# Topology and geometry: extremal and typical

## A Zoom seminar

This is an online seminar for the COVID era organized by Fedya Manin and Shmuel Weinberger. It will focus on whatever interests us, but mainly quantitative questions in geometry and topology.

Spring semester confirmed speakers include:

## Schedule and abstracts

The seminar will run on Mondays, roughly biweekly. Unless otherwise specified, the talks will be at noon Eastern Time, which is usually:
• 11am in Chicago
• 9am in Los Angeles
• 5pm in London (rarely 4pm)
• 6pm in Paris (rarely 5pm)
• 7pm in Tel-Aviv (rarely 6pm)
• 8pm (winter) or 7pm (summer) in Moscow
Our apologies to friends in Australia and East Asia for whom this is a terrible time.

Expand the items in this list to see abstracts.

August 17, 2020:
Robert Young (NYU)

How do you build a complicated surface? How can you decompose a surface into simple pieces? Understanding how to construct an object can help you understand how to break it down. In this talk, we will present some constructions and decompositions of surfaces based on uniform rectifiability. We will use these decompositions to study problems in geometric measure theory and metric geometry, such as how to measure the nonorientability of a surface and how to optimize an embedding of the Heisenberg group into L1 (joint with Assaf Naor).

August 31, 2020:
Panos Papasoglu (Oxford)

The Uryson width of an n-manifold gives a way to describe how closely it resembles an $(n-1)$-dimensional complex. It turns out that this is a useful tool to approach several geometric problems.

In this talk we will give a brief survey of some questions in ‘curvature-free’ geometry and sketch a novel approach to the classical systolic inequality of Gromov. Our approach follows up recent work of Guth relating Uryson width and local volume growth. For example we deduce also the following result of Guth: there is an $\epsilon_n>0$ such that for any $R>0$ and any compact aspherical n-manifold M there is a ball B(R)$of radius R in the universal cover of M such that$\operatorname{vol}(B(R))\geq \epsilon_n R^n$. September 14, 2020: Mark Pengitore (Ohio State) In this talk, we will relate homological filling functions and the existence of coarse embeddings. In particular, we will demonstrate that a coarse embedding of a group into a group of geometric dimension 2 induces an inequality on homological Dehn functions in dimension 2. As an application of this, we are able to show that if a finitely presented group coarsely embeds into a hyperbolic group of geometric dimension 2, then it is hyperbolic. Another application is a characterization of subgroups of groups with quadratic Dehn function. If there is enough time, we will talk about various higher dimensional generalizations of our main result. September 21, 2020 (note odd week): Alexey Balitskiy (MIT) The Urysohn d-width of a metric space quantifies how well it can be approximated by a d-dimensional simplicial complex. We discuss various questions of the following flavor: how does knowledge of the width of certain pieces of a riemannian manifold help us to estimate the total width of the whole manifold? Here are two examples. 1. A waist-type inequality: If the euclidean 3-ball is sliced into a 1-parametric family of (possibly singular) surfaces with$\operatorname{rank} H_1 \le b$then at least one of them has 1-width at least ~1/b (so it's "essentially 2-dimensional"). 2. The width can behave counterintuitively: it can happen that an n-manifold has substantial$(n-1)$-width but all its unit balls are "almost 1-dimensional" (that is, of small 1-width). Based on joint work with Sasha Berdnikov. October 12, 2020: Rina Rotman (Toronto) I am planning to present the following result of mine: Let$M^n$be a closed Riemannian manifold of dimension n and$\operatorname{Ric} \geq (n-1)$. Then the length of a shortest periodic geodesic can be at most$8\pi n$. The technique involves quantitative Morse theory on loop spaces. We will discuss some related results in geometry of loop spaces on Riemannian manifolds. October 26, 2020: Radmila Sazdanović (NCSU) A multitude of knot invariants, including quantum invariants and their categorifications, have been introduced to aid with characterizing and classifying knots and their topological properties. Relations between knot invariants and their relative strengths at distinguishing knots are still mostly elusive. We use Principal Component Analysis (PCA), Ball Mapper, and machine learning to examine the structure of data consisting of various polynomial knot invariants and the relations between them. Although of different origins, these methods confirm and illuminate similar substructures in knot data. These approaches also enable comparison between numerical invariants of knots such as the signature and s-invariant via their distribution within the Alexander and Jones polynomial data. Although this work focuses on knot theory the ideas presented can be applied to other areas of pure mathematics and possibly in data science. The hybrid approach introduced here can be useful for infinite data sets where representative sampling is impossible or impractical. November 9, 2020: Alex Nabutovsky (Toronto) We will discuss the isoperimetric inequality for Hausdorff content and compact metric spaces in (possibly infinite-dimensional) Banach spaces. We will also discuss some of its implications for systolic geometry, in particular, systolic inequalities of a new type that are true for much wider classes of non-simply connected Riemannian manifolds than Gromov’s classical systolic inequality. Joint work with Y. Liokumovich, B. Lishak, and R. Rotman. November 23, 2020: Dima Burago (Penn State) This is not quite a research talk. This is a collection of problems (in random order). Some of them arose from my research (often with collaborators), some are known or folklore known, and there is some progress in our work. The problems are followed by comments which often contain announcements of recent results of mine (with co-authors) and brief discussions. At many places, I may be rather vague and also omit known definitions and discussions of known results. December 7, 2020: Robin Elliott (MIT) How efficiently can we represent a large integer multiple kα of a given non-torsion element α of a homotopy group of a Riemannian manifold? Here efficiency is measured by the Lipschitz constant L of a representing map, and the question is quantitatively answered by bounding the asymptotics of the minimal L needed to represent kα. In this talk I will talk about related functions defined in terms of the (co)homology of the loop space of the Riemannian manifold. I will discuss results for producing general upper bounds and applications of these, as well as specific constructions for lower bounds. January 11, 2021: Sahana Vasudevan (MIT) Triangulated surfaces are compact hyperbolic Riemann surfaces that admit a conformal triangulation by equilateral triangles. They arise naturally in number theory as Riemann surfaces defined over number fields, in probability theory as conjecturally related to Liouville quantum gravity, and in metric geometry as a model to understand arbitrary hyperbolic surfaces. Brooks and Makover started the study of the geometry of random large genus triangulated surfaces. Mirzakhani later proved analogous results for random hyperbolic surfaces. These results, along with many others, suggest that the geometry of triangulated surfaces mirrors the geometry of arbitrary hyperbolic surfaces especially in the case of large genus asymptotics. In this talk, I will describe an approach to show that triangulated surfaces are asymptotically well-distributed in moduli space. January 25, 2021: Fedya Manin (UCSB) I will explain the following theorem. Let X be a finite complex ($S^m$is a good example to keep in mind). Then every nullhomotopic, L-Lipschitz map$X \to S^n$has a$C(X,n) \cdot (L+1)$-Lipschitz nullhomotopy. The proof is spread over several papers, and the full story has never been told in one place. Joint and separate work variously with Chambers, Dotterrer, Weinberger, Berdnikov, and Guth. February 8, 2021: Roman Sauer (KIT) We prove the macroscopic cousins of three conjectures: 1) a conjectural bound of the simplicial volume of a Riemannian manifold in the presence of a lower scalar curvature bound, 2) the conjecture that rationally essential manifolds do not admit metrics of positive scalar curvature, 3) a conjectural bound of$\ell^2$-Betti numbers of aspherical Riemannian manifolds in the presence of a lower scalar curvature bound. The macroscopic cousin is the statement one obtains by replacing a lower scalar curvature bound by an upper bound on the volumes of 1-balls in the universal cover. Group actions on Cantor spaces surprisingly play an important role in the proof. The talk is based on joint work with Sabine Braun. February 22, 2021: Leonid Polterovich (Tel Aviv) We argue that existence of symplectically rigid fibers of integrable systems can be put on an equal footing with big fiber theorems from other areas of mathematics such as the Centerpoint theorem from ‎combinatorics and the Gromov maximal fiber theorem from topology. Our approach involves a symplectic counterpart of ideal-valued measures, and a new cohomology theory by Umut Varolgüneş. Symplectic preliminaries will be explained. This is work in progress with Adi Dickstein, Yaniv Ganor, and Frol Zapolsky. March 8, 2021: Matthew Kahle (Ohio State) Various models of random simplicial complex have been studied extensively over the past 15 years or so. We will discuss two models for random cubical complex, and what we know so far about their expected topological behavior: March 22, 2021: Hannah Alpert (UBC) Gromov conjectured in 1983 that if an n-manifold has large positive scalar curvature at every point, then it can be mapped to an$(n-2)$-complex with every fiber of the map having small diameter, and he sketched a proof for n = 3. We consider an analogous conjecture, where the scalar curvature hypothesis is replaced by supposing that every ball of radius 10 has small volume and that every loop in a ball of radius 1 is null-homotopic in the concentric ball of radius 2. We prove some version of this conjecture for n = 3. Joint work in progress with Alexey Balitskiy and Larry Guth. April 5, 2021: Yuanan Diao (UNC Charlotte) The ropelength R(K) of a knot K is the minimum length of a unit thickness rope needed to tie the knot. If K is alternating, it is conjectured that$R(K) \geq a\operatorname{Cr}(K)$for some constant$a > 0$, where Cr(K) is the minimum crossing number of K. In this talk I will first give a brief introduction to the ropelength problem. I will then show that there exists a constant$a_0 > 0$such that$R(K) \geq a_0b(K)$for any knot K, where b(K) is the braid index of K. It follows that if$b(K) \geq a_1\operatorname{Cr}(K)$for some constant$a_1 > 0$, then $$R(K) ≥ a_0a_1\operatorname{Cr}(K) = a\operatorname{Cr}(K).$$ However if b(K) is small compared to Cr(K) (in fact there are alternating knots with arbitrarily large crossing numbers but fixed braid indices), then this result cannot be applied directly. I will show that this result can in fact be applied in an indirect way to prove that the conjecture holds for a large class of alternating knots, regardless what their braid indices are. April 19, 2021: Madhur Tulsiani (TTIC) I will explain a recent construction of explicit instances of optimization problems, which are hard for the family of optimization algorithms captured by so called “sum-of-squares” (SoS) hierarchy of semidefinite programs. The SoS hierarchy is a powerful family of algorithms, which captures many known optimization and approximation algorithms. Several constructions of random families of instances have been proved to be hard for these algorithms, in the literature on optimization and proof complexity (since the duals of these optimization algorithms can be viewed as proof systems). I will describe a recent construction, based on the Ramanujan complexes of Samuels, Lubotzky and Vishne, which yields the first explicit family instances, where the optimization problem is hard even to solve approximately (using SoS). Joint work with Irit Dinur, Yuval Filmus, and Prahladh Harsha. May 3, 2021: Antonio Lerario (SISSA) In this talk I will investigate the structure of the "moduli space" W(G, d) of a geometric graph G, i.e. the set of all possible geometric realizations in$\mathbb{R}^d$of a given graph G on n vertices. This moduli space is Spanier–Whitehead dual to a real algebraic discriminant. For example, in the case of geometric realizations of G on the real line, the moduli space W(G, 1) is a component of the complement of a hyperplane arrangement in$\mathbb{R}^n$. (Another example: when G is the empty graph on n vertices, W(G, d) is homotopy equivalent to the configuration space of n points in$\mathbb{R}^d\$.) Numerous questions about graph enumeration can be formulated in terms of the topology of this moduli space.

I will explain how to associate to a graph G a new graph invariant which encodes the asymptotic structure of the moduli space as d goes to infinity, for fixed G. Surprisingly, the sum of the Betti numbers of W(G, d) stabilizes as d goes to infinity, and gives the claimed graph invariant B(G), even though the cohomology of W(G, d) "shifts" its dimension. We call the invariant B(G) the "Floer number" of the graph G, as its construction is reminiscent of Floer theory from symplectic geometry.

Joint work with M. Belotti and A. Newman.

May 17, 2021:
Boris Apanasov (Oklahoma)

We present a new effect in the theory of deformations of hyperbolic manifolds/orbifolds or their uniform hyperbolic lattices (i.e. in the Teichmüller spaces of conformally flat structures on closed hyperbolic 3-manifolds). We show that such varieties may have connected components whose dimensions differ by arbitrary large numbers. This is based on our "Siamese twins construction" of non-faithful discrete representations of hyperbolic lattices related to non-trivial "symmetric hyperbolic 4-cobordisms" and the Gromov–Piatetski-Shapiro interbreeding construction. There are several applications of this result, from new non-trivial hyperbolic homology 4-cobordisms and wild 2-knots in the 4-sphere, to bounded quasiregular locally homeomorphic mappings, especially to their asymptotics in the unit 3-ball solving well-known conjectures in geometric function theory.