What is ... ? Seminar

Abstract: Smooth bump functions are infinitely differentiable, compactly supported functions. In this talk we explain some elementary, but versatile constructions involving bump functions. We use them both to prove some classical results about extending/combining smooth functions, and to construct some counterexamples.

Talk Notes

Saúl Rodríguez Martín

Abstract: The length of a curve is defined as the supremum of the lengths of inscribed polygonal lines. Serret (1868), in his calculus textbook, proposed an analogous definition of surface area. Schwarz (1880) and Peano (1882) discovered that Serret’s definition of surface area was wrong, because it can give an infinite result even for surfaces as simple as a sphere or an ordinary bounded cylinder, as I shall explain. Lebesgue (1902) proposed a modification of Serret’s definition that gives the right value for nice surfaces. But for an irregular surface, Lebesgue’s definition of surface area can give an inappropriate value. One formulation of the so-called isoperimetric inequality is the statement that among all bodies of a given surface area, a ball has the largest volume. But I shall present a simple example, due to Besicovitch (1945), of a surface of small Lebesgue area that encloses a large volume. Thus, for Lebesgue area, the isoperimetric inequality does not hold, at least for certain irregular surfaces. Next, I shall discuss better ways to define surface area, for which the isoperimetric inequality does hold, even for quite irregular surfaces. (These are particular cases of Hausdorff measures and of integral geometric measures.) Also, I shall present an intriguingly simple proof of a weak form of the isoperimetric inequality. Can this proof be converted into a proof of the actual isoperimetric inequality? Perhaps someone attending the talk may answer this question someday!

Abstract: Domino tilings are the source of many questions in enumerative combinatorics. In this talk, we will address some famous questions and theorems regarding domino tilings of m x n rectangular grids and Aztec Diamonds. Along with combinatorics, we will meet several other areas of mathematics, including linear algebra and statistical mechanics, that help shine light on some of the phenomena that arise when studying domino tilings.

Abstract: The Gamma function was defined by Euler in 1729 to interpolate the factorial function. It plays the same role in the world of difference equations, as Euler's exponential function does for differential equations. In this talk, I will define this function and present some of its fundamental properties such as: Weierstrass' infinite product formula, relation between Gamma and sine functions, and discuss its behavior as the argument goes to infinity (Stirling series) and how it relates to Bernoulli numbers.

Abstract: The separating hyperplane theorem is an elementary result in convex geometry which has surprising connections to other facts in geometry and in the theory of linear programing. In this talk, we will explore the way that these connections generalize in order to find an equivalent form of the axiom of choice and applications to economic theory.

Abstract: In the theory of finite groups, much can be said about the structure of a group $G$ only knowing its size. The Schur-Zassenhaus theorem follows along these lines: if a finite group $G$ has a normal subgroup $N$ such that $(|N|,|G/N|) = 1$, then this normal subgroup $N$ has a complement, meaning a subgroup $H$ where $G = NH$ and $N \cap H = \{e\}$. As $N$ is normal, this also implies $G$ is a semidirect product of $N$ and $H$. In this talk, we will attempt to prove this theorem, as well as present other analysis on how order imposes structure in the case of finite groups.

Abstract: In this talk we introduce the so-called “Gauss continued fraction map,” which conveniently allows one to phrase questions about continued fractions in the framework of dynamical systems. We mainly focus on the distribution of a continued fraction’s partial quotients, and we also mention connections to the Euclidean algorithm.

Abstract: Kuratowski's theorem characterizes all planar graphs as exactly the graphs that cannot be reduced to K5 or K3,3 by a sequence of edge deletions or contractions. This characterization of planar graphs compactifies a question about the non-existence of 'nice' ways to draw a graph into a question about whether a graph contains one of a finite number of substructures. In this talk, we will examine the graph minor theorem, a far-reaching generalization of Kuratowski's theorem, which was proved in Paper 20 of Robertson and Seymour's graph minor project. We will cover the necessary information to understand the motivation and statement of the graph minor theorem. Beyond that, we will examine Well Quasi-Orderings and Kruskal's theorem to gain insight into the ideas that motivated Robertson and Seymour's graph minor project.