# QG&T@OSU

## Regional Workshop in Quantitative Geometry & Topology April 27–28, 2019

Welcome! We (Fedya Manin and Hannah Alpert) are planning a weekend workshop on quantitative geometry and topology. We hope to provide a venue for young researchers in geometry and topology to learn about this subject and to trade problems and ideas. Some travel funding is available.

In addition to several invited talks, we want to provide as much time as possible, both structured and unstructured, for participants to get to know each other and strike up collaborations. This might include a problem session and/or a series of 5-minute lightning talks.

Quantitative geometry and topology refines the qualitative, discrete questions of algebraic and geometric topology into continuous ones. For example, we may see a loop in a space which is homotopically trivial and ask how difficult it is to trivialize. Depending on what we mean by "difficult", we might obtain different notions of isoperimetry; one common choice is the area of a filling disk, which leads to the definition of the Dehn function of a group.

A priori, such notions usually depend on the choice of a metric on the space; one can then analyze their dependence on the metric (quantitative geometry) or show results, for example asymptotic ones, which are independent of the choice (quantitative topology).

From a more global perspective, one can see this as the study of geometric functionals on spaces of geometric objects: the Lipschitz constant as a functional on the space of maps between two spaces; mass, for cycles in a space; Riemannian volume, for manifolds.

This circle of questions has ties to geometric group theory, minimal surface theory, and the theory of computation among other areas. Specific topics which can be viewed through this lens include:

• Isoperimetric functions
• Systolic geometry
• High-dimensional expanders
• Configuration spaces of hard disks

## Speakers and abstracts

We will have six invited talks. Expand to see abstracts.

#### Constructing optimal homotopies and sweepouts

Suppose that the boundary of a Riemannian disc can be contracted through closed curves of length less than L. Can we find such a contraction that consists of curves that are pairwise disjoint and are simple or constant? We will discuss the answer to this question, and will also describe how the methods involved in its resolution can be applied to a number of other problems, including finding minimal hypersurfaces in noncompact manifolds, and constructing optimal sweepouts of Riemannian 2-spheres. This talk will include work with Regina Rotman, Yevgeny Liokumovich, Erin W. Chambers, Tim Ophelders, and Arnaud de Mesmay.

#### A survey of k-dilation

The k-dilation of a map measures how much the map stretches k-dimensional volumes. The idea was first coined by Gromov and Lawson in the early 80s. It is a generalization of the Lipschitz constant (which is equal to the 1-dilation) but larger k behave differently in a lot of ways. We will survey some of the things that are known and also a lot of open questions.

#### Metrics on the knot concordance space

The knot concordance group (a non-finitely generated abelian group) is an important space in low-dimensional topology, related to questions about whether surgery works in 4-dimensions, as well as the smooth 4-dimensional Poincare conjecture. For many decades, people studied the set as a group. We discuss some natural metrics on the set of knots up to concordance and use these to give evidence that the knot concordance group has the structure of a fractal space. We will give all the necessary background on knot concordance. This is joint work with Tim Cochran, Mark Powell, and Aru Ray.

#### Filling metric spaces: Guth's conjecture on Urysohn width and Hausdorff content

Hausdorff content and Urysohn width are two ways of measuring the size of a metric space. Urysohn width measures how well a metric space X can be approximated by an $(m-1)$-dimensional space. Hausdorff content is defined like the Hausdorff measure except that we do not take the limit with the radii of balls in the covering going to 0. Larry Guth conjectured an inequality relating the two, which generalizes his and Gromov's results about filling Riemannian manifolds. I will talk about a proof of Guth's conjecture and how it can be used to strengthen and generalize Gromov's systolic inequality for essential manifolds. This is joint work with Boris Lishak, Alexander Nabutovsky and Regina Rotman.

#### The Unbearable Hardness of Unknotting

Given a diagram of an unknot and a natural number n, is it possible to eliminate all crossings in the diagram using n Reidemeister moves? Is it possible to change a non-trival link to the unlink by changing n crossings? Given a 2-complex, does it embed in $\mathbb{R}^3$? We will discuss the authors' work that shows that these (and other related) problems are NP-hard, that is, at least as hard as any in a class of problems (NP) that includes classically "hard" problems such as the Traveling Salesman Problem and the Hamiltonian Cycle Problem. Joint work with Arnaud de Mesmay, Yo'av Rieck and Martin Tancer.

#### Self-similar solutions to extension and approximation problems

In 1979, Kaufman constructed a remarkable surjective Lipschitz map from a cube to a square whose derivative has rank $1$ almost everywhere. In this talk, we will present some higher-dimensional generalizations of Kaufman's construction that lead to Lipschitz and Hölder maps with wild properties, including: topologically nontrivial maps from $S^m\to S^n$ with derivative of rank $n-1$, $(\frac{2}{3}-\delta)$–Hölder approximations of surfaces in the Heisenberg group, and Hölder maps from the disc to the disc that preserve signed area but approximate an arbitrary continuous map. This is joint work with Stefan Wenger and Larry Guth.

## Schedule

 Friday, April 26 6:00–8:00pm Drop-in board games and registration MW 154 To find the room, look to the left as you enter the Mathematics Tower. You can pick up a name tag and meet up with other participants, before or after you eat dinner. Saturday, April 27 All talks will be in room CH 240. To find the room, go to the second floor of Math Tower or Math Building and head west, past the kitchen area where coffee breaks will be held. 8:30am Coffee and bagels outside CH 240 9:00–10:00 Shelly Harvey: Metrics on the knot concordance space CH 240 10:30–11:30 Lightning talks: Zhang, Wan, Vasudevan, Okutan, Newton, Mirth, Lowe, Lim, Kumanduri, Kim — lunch on your own — 1:00–2:00 Eric Sedgwick: The unbearable hardness of unknotting CH 240 2:15–3:15 Lightning talks: Li, Grindstaff, Gómez, Elliott, Duncan, Bush, Berdnikov, Balitskiy, Altawallbeh — coffee break — 4:00–5:00 Larry Guth: A survey of k-dilation — dinner together at Lalibela Restaurant — Sunday, April 28 8:30am Coffee and bagels outside CH 240 9:00–10:00 Greg Chambers: Constructing optimal homotopies and sweepouts CH 240 10:30–11:30 Robert Young: Self-similar solutions to extension and approximation problems — lunch on your own — 1:00–2:00 Yevgeny Liokumovich: Filling metric spaces: Guth's conjecture on Urysohn width and Hausdorff content CH 240

## Registration & Funding

The registration form is available; please register at least a week prior to the workshop so we can get enough coffee for everyone! We will have partial lodging and travel support for graduate students and other early career participants who have registered by March 15. We will try to get back to you soon afterwards with estimates of the maximum travel cost we can reimburse.

We would like to acknowledge the funding support of the National Science Foundation via RTG grant DMS-1547357 and of the Mathematics Research Institute at OSU.

## Travel Info

Invited speakers will stay at the Blackwell, Ohio State's on-campus hotel. Unfortunately, the Blackwell is almost fully booked for those dates and we cannot give the opportunity of staying there to other participants. Funded participants need to book their own accommodations and will be reimbursed up to some amount, likely roughly \$80/night. We are happy to match people who want to find an AirBnB or hotel room to share.

Columbus Airport (CMH) is the nearest airport, and the most convenient way to get to Columbus from outside driving distance. It's about 15 minutes to campus by Uber/Lyft; the Blackwell also offers a free shuttle. Participants from the Great Lakes region may want to drive or take a bus. In addition to Greyhound, Barons Bus is a regional company serving Columbus, including some buses to Ohio State campus.