability

Abstract:  Hyers-Ulam stability is defined and illustrated using a first-order linear differential equation. The results are extended to the equation x'=Ax for 2x2 constant matrices. The exponential equation is discretized in various ways, each of which is investigated for Hyers-Ulam stability. For discretization using the forward difference operator, the Hilger circle in the complex plane plays a crucial role. For the discrete version using the diamond-alpha operator, a new object, the diamond-alpha ellipse, determines the Hyers-Ulam stability regions in the complex plane. These results are then compared with new Lyapunov stability results for the same equation.


Tuesday, November 15          Tri-College Colloquium
 Michael Marmorstein          at Concordia 

Asymptotic Properties of Generalized Symbolic Powers

TIME and ROOM : 3:00 pm, Integrated Science Center 338
(Refreshments: 2:30 pm, the Integrated Science Center 362)

Abstract:  In 1921, Emmy Noether proved, in its fullest generality, the Lasker-Noether primary decomposition theorem. Out of this classic result arose a particularly important construction, the symbolic powers of an ideal, which can be generalized.

The nth generalized symbolic power of an ideal $I$ with respect to an ideal $L$ is defined to be the ideal:

$$\bigcup_{k \in \mathbb{N}} (I^n:L^k) $$

The talk will be divided into two parts. In the first, we motivate the study of generalized symbolic powers by laying down the important definitions, properties and connections to the usual symbolic powers. In the second part, we explore the stability of the generalized Symbolic Powers for large values of $n$.


Tuesday, December 6             Azer Akhmedov (NDSU)

The use of epsilon-delta language in teaching calculus.

Abstract: Calculus courses (Calculus I-III) are taught to a broader audience (as we expect the basic calculus knowledge to be in the culture of any educated person) and also because of this reason, we use very little (if any) epsilon-delta language in teaching it. Indeed, the use of epsilon-delta language is debatable at this level of teaching mathematics. In the talk, I'll identify/mention several traditional approaches to teaching calculus and discuss them from the point of view of using "epsilon-delta" in particular, and mathematically rigorous language in general. I'll also briefly discuss approaches to teaching courses beyond calculus such as Math 270 (Intro to Abstract Math) , Math 346 (Metric Space Topology) and Math 450 (Real Analysis I).  


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