The New York City
Category Theory Seminar
September 27th (TUESDAY), 2011. Olivia Caramello,
Title: The Unification of Mathematics via Topos Theory
Abstract: I will propose a new view of Grothendieck toposes as unifying spaces in Mathematics being able to serve as 'bridges' for transferring information between distinct mathematical theories. This approach, first introduced in my Ph.D. dissertation, has already generated ramifications into different mathematical fields and points towards a realization of Topos Theory as a unifying theory of Mathematics.
In the talk, I will explain the fundamental principles that characterize my view of toposes as unifying spaces, and demonstrate the technical usefulness of these methodologies by providing applications in several distinct areas including Model Theory, Algebra, Geometry and Topology.
Wednesday, November 9, 2011. Andrei Rodin,
Title: Why Category Theory Is “Unreasonably Effective”?
Abstract: In the first part of my talk I briefly present a neo-Kantian view on mathematics and its relationships with natural sciences, according to which the primary task of doing mathematics is not establishing certain truths as such but making coherent conceptual constructions playing a crucial role in the exploration of our environments that we call (experimental) science. The notion of mathematical (“a priori”) truth serves this latter purpose: remind Vladimir Arnold’s provocative motto “Mathematics is a part of Physics”. Making explicit the way, in which pure mathematics is deeply involved into sciences, dissolves Wiegner’s puzzle about the “unreasonable effectiveness” of mathematics in these sciences. The effecitiveness of mathematics is not unreasonable because the same human reason operates in all these disciplines.
In the second part of my talk I take the case of Category theory and try to show why this particular mathematical theory is and should be “unreasonably effictive” in a sense in which Set theory is not. Here I analyze the mathematical notion of *map* (which I assume to be central for Category theory) through its history, demonstrate its epistemic significance, and consider its role in the modern science. In this context I compare geometrical groups with more general geometrical categories and discuss some epistemic aspects of such a generalization.