Asaf Horev:

The inductive definition of C⁰ stratified spaces, involving cones. An indication of conically smooth stratified spaces -- notably so that strata, as well as links, are equipped with smooth structure. Unzips (aka blow-ups). The simplicially enriched category Strat of conically smooth stratified spaces.

Ref: Local structures on stratified spaces.

Shay Ben Moshe:

The internal definition of the oo-category of oo-categories as an oo-subcategory of Fun(\Delta^{\op}, Spaces). Construction of functor \Delta ---> Strat sending [p] to \Delta^p=Cone^p(point) with the standard stratification. Exit-path oo-categories as the restricted Yoneda functor \Delta ---> Strat. Spaces of objects in Exit(X) as strata, and morphisms in Exit(X) as links between strata. (Optional: discuss the identification Fun(Exit(X),Spaces) = Shv^{constructible}(X) , which contextualizes Exit(X) .)

Ref: A stratified homotopy hypothesis.

Shachar Carmeli:

This talk is an opportunity to discuss oo-categories in more depth beyond the definition. Here are some specific points to address. Correspondences (as opposed to spans). The (flagged) oo-category Corr whose objects are oo-categories and whose morphisms are correspondences. Pushforward along an exponentiable fibration.

Ref: Fibrations of oo-categories.

Samik Basu:

Definition of Theta_n and of the oo-category of (oo,n)-categories as an oo-subcategory of Fun(\Theta_n^{\op}, Spaces).

Ref: Rezk, A cartesian presentation of weak n-categories. Factorization homology I.

Ariel Davis:

The oo-category Vect^{inj} of (finite-dimensional) vector spaces (over R) and injections among them. Tangential structures as oo-categories over Vect^{inj}. Vari-framings. Framed suspension. Solid n-framings. Examples of vari-framed, and solid n-framed, stratified spaces, including the hemispherically-stratified n-disk D^n. Aut^{vfr}(D^n) is discrete.

Ref: Factorization homology I.

Lior Yanovsky:

Definition of the symmetric monoidal oo-category Mfld_n; likewise, the definition of Mfld_n^{fr}. The definition of Disk_n, as well as Disk_n^{fr}. The definition of Disk_n-algebras, and E_n-algebras, in these terms. The definition of ``\otimes-excisive'' functors from Mfld_n, and Mfld_n^{fr}. The classification of such as Disk_n-algebras, and E_n-algebras, respectively, via factorization homology. Non-abelian Poincare' duality.

Ref: Factorization homology of topological manifolds.

Shaul Barkan:

Proof of the Pontryagin–Thom theorem: inverse images of regular values; Pontryagin–Thom collapse maps of open embeddings.

Ref: Milnor, Topology from the differentiable viewpoint.