応用特異点論ラボセミナー：Thom Polynomials in Enumerative Geometry(Soeren Nekarda), Poincaré completions and Intersection Spaces(J. Timo Essig)

time：15:00-17:00

Place：理学部3号館3-204

Speaker：Soeren Nekarda (Univ. Halle), J. Timo Essig

Organizer：石川剛郎

Timetable

15:00-15:50　Soeren Nekarda (Univ. Halle)

Title：Thom Polynomials in Enumerative Geometry

Abstract：
We sketch how the theory of universal polynomials for singularity classes can be applied to study problems in enumerative geometry of projections. First, we showcase the general idea using plane curves, then recall some notions on Thom polynomials for multisingularities and its generalizations. The theory is then applied to projections of generic space curves. Finally, we highlight the progress in an ongoing research project concerning projections of generic surfaces in the projective space P3.

16:10-17:00　J. Timo Essig

Title：Poincaré completions and Intersection Spaces

Abstract：
A Poincaré pair of dimension $n$ is a pair of spaces $(X,A)$ such that there is a fundamental class $\nu \in H_n(X,A)$ with the property that $_ \cap \nu : H^k (X) \to H_{n-k} (X,A)$ is an isomorphism. A Poincaré space $X$ is a Poincaré pair $(X, \emptyset)$. The concept goes back to Browder, who introduced Poincaré pairs as an important tool for the classification of high dimensional manifolds via surgery theory. I present a procedure to transform space pairs into Poincaré pairs by gluing cells. The central result covers the conditions for achieving this goal by gluing only one pair of cells. An application is intersection spaces: Using the procedure, Poincaré spaces can be assigned to certain singular spaces via intersection spaces. The present approach is a formalization of existing results by Klimczak and Wrazidlo and an extension to stratification depths greater than one.