Algebra sits at the crossroads of mathematics and physics — a shared language that appears many places and often allows tackling computations. The NBIA Algebra Day will bring together mathematicians and physicists for a day of talks exploring this common ground, held in the historical Auditorium A of the Niels Bohr Institute on Blegdamsvej. The talks this year will cover a range of topics, from classical questions about systems of linear equations in polynomials to the algebra behind deformations of shapes.
All participants are invited for lunch at Restaurant Post
Timetable:
9:45 Coffee, tea, and pastries
10-10:45 David Eisenbud
10:45-11 Coffee break
11-11:45 Elisenda Feliu
12-13:30 lunch at Restaurant Post
13:30-14:00 Emil Bjerrum-Bohr: History of Niels Bohr Institute
14:00-14:45 Nathalie Wahl
Talk titles and abstracts:
Speaker: David Eisenbud
Title: Linear Equations in polynomials.
Abstract: Given polynomials $a_1,...,a_n$ with complex coefficients, in $d$ variables, what can you say about the solutions of the equation $\sum_{i=1}^n a_i(x) f_i = 0$ ? Here the task is to find enough vectors $F^j := (f_1^j, f_2, ..., f_n^j)$ of polynomials solving the equation so that every vector $G = (g_1,...,g_n)$ of polynomials solving the equation can be written in the form $G = \sum_j h_j(x) F^j$ for some polynomials $h_j(x)$. If the $F^j, G$, and the $h_j$ were allowed to be rational functions, this problem would look like standard linear algebra; the difficulty is that the quotient of two polynomials, such as $x/y$, isn't usually a polynomial. Understanding how to think about this problem was part of what David Hilbert did around 1890 in the course of solving the main problems of the invariant theory of the time. In this overview talk I'll describe what Hilbert did, and why, and show some of the modern developments in the theory.
Speaker: Elisenda Feliu
Title: Algebra and reaction networks.
Abstract: Biochemical reaction networks describe how interacting (bio)chemical species evolve over time. Their dynamics are commonly modelled by systems of ordinary differential equations whose right-hand sides are polynomial or rational functions. Fundamental questions include whether a network can admit multiple equilibria, whether these equilibria are stable, and how qualitative changes in the dynamics (bifurcations) can occur as parameters vary. Many of these questions can be reformulated as problems about the solutions of systems of polynomial equations and inequalities, making them amenable to methods from algebra and real algebraic geometry. This talk will introduce the mathematical framework of reaction networks and illustrate how algebraic techniques can be used to study several aspects of their dynamics.
Speaker: Nathalie Wahl
Title: When are two shapes really the same?
Abstract: Two topological spaces are "the same" (homeomorphic) if there is a continuous map with continuous inverse relating them, while they are homotopic if one can be deformed into the other. Whitehead torsion is an algebraic invariant that, in some sense, measures how far a homotopy equivalence is from being a homeomorphism. The talk will introduce this invariant, and explain how studying the behavior of strings in manifolds lead to the definition of a geometric shadow of this invariant in the world of manifolds.
Organizers: David Eisenbud, Nathalie Wahl, and Poul H. Damgaard