Mathematics Colloquium

Fall 2023

Location and Time: 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, September 19          
Nikita Barabanov, NDSU

Smooth Invariant Binary Operations

Abstract:  The motivation to study this subject comes from the Einstein addition in the theory of relativity and the algebraic theory of gyrogroups. We show the one-to-one correspondence between smooth invariant binary operations in open balls and metric tensors parametrized by two scalar functions. The operations occur to be compositions of exponential mappings, parallel transports, and logarithmic mappings. The unique operations of scalar multiplications allow to introduce the gyro vector spaces. We use the definition and conditions for isomorphic operations to find all operations isomorphic to given operation. Then we consider sets of co-gyrolines, and binary operations that generate such sets. Finally, we calculate the sectional Gaussian curvatures and find conditions necessary and sufficient for smooth binary operations to be isomorphic to the Einstein addition.


Tuesday, October 10           
Christian Gaetz, Cornell

Hypercube decompositions and combinatorial invariance for Kazhdan--Lusztig polynomials

Abstract:  Kazhdan--Lusztig polynomials are of foundational importance in geometric representation theory. Yet the Combinatorial Invariance Conjecture suggests that they only depend on the combinatorics of Bruhat order. I'll describe joint work with Grant Barkley in which we adapt the hypercube decompositions introduced by Blundell-Buesing-Davies-Veličković-Williamson to prove this conjecture for Kazhdan-Lusztig R-polynomials in the case of elementary intervals in the symmetric group. This significantly generalizes the main previously-known case of the conjecture, that of lower intervals.


Thursday, October 12           
Oliver Pechenik, University of Waterloo

Chromatic and Kromatic symmetric functions 

Abstract: One of the guiding problems of graph theory is the 4-color theorem, claiming that the vertices of any planar graph can be colored from a palette of size 4 with adjacent vertices receiving different colors. In 1912, in an attempt to prove this theorem, Birkhoff introduced the chromatic polynomial of a graph, which was intended to make the problem amenable to analytic techniques. This attempt failed, but the chromatic polynomial remains a fascinating and powerful graph invariant. Even richer is Stanley’s 1995 refinement into the chromatic symmetric function, which simultaneously encodes information about stable decompositions, the contraction lattice, and acyclic orientations. Finally, we’ll discuss a new K-theoretic deformation of the chromatic symmetric function and the mysteries of what it might know about graphs. (Joint work with Logan Crew and Sophie Spirkl.)


Tuesday, October 17         
Andriy Prymak,  University of Manitoba

On illumination of convex bodies by external light sources

Abstract: The well-known Levi-Hadwiger-Gohberg-Markus conjecture states that any convex body in the $n$-dimensional Euclidean space can be covered by at most $2^n$ of its smaller homothetic copies, with equality only for parallelepipeds. Equivalently, this conjecture can be restated in terms of illumination of the boundary of a convex body by external light sources. The conjecture has been confirmed only for the 2-dimenisonal case, and is far from being solved in general. We will give a survey of known results for the cases of large and small dimensions, including a few very recent results. Certain ideas of the proofs will be discussed.


Tuesday, October 31

*Special Tri-College Colloquium at MSUM - please note the different location & time

Nikita Barabanov, North Dakota State University

Theory of smooth binary operations that generalize Einstein addition.

Location & Time: MSUM, 3:00-3:50

Reception: "Presidents" ITV Conference Room; MacLean 268 (at 2:30)
Talk: Bridges 268

Abstract: Gyrolines defined by Einstein addition have properties standard for spaces with negative curvature. Cogyrolines defined by Einstein addition have properties standard for spaces with zero curvature. Why there is a difference? We describe a recently developed theory of binary operations that is based on a differential geometry approach. New concepts, problems, approaches, and results will be presented and discussed.  The directions for future research will also be pointed out.


Tuesday, November 14

*Special Tri-College Colloquium at NDSU

Greg Tanner, Concordia College

Title: Runge-Kutta Methods for Stiff ODEs

Abstract: In this talk, I will introduce the concept of stiffness for ordinary differential equations and discuss how Runge-Kutta methods can be selected to efficiently and effectively integrate stiff ODEs. Order conditions and simplifying assumptions will be discussed using the theory of Butcher trees. Stability analysis will also be examined using the Dahlquist and Prothero-Robinson test problems.


Tuesday, December 5        
César  Lozano Huerta,  Universidad Nacional Autónoma de México - Oaxaca

The weak Lefschetz Principle in birational geometry

Abstract: Our departing point will be the influential work of Solomon Lefschetz started in 1924. We will look at the original formulation of the Lefschetz hyperplane theorem in algebraic topology and build up to recent developments of it in birational geometry.

The goal of the talk is to convey the following slogan: There are many scenarios in geometry in which analogous versions of the Lefschetz hyperplane theorem hold. Such scenarios can be unexpected and have had a profound impact in mathematics.


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