The seminar on differential geometry will continue with this lecture:

May 31, 10am, online on MS Teams and the seminar room on the second floor

Join via this LINK.

Marcus Dafinger (University of Jena):

On formal calculus of variations, Cartan formula and Noether's theorem

Abstract:

We give an introductory talk on the formal calculus of variations. Concepts from differential geometry, like Lie-derivative, exterior derivative and Cartan formula will be related to first variation, Helmholtz map and a kind of related Cartan formula. With the help of these operations we then formulate Noether's theorem and present some results on the so-called Takens' problem.

We will continue online on Thursday, May 13th, at 13.00 CEST on ZOOM platform (for information how to acces seminar and next programme visit this page) by the talk:

Nathanael Arkor

Higher-order algebraic theories and relative monads

Abstract:
There have traditionally been two ways to reason about universal algebraic structure categorically: via algebraic theories, and via monads. It is well known that the two are tightly related: in particular, there is a correspondence between algebraic theories and a class of monads on the category of sets.

Motivated by the study of simple type theories, Fiore and Mahmoud introduced second-order algebraic theories, which extend classical (first-order) algebraic theories by variable-binding operators, such as the existential quantifier ∃x of first-order logic; the differential operators d/dx analysis; and the λ-abstraction operator of the untyped λ-calculus. Fiore and Mahmoud estab- lished a correspondence between second-order algebraic theories and a second-order equational logic, but did not pursue a general understanding of the categorical structure of second-order algebraic theories. In particular, the possibility of a monad–theory correspondence for second- order algebraic theories was left as an open question.

In this talk, I will present a generalisation of algebraic theories to higher-order structure, in particular subsuming the second-order algebraic theories of Fiore and Mahmoud, and describe a universal property of the category of nth-order algebraic theories. The central result is a correspondence between (n + 1)th-order algebraic theories and a class of relative monads on the category of nth-order algebraic theories, which extends to a monad correspondence subsuming that of the classical setting. Finally, I will discuss how the perspective lent by higher-order algebraic theories sheds new light on the classical monad–theory correspondence.

This is a report on joint work with Dylan McDermott.

The seminar on differential geometry will continue with this lecture:

May 17, 10am, online on MS Teams

Join via this LINK.

Radoslaw Kycia (Masaryk university):

CoPoincare lemma and applications to physics

Abstract:

I will outline the construction of the homotopy operator for codifferential defined on Riemannian manifolds. This notion can be used to solve, in a star-shaped open subset, many equations of mathematical physics including Dirac, Maxwell and string theory problems. I will also present an intriguing correspondence between (co)homotopy operator and Clifford algebra. I will also discuss various incarnations of spinors that appear in the literature. The talk is based on the draft [2] and [1].

[1]Radoslaw Kycia, The Poincare lemma, antiexact forms, and fermionic quantum harmonic oscillator, Results in Mathematics 75, 122 (2020)
[2] Radoslaw Kycia, The Poincare lemma for codifferential, anticoexact forms, and applications to physics, arXiv: 2009.08542 [math.DG]

We will continue online on Thursday, May 6th, the special time of 14.00 CEST on ZOOM platform (for information how to acces seminar and next programme visit this page) by the talk:

Walter Tholen

Spaces vs Categories, Perfect Maps vs Discrete Cofibrations

Abstract:
We consider perfect maps of topological spaces and discrete cofibrations of categories to guide us into Burroni's notions of T-category and T-functor. In that environment we establish a so-called comprehensive factorization system that entails the classical Street-Walters system, as well as the (antiperfect, perfect)-system for continuous maps of Tychonoff spaces known since the 1960s.

(Based on joint work with Leila Yeganeh)