Classical Algebraic Geometry is a branch of mathematics that studies different sets of solutions to polynomial equations in several variables with coefficients over different kinds of number systems. Let's call this type of set an algebraic set. For example, if the number of variables is 2 and the number system is the real numbers, then we are talking about points, and curves in the real coordinate plane. Linkage theory studies classes of algebraic sets that can be “linked” by using another algebraic set which is “simpler” and better understood.
The winding number of a continuous function on the unit circle counts how many times a graph of the function loops around the origin. It is homotopy invariant and has applications to several areas of Mathematics.
Toeplitz operators with continuous symbol are bounded linear operators on the Hardy Space involving multiplication by a continuous function. The index of such a Toeplitz operator is closely connected to the winding number of its symbol.
This connection is examined and then extended for Toeplitz operators with crossed product symbols.
Personality psychologists often apply clustering techniques on questionnaire data to model personality structure. Inspired by this work, we apply techniques from topological data analysis (TDA) to understand the structure of this data. The data comes from Cattell’s Sixteen Personality Factor Questionnaire (collected by Bell, Rose, & Damon in 1972). Subjects were 969 adult male volunteers divided into three age groups: 25 to 34, 35 to 54, and 55 to 82. We use persistent homology (a TDA tool) to cluster the data and identify that personality structure is slightly different between the age groups. It is also curious to note that data from the youngest age group appears to have a topological “hole”, which raises questions of the psychological significance. This work suggests that additional research, including applying TDA tools to other questionnaire data sets can provide insights to the study of personality.
An algebraic set is the solution set of a system of polynomial equations. A variety is a special type of algebraic set. In the two dimensional Euclidean plane the solution set of the polynomial equation x=0 is nothing but the Y-axis, which is an example of a variety. Historically people studied varieties and their relations very carefully. It turns out that instead of studying varieties individually it is more fruitful to study them in a 'family'. But in this set up degeneracies happen naturally. For example, in the two dimensional Euclidean plane if we take the unit circle centered at the origin and intersect it by the vertical lines x=a, where a is any number from the interval [-1, 1], we always get two distinct points unless a is -1 or 1, i.e., degeneracy happens at -1 and 1. In those two degenerate cases the solution sets consist of just one point, but in the algebraic point of view they come twice. So we can say these algebraic sets have multiplicity 2. It was very difficult to study these multiplicity structures using the classical tools of algebraic geometry. But after Alexander Grothendieck introduced and developed his theory of 'Schemes' in the 1950s , we can study such structures closely and talk about their geometries.
An algebraic integer is a complex number that is a root of a monic polynomial with integer coefficients. It is well-known that there is not always a single algebraic integer that can generate the ring of algebraic integers contained in a field extension of the rational numbers. The index of an algebraic integer is a natural number that measures how far a ring of integers is from having such a "primitive element." We investigate these indices in cubic fields and determine which natural numbers occur as indices in given families.