Analysis Seminar

Speaker: Janusz Adamus (Western)

"On relative Nash approximation of complex analytic sets (Part I)"

Time: 15:30 - 16:30

Room: MC 108

A basic problem in complex analysis is to approximate holomorphic maps by algebraic ones. This problem has a natural generalization in complex analytic geometry. Namely, one can ask whether a complex analytic set can be approximated by branches of algebraic sets (so-called Nash sets). In the case when the analytic set has only isolated singularities, this question is closely related to the classical problem of transforming an analytic set onto a Nash set by a biholomorphic map. The situation is quite different when the singular locus is of higher dimension, as there exist analytic set germs which are not biholomorphically equivalent to any Nash set germ. A major progress in this direction was allowed by the use of the so-called Neron desingularization.

In this talk, we will report on the recent developments in Nash approximation of analytic sets and mappings. Particularly, on the problem of relative approximation along arbitrary subsets.

Speaker's homepage: http://www.math.uwo.ca/~jadamus/

Homotopy Theory

Speaker: Chris Kapulkin (Western)

"Motivation and Basic Concepts of Higher Category Theory"

Time: 13:00 - 14:30

Room: MC 107

I will present the motivation to study higher category theory coming from different areas (homotopy theory, TQFTs, derived algebraic geometry) and introduce basic concepts of quasicategory theory, such as: homotopies and equivalences, proving some of their properties.

Geometry and Topology

Speaker: Fosco Loregian (Western)

"$t$-structures on stable $(\infty,1)$-categories"

Time: 15:30 - 16:30

Room: MC 107

We characterize $t$-structures in stable $\infty$-categories as suitable quasicategorical factorization systems. More precisely we show that a $t$-structure $t$ on a stable $\infty$-category $C$ is equivalent to a "normal torsion theory" $F$, i.e. to a factorization system $F=(E,M)$ on $C$ where both classes satisfy the $3$-for-$2$ cancellation property, and a certain compatibility with respect to pullbacks/pushouts.

This paves the way for a treatment of $t$-structures in different models for $(\infty,1)$-categories.

Analysis Seminar

Speaker: Janusz Adamus (Western)

"On relative Nash approximation of complex analytic sets (Part II)"

Time: 15:30 - 16:30

Room: MC 108

On relative Nash approximation of complex analytic sets. Abstract: A basic problem in complex analysis is to approximate holomorphic maps by algebraic ones. This problem has a natural generalization in complex analytic geometry. Namely, one can ask whether a complex analytic set can be approximated by branches of algebraic sets (so-called Nash sets). In the case when the analytic set has only isolated singularities, this question is closely related to the classical problem of transforming an analytic set onto a Nash set by a biholomorphic map. The situation is quite different when the singular locus is of higher dimension, as there exist analytic set germs which are not biholomorphically equivalent to any Nash set germ. A major progress in this direction was allowed by the use of the so-called Neron desingularization.

In this talk, we will report on the recent developments in Nash approximation of analytic sets and mappings. Particularly, on the problem of relative approximation along arbitrary subsets.

Speaker's homepage: http://www.math.uwo.ca/~jadamus/

Homotopy Theory

Speaker: Dan Christensen (Western)

"The Joyal model structure on simplicial sets"

Time: 13:00 - 14:30

Room: MC 107

I will discuss the class of "weak categorical equivalences" between simplicial sets, which form the weak equivalences in the Joyal model structure, whose fibrant objects are the quasicategories.

Geometry and Topology

Speaker: Boris Chorny (Haifa)

"Homotopy theory of homotopy presheaves"

Time: 15:30 - 16:30

Room: MC 107

We will discuss the category small functors from a combinatorial model category to simplicial sets and introduce a model structure in which the fibrant objects are the levelwise fibrant homotopy functors. This model structure is called the homotopy model structure.

Next, we will discuss the conditions under which a Quillen equivalence of model categories induces a Quillen equivalence of the homotopy model structures. If the time permits, then we will describe an alternative model structure Quillen equivalent to the homotopy model structure.

This work is joint with D. White.

Analysis Seminar

Speaker: Almut Burchard (Toronto)

"Symmetrization and sharp functional inequalities"

Time: 15:30 - 16:30

Room: MC 108

Symmetric decreasing rearrangement replaces a given function $f$ on $\mathbb{R}^d$ by a radially decreasing function $f^*$ that is equimeasurable to $f$. Symmetrization techniques have been used to determine the sharp constants in classical functional inequalities such as the Sobolev inequality, and for solving minimization problems in Geometry and Mathematical Physics. Symmetrization also motivates the definition of rearrangement-invariant function spaces.

I will describe recent work with A. Ferone on the extremals of the Polya-Szego inequality. The inequality says that the $p$-norms of the gradient decrease under symmetrization. It is known that there are non-trivial cases of equality, even when $p>1$. We use Ryff's polar factorization to describe these equality cases.

Speaker's homepage: http://www.math.toronto.edu/almut/

Homotopy Theory

Speaker: Karol Szumilo (Western)

"The Joyal model structure on simplicial sets II"

Time: 13:00 - 14:30

Room: MC 107

I will construct the Joyal model structure.

Colloquium

Speaker: Olivier Haution (Munich)

"p-group actions on smooth projective varieties"

Time: 15:30 - 16:30

Room: MC 107

I will discuss how the geometry of an algebraic variety restricts the possible p-group actions on it, especially concerning fixed points."