We introduce the concept of (coarse) n-point bottlenecking in graphs and study the coarse geometry of graphs in terms of bottlenecking in their coarse skeletons. We examine the connections of bottlenecking with coarse planarity. This is joint work with Atish Mitra and Heidi Steiger.

Suppose we are handed a map of smooth manifolds and would like to know if it is homotopic to an immersion. In general this problem is undecidable, indeed even immersibility of an arbitrary manifold into $\mathbb{R}^n$Rn is undecidable. In this talk we will see how to use techniques from rational homotopy theory, and the h-principle of Hirsch and Smale to provide an algorithm for this problem whenever the codimension of the manifolds is odd.

Gromov defined Macroscopic Dimension for manifolds as a measure of largeness, in order to study topology of manifolds admitting Positive Scaler Curvature metric. He conjectured universal cover $\tilde M$M~, of closed $n$n-manifold $M$M admitting Positive Scaler Curvature metric, should have $dim_{mc}\tilde M\le n-2$dimmc​M~≤n−2, for the metric on $\tilde M$M~ lifted from $M$M. This conjecture heavily depends on $\pi_1(M)$π1​(M). Recently we could prove the conjecture for closed spin $n$n-manifolds $M$M, having $\pi_1(M)\in RAAG$π1​(M)∈RAAG. We will try to see a brief history of this Conjecture and an overview of our proof method.

In 1987 Gromov and Thurston developed the first Riemannian manifolds that are not homotopy equivalent to a hyperbolic manifold but admit a Riemannian metric that is ϵ-pinched for any given ϵ>0. The manifolds that they construct are branched covers of hyperbolic manifolds, and to construct the metric they perform a sort of "geometric surgery" about the ramification locus. In 2022 Stover and Toledo proved the existence of similar branched cover manifolds built out of complex hyperbolic manifolds, and via a result of Zheng these manifolds admit a negatively curved Kahler metric. In this talk we will discuss how to construct a (not Kahler) Riemanain metric on these Stover-Toledo manifolds which is ϵ-close to being 1/4-pinched for any prescribed ϵ>0. These provide the first known examples of Kahler manifolds that are not homotopy equivalent to a complex hyperbolic manifold but admit a Riemannian metric that is ϵ-close to being 1/4-pinched.

Let $n$n be a positive integer. We provide an explicit geometrically motivated 1-Lipschitz map from the space of persistence diagrams on n points (equipped with the Bottleneck distance) into Hilbert space. Such maps are a crucial step in topological data analysis, allowing the use of statistics (and thus data analysis) on collections of persistence diagrams. The main advantage of our maps as compared to most of the other such transformations is that they are coarse and uniform embeddings with explicit distortion functions. Furthermore, we provide an explicit 1-Lipschitz map from the space of persistence diagrams on $n$n points on a bounded domain into a Euclidean space with an explicit distortion function. Our ideas come from geometric topology and dimension theory, and our methods are best described as quantitative dimension theory. This is joint work with Ziga Virk.

A plane set admits an inscribed rectangle if every homeomorphic copy of it in $\mathbb{R}^2$R2 contains the 4 vertices of at least one Euclidean rectangle. Vaughan proved that $S^1$S1 admits an inscribed rectangle by reducing the problem to the non-embeddability of the projective plane in $\mathbb{R}^3$R3 (it is not known if $S^1$S1 admits an inscribed rectangle of aspect ratio 1:1 i.e. a square). In this talk, using the non-embeddability of the Cone($K_5$K5​) and the Cone($K_{3,3}$K3,3​) in $\mathbb{R}^3$R3 we classify plane compact connected locally-connected sets that admit inscribed rectangles. Using similar topological techniques, we also present a one-dimensional non-connected set such that every copy of it in $\mathbb{R}^2$R2 admits an inscribed rectangle with at least one vertex in each component.

Essential families can be used to provide a simple characterization of the dimension of a normal space (and with a small adjustment, also for a regular space). For example, a normal space $X$X has dimension $n\in\mathbb{N}$n∈N if and only if it has an essential family of cardinality $n$n and for all $m>n$m>n, it has no essential family of cardinality $m$m. A space is {\it strongly infinite-dimensional} if it has a countably infinite essential family. The Hilbert cube, $I^\infty$I∞, is strongly infinite-dimensional; however, one might wonder if it has an uncountable essential family. Going even further, can a separable metrizable space have an uncountable essential family?

In this talk we will define essential families as they are used in this setting and then present the following theorem which establishes an upper bound on the cardinality of essential families in normal or regular ($\mathrm{T}_1$T1​ not required) spaces.

{\bf Theorem.} Let $X$X be a regular or normal space of infinite weight and $\mathcal{C}$C be an essential family in $X$X. Then $\mathrm{card}\,\mathcal{C}\leq\mathrm{wt}X$cardC≤wtX.

We employ a proof by contradiction in which we assume that there is an essential family of higher cardinality than the weight of the given space and then by a transfinite construction, which we will not try to present, arrive at a contradiction. But we will give a clue as to how one can ``finesse'' this supposedly essential family in order to detect that it is not essential.

The Deligne category of symmetric groups is the additive Karoubi closure of the partition category. It is semisimple for generic values of the parameter t while producing categories of representations of the symmetric group when modded out by the ideal of negligible morphisms when t is a non-negative integer. The partition category may be interpreted, following Comes, via a particular linearization of the category of two-dimensional oriented cobordisms. The Deligne category and its semisimple quotients admit similar interpretations. This viewpoint coupled to the universal construction of two-dimensional topological theories leads to multi-parameter monoidal generalizations of the partition and the Deligne categories, one for each rational function in one variable.

Let $C$C be a compact Riemann surface, and $X$X be smooth projective variety. We will consider the space of holomorphic maps $C \to X$C→X.

When $X = \mathbb P^n$X=Pn, Segal demonstrated a remarkable stabilization phenomenon: as $d$d increases, the homology of the component of degree $d$d holomorphic maps converges to homology of the component of degree $d$d continuous maps $C \to X$C→X. Ellenberg-Venkatesh and others have observed that this phenomenon is related to arithmetic conjectures about rational points on Fano varieties due to Batyrev and Manin. This suggests that this stabilization phenomenon may hold more generally.

I will talk about joint work with Ronno Das using the Vassiliev method to study the case of blowups of projective space at finitely many points (in particular del Pezzo surfaces).

The Geometrization Theorem of Thurston and Perelman provides a roadmap to understanding topology in dimension 3 via geometric means. Specifically, it states that every closed 3-manifold has a decomposition into geometric pieces, and the zoo of these geometric pieces is quite constrained: each is built from one of eight homogeneous 3-dimensional Riemannian model spaces (called the Thurston geometries).

In this talk, we will approach the question of “what does a 3-manifold look like” from the perspective of geometrization. Through animations of simple examples in dimensions 2 and 3 we review what it means to put a (complete, homogeneous) geometric structure on a manifold, and construct an example admitting each of the Thurston geometries.

Using software written in collaboration with Remi Coulon, Sabetta Matsumoto and Henry Segerman, we will explore these manifolds ``from the inside'' - that is, simulating the view one would have in such a space by raytracing along geodesics. Finally we will explore the re-assembly of these geometric pieces and understand an “inside view” of general 3-manifolds.

For a surface S, Thurston asked if the natural surjection Homeo(S) → π_0 Homeo(S) splits, i.e. if there a natural action of the mapping class group Mod(S):= π_0 Homeo(S) on S. Markovic showed that no such action exists. On the other hand, there is a natural action of Mod(S) on the unit tangent bundle of S. More generally, for a 3-manifold M that fibers as a circle bundle over S, there is natural surjection Homeo(M) → Mod(S). We study when this surjection splits. This is joint work with Lei Chen and Alina al Beaini.

In dimension 4, there exist simply-connected manifolds which are homeomorphic but not diffeomorphic; the difference between the distinct smooth structures can be localized using corks. Similarly, there exist diffeomorphisms of simply-connected 4-manifolds which are topologically but not smoothly isotopic. In this talk, I will discuss some preliminary results towards an analogous localization of this phenomena using corks for diffeomorphisms. This project is joint work with Slava Krushkal, Anubhav Mukherjee, and Mark Powell.

We call a knot K a complete Alexander neighbor if every possible Alexander polynomial is realized by a knot one crossing change away from K. It is unknown whether there exists a complete Alexander neighbor with nontrivial Alexander polynomial. I will discuss how to eliminate infinite families of knots with nontrivial Alexander polynomial from having this property and possible strategies for unresolved cases. I will also discuss how a related condition on determinants of knots one crossing change away from unknotting number one knots gives an obstruction to unknotting number one. This obstruction appears similar to an obstruction introduced by Lickorish, but Lickorish’s obstruction does not subsume the obstruction coming from the condition on determinants.

In this talk, we will address two conjectures. Firstly, we will present the proof of "0-shake slice knots are slice", which was a collaborative effort with Selman Akbulut. Secondly, we will discuss how the progress made in the first problem can assist in tackling the "Smooth 4-D Poincaré conjecture". If time allows, we will delve into this further.