Homotopy Theory

Speaker:

"Organizational meeting"

Time: 13:00

Room: MC 106

We will choose a topic for the semester. All are welcome.

Colloquium

Speaker: Ajneet Dhillon (Western)

"Essential Dimension and Algebraic Stacks"

Time: 15:30

Room: MC 108

This will be an expository talk on the work of Brosnan-Fakhuriddin-Reichstein-Vistoli. We will relate essential dimension to dimension and discuss the discrepency between the two for algebraic stacks. If time permits applications to vector bundles on curves will be given.

Graduate Seminar

Speaker: Josue Rosario-Ortega (Western)

"Special Lagrangian Submanifolds"

Time: 11:20

Room: MC 106

Let $X$ be a Calabi-Yau manifold of dimension $2n$. A special Lagrangian submanifold (SL-submanifold) is an oriented, embedded real $n$-submanifold $\psi: N \rightarrow X$ calibrated by the real part of the holomorphic volume form $\Omega$. In this talk I will explain the basics of SL-submanifolds when $X= \mathbb{C}^N$ and from the point of view of minimal submanifolds and calibrated geometry (I will explain what is a minimal submanifold and calibrations). Just basic background in differential geometry is enough to follow the talk.

Homotopy Theory

Speaker: Chris Kapulkin (Western)

"Survey of Homotopy Type Theory"

Time: 13:00

Room: MC 107

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.

Algebra Seminar

Speaker: Enxin Wu (University of Vienna)

"Homological algebra of diffeological vector spaces, with application to analysis"

Time: 14:30

Room: MC 107

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.

Graduate Seminar

Speaker: Wayne Grey (Western)

"Embeddings among mixed norm Lebesgue spaces"

Time: 11:30

Room: MC 106

Mixed-norm Lebesgue spaces generalize standard Lebesgue ($L^p$) spaces. A mixed norm of a function of multiple variables is computed similarly to an $L^p$ norm, but with a different exponent associated with each variable.

When two Lebesgue spaces are defined (with different exponents and measures) for functions on a common domain, there are well-known conditions to determine whether any embedding exists between them, ultimately depending on properties of the Radon-Nikodym derivative. For most cases, we've found necessary and sufficient conditions for embeddings of mixed-norm spaces, relying on the inequalities of Holder and Minkowski.

This talk should be accessible to anyone familiar with Lebesgue spaces. However, the necessary ideas will be summarized, hopefully to make it somewhat understandable given only knowledge of normed vector spaces.

Homotopy Theory

Speaker: Martin Frankland (Western)

"Inductive types and identity types"

Time: 13:00

Room: MC 107

We will discuss in more detail the rules that define types, along with examples. The focus will be on inductive types, which are characterized by their constructors. An important example of inductive types is given by identity types, which play the role of path spaces in the homotopical interpretation. We will discuss path induction and the higher groupoid structure obtained from iterated identity types.

Algebra Seminar

Speaker: Cihan Okay (Western)

"Nilpotent groups and colimits"

Time: 14:30

Room: MC 107

A natural way to study nilpotency in a group $G$ is to consider the colimit of the nilpotent subgroups of certain class. For a fixed nilpotency class $q$ there groups appear as the fundamental group of a certain subspace $B(q,G)$ of the usual classifying space $BG$, which are introduced in a paper by Adem, Cohen, and Torres Giese. I will discuss some properties of these colimits, and describe them for certain groups in the case of $q=2$ which corresponds to the colimit of the abelian subgroups.

Graduate Seminar

Speaker: Tyson Davis (Western)

"Essential Dimension of Moduli stacks"

Time: 11:20

Room: MC 106

Geometry and Topology

Speaker: Karol Szumilo (Western)

"Cofibration categories and quasicategories"

Time: 15:30

Room: MC 107

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.

Analysis Seminar

Speaker: Myrto Manolaki (Western)

"Zero sets of real analytic functions and the fine topology"

Time: 14:30

Room: MC 107

In this talk we will discuss some results concerning the zero sets of real analytic functions on open sets in $\mathbb{R}^n$. We will consider the related notion of analytic uniqueness sequences and, as an application, we will show that the zero set of every non-constant real analytic function on a domain has always empty interior with respect to the fine topology (which strictly contains the Euclidean one). Further, we will see that for a certain category of sets $E$ (containing the finely open sets), a function is real analytic on some open neighbourhood of $E$ if and only if it is real analytic ''at each point'' of $E$. (Joint work with Andre Boivin and Paul Gauthier.)

Homotopy Theory

Speaker: Karol Szumilo (Western)

"Univalence Axiom"

Time: 13:00

Room: MC 107

We will introduce the Univalence Axiom and discuss a few of its immediate consequences such as existence of types that are not sets, function extensionality or preservation of n-types by dependent products.