The Mathematical and Computational Sciences at Western are represented by four separate departments: Applied Mathematics, Computer Science, Mathematics, and Statistical & Actuarial Sciences. The Department of Mathematics has established research groups in several areas of contemporary mathematics including algebra, analysis and analytic geometry, homotopy theory, and noncommutative geometry.
Each week the Department of Mathematics hosts a wide variety of seminars and events. For a comprehensive list of events, please consult our departmental calendar.
I will give an overview of the field of Homotopy Type Theory. This relatively new field of mathematics is based on a realization that the formal logical system of dependent type theory can be interpreted in various homotopy-theoretic settings. After briefly discussing type theory, I will sketch the idea of its homotopical interpretation and its connection to higher category theory. In the last part of the talk, I will highlight some recent results.
Homological algebra of vector spaces is well understood. In functional analysis, many infinite dimensional vector spaces also contain analysis information. A diffeological vector space is a vector space with a compatible (generalized) smooth structure. In this talk, I will present a non-trivial example from functional analysis under the framework of diffeological vector spaces, see how the generalized smooth structure can be used to generalize a known result from analysis, as a motivation for the development of homological algebra of diffeological vector spaces. Then I will talk about the similarity and difference between this homological algebra and the homological algebra of R-modules. If time permits, some open questions will be discussed at the end.
Approaches to abstract homotopy theory fall roughly into two types:
classical and higher categorical. Classical models of homotopy theories
are some structured categories equipped with weak equivalences, e.g. model
categories or (co)fibration categories. From the perspective of higher
category theory homotopy theories are the same as (infinity,1)-categories,
e.g. quasicategories or complete Segal spaces. The higher categorical
point of view allows us to consider the homotopy theory of homotopy
theories and to use homotopy theoretic methods to compare various notions
of homotopy theory. Most of the known notions of (infinity,1)-categories
are equivalent to each other. This raises a question: are the classical
approaches equivalent to the higher categorical ones? I will provide a
positive answer by constructing the homotopy theory of cofibration
categories and explaining how it is equivalent to the homotopy theory of
(finitely) cocomplete quasicategories. This is achieved by encoding both
these homotopy theories as fibration categories and exhibiting an explicit
equivalence between them.