**Homotopical cocompleteness of enriched categories**

Abstract:

We will consider categories enriched in a monoidal model category V that are accessible and admit certain limits, and will prove that they admit weighted homotopy colimits. As a special case, we will apply our results to the infinity cosmoi of Riehl and Verity. This is a report on ongoing work with Steve Lack and Lukas Vokrinek.

]]>**Homotopical cocompleteness of enriched categories**

Abstract:

We will consider categories enriched in a monoidal model category V that are accessible and admit certain limits, and will prove that they admit weighted homotopy colimits. As a special case, we will apply our results to the infinity cosmoi of Riehl and Verity. This is a report on ongoing work with Steve Lack and Lukas Vokrinek.

]]>**Bifurcation of solutions for generalized ordinary differential equations**

**Stability shaped by the noise in difference equations**

Abstract:

An active role of noise perturbations in forming stability properties of solutions of difference equations is explored. Some of our models are inspired by mathematical biology where noise enters naturally through the influence of the environment. We discuss how different types of stochastic perturbations change stability properties of the solution of the deterministic counterpart. We also consider a highly nonlinear stochastic differential equation where stability of the equilibrium is induced by the noise term. It is often challenging to retain such stability in a numerical simulation. We solve this issue by designing an adaptive timestepping discretization scheme which faithfully reproduces stability properties of the solution of the original differential equation.

You are cordially invited to the 21st mathematical hike. It will take place on

Start at **08:07 at the tram stop "Kamenolom". **

We plan a 13 km hike - go over the hills in the north part of Brno and then go to Bystrc. You can modify the path to finish at the starting point. We expect the hike to end fairly early. Both the start and the end are planned in the zone 101 of Brno public transport.

Have a nice start of the semester, we're looking forward to seeing you,

Jonatan Kolegar a Jana Bartoňová, organizers,

Jan Slovák, Director of the Department of Mathematics and Statistics

**Cartesian-Enriched Quasi-categories, the Isofibration theorem and Yoneda's lemma**

Abstract:

In Joyal's theory of quasi-categories, there is a very nice characterization of the fibrant objects and the fibrations between them as the inner-fibrant objects and isomorphism-lifting inner fibrations respectively. Given a nice-enough Reedy category C, we construct a horizontal model structure on the category of presheaves of sets on Θ[C] that shares a variant of this characterization. Moreover, given a Cartesian presentation with respect to simplicial presheaves on C, we show that the horizontal model structure Psh(Θ[C]) admits a left-Bousfield localization that agrees with Rezk's model structure on sPsh(Θ[C]). It will follow by general facts about left-Bousfield localization that the model fibrations between the local objects are exactly the horizontal isofibrations. We will also briefly describe the generalization of the homotopy-coherent nerve and realization for these enriched Quasi-categories and sketch a proof of Yoneda's lemma in this setting, if time permits.

**On two extensions of the notion of oo-topos**

Abstract:

The two notions of elementary oo-topos and oo-pretopos as in the title should hopefully be the correct analog of their counterparts in ordinary topos theory. The focus here will be put on their precise relationship with the much better known oo-toposes. In particular, under suitable set-theoretical assumptions, both of these classes of objects contain the class of geometric oo-toposes, and they do so strictly, for which explicit counterexamples can be provided.

]]>**The Best Possible Shapes of Surfaces**

Abstract: Much of classical mathematics involves finding a configuration or shape that provides an optimum solution to a problem. For example, it has long been known (though a rigorous proof took quite a while to find) that the surface of least area enclosing a given volume is a round sphere. There are many other ways to measure surfaces, though, and finding 'the' surface that optimizes a given 'measurement' (subject to some given constraints) remains a challenging problem that has motivated some of the deepest recent work in the mathematics of geometric shapes.

In this talk, I will explain some of the classic ways to measure shapes of surfaces and relate this to classical problems involving surface area (soap films and bubbles) and total curvature as well to as recent progress by myself and others on these important optimization problems.

You are cordially invited to the 20th mathematical hike. It will take place on

Start at

Have a nice December,

Jana Bartoňová and Jonatan Kolegar, organizers,

Jan Slovák, Director of the Department of Mathematics and Statistics]]>