We will start our seminar on Thursday, January 24, in M5 at 1pm.

Harry Gindi

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.

We will start our seminar on Thursday, January 24, in M5 at 1pm.

Giulio Lo Monaco

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.

PF 2019

December 12, 2018 from 5:00 PM at Refectory of Augustinian Abbey at Mendel Square - Mendel Museum

Robert Bryant

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.