Kensuke Arakawa (Kyoto University)
Abstract:
Manifold calculus is a homotopy-theoretic technique to study presheaves on manifolds, which decomposes them into successive approximations called polynomial approximations. First invented by Weiss to study embedding spaces, it has become an important toolset for homotopical study of manifolds. Like ordinary calculus, manifold calculus has two "fundamental theorems," one which classifies polynomial presheaves, and the other that classifies homogeneous presheaves. Consistent with his goal to study embedding spaces, Weiss established these theorems for space-valued presheaves. From the perspective of studying manifold invariants, it is extremely natural to develop manifold calculus for presheaves with more general values, such as spectra and chain complexes. However, Weiss\'s proof of the fundamental theorems relies on ad-hoc constructions on spaces, which do not seem to generalize easily. In this talk, I will explain that the two fundamental theorems do not depend on space-level constructions. Consequently, they extend to presheaves valued in essentially any infinity category. This talk is based on my paper "A context for manifold calculus" (arXiv:2403.03321).
Dr. Jutta Rath (Universität Klagenfurt)
On the Asymptotic Structure of Powers of Monomial Ideals abstract
Abstract:
<p>After introducing monomial ideals and their connections to combinatorial objects such as graphs and simplicial complexes, we investigate some of their invariants and their asymptotic behaviour. In particular, we use Newton polyhedra to give an explicit description of the minimal generating sets of large powers of bivariate monomial ideals. Moreover, we establish bounds on the power beyond which the sets of associated primes of monomial ideals in three or more variables stabilize.</p>
11:30 • Uni St. Gallen, 64-110
Lukas Klawuhn (Universität Paderborn)
Why I love association schemes - an introduction to Delsarte Theory abstract
Abstract:
<p>Interesting combinatorial structures can often be characterised as special subsets of association schemes. In his PhD thesis, Philippe Delsarte developed powerful linear programming techniques to prove non-existence and uniqueness results for such structures. In particular, this applies to error-correcting codes. Ideas of this type were fundamental in the work for which Maryna Viazovska was awarded the Fields medal in 2022. This talk will begin with an overview of Delsarte theory.</p> <p>We will use this theory to study perfect matchings. We show that 1-factorisations of the complete graph are special subsets in the sense of Delsarte. The same is true for hyperfactorisations and other generalisations of 1-factorisations. The characterisation of these structures as a special subset of an association scheme gives rise to divisibility conditions and non-existence results. We also give a construction of hyperfactorisations using finite geometry.</p> <p>No particular knowledge of association schemes will be required to appreciate this talk.</p>
14:00 • Uni St. Gallen, 64-110