Page 254 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 254
APPLIED COMBINATORIAL AND GEOMETRIC TOPOLOGY (MS-34)
Invariants for tame parametrised chain complexes
Claudia Landi, claudia.landi@unimore.it
Università di Modena e Reggio Emilia, Italy
Coauthors: Wojciech Chachólski, Barbara Giunti
Data analysis is often about simplifying, ignoring most of the information available and extract-
ing what might be meaningful to a task at hand. This strategy of making sense by disregarding
some information and focusing on aspects that might be relevant is very common across math-
ematics. Colocalization in homotopy theory is an example of such a process. In colocalization,
simplification is achieved by approximating arbitrary objects by other objects that one considers
simpler and more manageable. For instance, by approximating a given space by n-connected
spaces, one obtains its n-connected cover. The aim of our presentation is to explain why ex-
tracting persistence invariants in Topological Data Analysis (TDA) is an example of the homo-
topical colocalization process. This allows us to extract computable invariants also in certain
cases, such as commutative ladders, that have not been covered by more standard approaches.
Furthermore, it allows for a comprehensive theory including several cases that standard persis-
tence theory handles separately, such as persistence modules, zigzag modules, and commutative
ladders.
Transitive factorizations of pairs of permutations and three-dimensional
constellations
Luca Lionni, luca.lionni@ru.nl
Radboud University, Nijmegen, Netherlands
Factorizations of pairs of permutations that generate the full symmetric group on n-symbols are
shown to be in bijection with a certain family of three-dimensional 4-colored triangulations that
generalize constellations. These spaces are ramified coverings of the three-dimensional sphere,
branched over the n-unlink. We will present a generalization of the Riemann-Hurwitz formula,
and identify the spaces that maximize the number of branching edges for a fixed number of
sheets.
Explicit computation of some families of Hurwitz numbers
Carlo Petronio, petronio@dm.unipi.it
Università di Pisa, Italy
I will describe the computation (based on dessins d’enfant) of the number of equivalence classes
of surface branched covers matching a given branch datum belonging to a certain very specific
family.
252
Invariants for tame parametrised chain complexes
Claudia Landi, claudia.landi@unimore.it
Università di Modena e Reggio Emilia, Italy
Coauthors: Wojciech Chachólski, Barbara Giunti
Data analysis is often about simplifying, ignoring most of the information available and extract-
ing what might be meaningful to a task at hand. This strategy of making sense by disregarding
some information and focusing on aspects that might be relevant is very common across math-
ematics. Colocalization in homotopy theory is an example of such a process. In colocalization,
simplification is achieved by approximating arbitrary objects by other objects that one considers
simpler and more manageable. For instance, by approximating a given space by n-connected
spaces, one obtains its n-connected cover. The aim of our presentation is to explain why ex-
tracting persistence invariants in Topological Data Analysis (TDA) is an example of the homo-
topical colocalization process. This allows us to extract computable invariants also in certain
cases, such as commutative ladders, that have not been covered by more standard approaches.
Furthermore, it allows for a comprehensive theory including several cases that standard persis-
tence theory handles separately, such as persistence modules, zigzag modules, and commutative
ladders.
Transitive factorizations of pairs of permutations and three-dimensional
constellations
Luca Lionni, luca.lionni@ru.nl
Radboud University, Nijmegen, Netherlands
Factorizations of pairs of permutations that generate the full symmetric group on n-symbols are
shown to be in bijection with a certain family of three-dimensional 4-colored triangulations that
generalize constellations. These spaces are ramified coverings of the three-dimensional sphere,
branched over the n-unlink. We will present a generalization of the Riemann-Hurwitz formula,
and identify the spaces that maximize the number of branching edges for a fixed number of
sheets.
Explicit computation of some families of Hurwitz numbers
Carlo Petronio, petronio@dm.unipi.it
Università di Pisa, Italy
I will describe the computation (based on dessins d’enfant) of the number of equivalence classes
of surface branched covers matching a given branch datum belonging to a certain very specific
family.
252
