# Algebraic Foundations For Applied Topology And ...

The school is a five-day event with the aim of educating researchers to work at the interface of Topological Data Analysis (TDA) and Spatial Statistics. The principal target group are PhD students and postdocs in applied topology, statistics, and related subjects. Although dealing with similar problems, until recently there has been little interaction between TDA and spatial statistics. The summer school will thus be a major stepping stone for networking and knowledge sharing between these branches of applied topology and statistics.

## Algebraic Foundations for Applied Topology and ...

In Winter 2023, we will offer an online graduate course on introductory algebraic topology. This course could be well-suited to some students, especially if their institutions are not offering introductory algebraic topology this year. Our target audience includes first-year PhD students with interests in algebra, geometry, or topology; and masters students who are preparing to enter a PhD program soon. We are also open to enrolling advanced undergraduate students under certain circumstances.

See -courses/algebraic-topology-homology-and-cohomology-winter-2023/for more information, including a syllabus, grading policies, prerequisites, and application instructions. We do not charge tuition, but students are generally expected to enroll for independent study credits at their home institutions.

TMATH 342 Applied Topology (5) RSNEngages with varied topics which will be chosen from differential topology, knot theory, or algebraic topology. Applications (such as chemistry, physics or engineering) will be emphasized throughout the course. Prerequisite: minimum grade of 2.0 in TMATH 324.View course details in MyPlan: TMATH 342

There will be parallel sessions in the following fields: algebraic topology; applied topology and geometry; differential geometry and geometric analysis; rings and algebras; and stochastic analysis and applications. See below for descriptions of each session.

There are five categories: algebra, combinatorics and number theory; logic and foundations; analysis; geometry and topology; applied mathematics. From three of the five categories, at least two courses in each must be completed. Yale course search provides current listing and descriptions of the courses. It lists the attributes for each course, and can be used to search for courses with a particular attribute. (A sample list of courses that could be offered, by category, can be found below.)

After their 1945 paper, it was not clear that the concepts of categorytheory would amount to more than a convenient language; this indeedwas the status quo for about fifteen years. Category theory wasemployed in this manner by Eilenberg & Steenrod (1952), in aninfluential book on the foundations of algebraic topology, and byCartan & Eilenberg (1956), in a ground breaking book onhomological algebra. (Curiously, although Eilenberg & Steenroddefined categories, Cartan & Eilenberg simply assumed them!) Thesebooks allowed new generations of mathematicians to learn algebraictopology and homological algebra directly in the categorical language,and to master the method of diagrams. Indeed, without the method ofdiagram chasing, many results in these two books seem inconceivable,or at the very least would have required a considerably more intricatepresentation.

The history of category theory offers a rich source of information toexplore and take into account for an historically sensitiveepistemology of mathematics. It is hard to imagine, for instance, howalgebraic geometry and algebraic topology could have become what theyare now without categorical tools. (See, for instance, Carter 2008,Corfield 2003, KrÃ¶mer 2007, Marquis 2009, McLarty 1994, McLarty2006.) Category theory has lead to reconceptualizations of variousareas of mathematics based on purely abstract foundations. Moreover,when developed in a categorical framework, traditional boundariesbetween disciplines are shattered and reconfigured; to mention but oneimportant example, topos theory provides a direct bridge betweenalgebraic geometry and logic, to the point where certain results inalgebraic geometry are directly translated into logic and vice versa.Certain concepts that were geometrical in origin are more clearly seenas logical (for example, the notion of coherent topos). Algebraictopology also lurks in the background. (See, for instance, Caramello2018 for a systematic exploitation of the idea of toposes as bridgesin mathematics.) On a different but important front, it can be arguedthat the distinction between mathematics and metamathematics cannot bearticulated in the way it has been. All these issues have to bereconsidered and reevaluated.

We have an award-winning graduate faculty of 27, with current areas of research interest in algebra, algebraic geometry, analysis, combinatorics and lie theory, differential equations, differential geometry, logic and foundations, number theory, probability, topology and related areas. The department hosts numerous visiting scholars and postdoctoral fellows. There is an active seminar culture, including many seminars run by students. Our course offerings include a regularly scheduled rotation of algebra, analysis, topology, geometry, foundations, number theory, etc., as well as topics courses whose more advanced subject matter varies year-to-year based on the interests of faculty and students. Students and faculty also interact with other departments on campus such as Applied Mathematics and Computer Science, including joint PhD/MA/MS advising. We have mathematical relationships, including shared seminars, with nearby Colorado State University and other front range universities. Our graduate students, of which there are approximately 60, have excellent opportunities for research, including a first-year summer research experience.

Faculty and students actively pursue both basic and applied research in mathematics from traditional areas of algebra, analysis, applied and computational mathematics, dynamical systems and ergodic theory, geometry and topology, mathematical logic and foundations, number theory, statistics. 041b061a72