Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userStructure and Interleavings of Relative Interlevel Set Cohomology
194
0
0.0
(
0
)
Added by
Benedikt Fluhr
Publication date
2021
fields
Informatics Engineering
and research's language is
English
Ask ChatGPT about the research
No Arabic abstract
The relative interlevel set cohomology (RISC) is an invariant of real-valued continuous functions closely related to the Mayer--Vietoris pyramid introduced by Carlsson, de Silva, and Morozov. We provide a structure theorem, which applies to the RISC if it is pointwise finite dimensional (pfd) or, equivalently, $q$-tame. Moreover, we provide the notion of an interleaving for RISC and we show that it is stable in the sense that any space with two functions that are $delta$-close induces a $delta$-interleaving of the corresponding relative interlevel set cohomologies.
rate research
Read More
We give a self-contained treatment of the theory of persistence modules indexed over the real line. We give new proofs of the standard results. Persistence diagrams are constructed using measure theory. Linear algebra lemmas are simplified using a new notation for calculations on quiver representations. We show that the stringent finiteness conditions required by traditional methods are not necessary to prove the existence and stability of the persistence diagram. We introduce weaker hypotheses for taming persistence modules, which are met in practice and are strong enough for the theory still to work. The constructions and proofs enabled by our framework are, we claim, cleaner and simpler.
We prove duality results for absolute and relati
Suppose $k$ is a field of characteristic 2, and $n,mgeq 4$ powers of 2. Then the $A_infty$-structure of the group cohomology algebras $H^*(C_n,k)$ and $H^*(C_m,k)$ are well known. We give results characterizing an $A_infty$-structure on $H^*(C_ntimes C_m,k)$ including limits on non-vanishing low-arity operations and an infinite family of non-vanishing higher operations.
We establish a system of PDE, called open WDVV, that constrains the bulk-deformed superpotential and associated open Gromov-Witten invariants of a Lagrangian submanifold $L subset X$ with a bounding chain. Simultaneously, we define the quantum cohomo
logy algebra of $X$ relative to $L$ and prove its associativity. We also define the relative quantum connection and prove it is flat. A wall-crossing formula is derived that allows the interchange of point-like boundary constraints and certain interior constraints in open Gromov-Witten invariants. Another result is a vanishing theorem for open Gromov-Witten invariants of homologically non-trivial Lagrangians with more than one point-like boundary constraint. In this case, the open Gromov-Witten invariants with one point-like boundary constraint are shown to recover certain closed invariants. From open WDVV and the wall-crossing formula, a system of recursive relations is derived that entirely determines the open Gromov-Witten invariants of $(X,L) = (mathbb{C}P^n, mathbb{R}P^n)$ with $n$ odd, defined in previous work of the authors. Thus, we obtain explicit formulas for enumerative invariants defined using the Fukaya-Oh-Ohta-Ono theory of bounding chains.
Building on work of Livernet and Richter, we prove that E_n-homology and E_n-cohomology of a commutative algebra with coefficients in a symmetric bimodule can be interpreted as functor homology and cohomology. Furthermore we show that the associated Yoneda algebra is trivial.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
