MPS Conference on Quantitative Topology and Related Areas

The MPS Conference on Quantitative Topology and Related Areas shared recent developments and highlighted interesting questions in quantitative topology. Quantitative topology is broad, connecting topics from “pure” mathematics (differential geometry, cohomology, etc.) to more “applied” concerns (asymptotic estimates, algorithms and high-dimensional data). The conference reflected this by offering talks in many different areas, such as systolic geometry, metric geometry, persistent homology and geometric analysis.
 

Systolic and isoperimetric geometry

The systole of a space measures the length of the shortest non-contractible closed curve in the space. Systoles and related invariants are fundamental to quantitative topology, for instance, in the study of min-max methods used to construct minimal surfaces or in studying Urysohn width, which measures how well a space can be approximated by some lower dimensional.

In this area, Hannah Alpert gave a notable talk on counterexamples in the study of Urysohn width, giving examples of spaces which cannot be approximated by graphs, but which have covers that can be approximated. Alex Lubotzky’s talk addressed a different sort of width, the phenomenon of topological expanders, which are complexes such that any continuous map to R^n must have large fibers. Such complexes can be used to produce error-correcting codes, and Lubotzky related these complexes to the notion of sofic groups. Alex Nabutovsky described recent work on generalizations of the boxing inequality, which describes the relationship between the size of a subset of a Banach space and the size of the smallest contractible subset containing that set. Finally, Sahana Vasudevan’s talk introduced new questions on embeddings of tori into Euclidean space. She presented a remarkable result that a highly twisted embedded torus must be “thin” — it must contain nontrivial closed curves that stay in a small ball.
 

Metric geometry and embeddings

Several talks dealt with the question of how well one space embeds into another, or what sorts of maps can be constructed from one space to another. Assaf Naor’s talk dealt with the question of the Lipschitz extension modulus of a space; this measures the difficulty of taking a map defined on a subset of the space and extending it to the whole space. Naor described a dual version of this problem based on the difficulty of partitioning a space and presented new results on this modulus. Continuing the theme of Lipschitz maps, Guth discussed this question: given two closed n-manifolds M and N, what is the largest possible degree of an L-Lipschitz map f : M → N in terms of L? Fourier analysis comes into play, along with the “shadowing principle” that relates these questions to finding differential forms satisfying certain conditions.
 

Mathematical applications of persistent homology

Omer Bobrowski and Leonid Polterovich gave talks about applications of persistent homology, a tool most frequently used in topological data analysis to study the topology of an underlying space from which samples are drawn. Bobrowski’s talk concerned the topology of sampling noise: given a random sample of points from a topologically trivial object, how does the topology of the sample depend on various parameters? Results of this type give a theoretical underpinning to the methods of applied topology. Polterovich discussed counting problems relating to functions with special properties, such as bounding the number of zeros of a holomorphic function or the number of nodal domains of linear combinations of Laplace eigenfunctions. The theme of the talk was that these counts do not actually obey the expected bounds; however, applying the framework of persistent homology to ignore small oscillations allows one to generalize classical counting theorems to settings where the “obvious” generalizations do not hold.
 

Geometric analysis

Guoliang Yu and Mikolaj Fraczyk gave talks on geometric analysis. Yu surveyed progress on manifolds with positive scalar curvature metrics, and in particular on the question of which manifolds do (or do not) admit positive scalar curvature metrics. He discussed a result showing that closed aspherical manifolds whose fundamental groups embed discretely into diffeomorphism groups of compact manifolds do not admit such metrics, generalizing to a non-linear setting results that had been established earlier for subgroups of linear groups. He also discussed results on existence and rigidity of positive scalar curvature metrics with prescribed boundary conditions on manifolds with corners and a proof of Gromov’s dihedral extremality conjecture.

Fraczyk discussed submanifolds of octonionic hyperbolic manifolds with codimension ≤ 3, relating the growth rate of such submanifolds and the spectral gap of the Laplace operator, and thereby showing that minimal submanifolds of codimension ≤ 3 must fill a positive proportion of the ambient space. A significant application is that closed octonionic hyperbolic manifolds form a family of higher topological expanders.
 

Other areas

Sylvain Cappell discussed work on Euler-Maclaurin formulae which give methods for counting the number of lattice points in a polytope via algebraic topological invariants of toric varieties associated to the polytopes. Sergey Avvakumov discussed the theorem that a convex body in the plane can be equipartitioned into m≥2 convex parts of equal areas and perimeters; this was previously known only for prime powers m=p^k. In the theorem, “area” can be replaced by a probability measure (i.e., something additive) and “perimeter” can be replaced by a continuous function of the convex body (e.g., something which isn’t additive).
 

 
The breadth of quantitative topology makes conferences like this essential for spreading new methods, ideas and questions, and we hope that it has sparked new collaborations and new research directions for the speakers and attendees alike.

Hannah Alpert
Auburn University

Unintuitive Properties of Urysohn 1-Width
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A metric space has small Urysohn 1-width if it admits a continuous map to a 1-dimensional complex where the preimage of each point has small diameter. An open problem is, if a space’s universal cover has small Urysohn 1-width, must the original space also have small Urysohn 1-width? Naively we would guess yes, but various strange examples suggest maybe not.

Joint work with Panos Papasoglu, Arka Banerjee, Alexey Balitskiy and Larry Guth.
 

Sergey Avvakumov
Weizmann Institute of Science

Convex Equipartition
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As a simple application of the intermediate value theorem, one can see that any convex body in the plane can be partitioned into m=2 convex pieces of equal area and equal perimeter. Using the equivariant obstruction method this result was generalized to m=3, then m=2^k, and finally to m=p^k for all prime p. The “area” and “perimeter” in the statement can be substituted by a probability measure and a continuous function, respectively. More functions can be equalized in higher dimensions.

Sergey Avvakumov will discuss a proof for arbitrary m, which uses new ideas allowing us to overcome the inherent limitations of the previously used methods.

Joint work with Arseniy Akopyan and Roman Karasev.
 

Omer Bobrowski
Queen Mary University of London

Homological Connectivity in Random Geometric Complexes & Topological Crackle
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Random geometric complexes are simplicial complexes whose vertex set is a random point process in a metric space. In this talk, Omer Bobrowski will focus on the topology of random geometric complexes, addressing two parallel lines of research.

The first is about “homological connectivity” — the conditions for the random complex to recover the homology of the underlying space. The second is about “topological crackle” — the topological structure of outliers generated by distributions with non-compact supports (e.g., Gaussian).

Bobrowski will review the main advances in these studies and propose how they might actually cross paths.
 

Sylvain Cappell
New York University

Comparisons of Discrete and Continuous Summations

Fundamental questions in both pure and applied mathematics involve comparisons of discrete summations and integrations. Sylvain Cappell will present new comparisons of these processes in the geometrical settings of convex lattice polytopes and develop some related identities.

This is a report of joint research of the speaker with Laurentiu Maxim, Julius Shaneson and Joerg Schurmann.
 

Mikolaj Fraczyk
Jagiellonian University

Minimal Submanifolds in Locally Symmetric Spaces

Mikolaj Fraczyk will talk about a curious dichotomy that holds for any compact octonionic hyperbolic manifold M. A submanifold S of M of codimension at most 3 either must approximately fill a positive proportion of M, or it can be continuously deformed to a lower dimensional subset. In particular, the are no “small” minimal submaniflods of M of codimensions 1,2,3. This result leads to new waist inequalities for hyperbolic octonionic manifolds, new families of bounded geometry higher topological expanders and new families of manifolds with power law Z2-systolic freedom.

Based on a joint work (in progress) with Ben Lowe.
 

Larry Guth
MIT

Fourier Analysis in Quantitative Homotopy Theory

Larry Guth will consider the following problem in quantitative homotopy theory. Given two closed Riemannian manifolds M, N (of same dimension), estimate the largest possible degree of an L-Lipschitz map f: M to N, focusing on the asymptotics as L goes to infinity.

A few years ago, Berdnikov and Manin proved some striking new results about this problem. For example, suppose that M and N are both connected sums of k copies of S^2 x S^2. If k is at most 3, they constructed L-Lipschitz maps of degree c L^4, which is optimal up to a constant factor. But if k is at least 4, then they proved that the degree of an L-Lipschitz map is o(L^4). In recent work, Berdnikov, Manin, and I quantified this further, showing that when k is at least 4, the degree is at most C L^4 (log L)^{-1/2}. The proof of this quantitative bound is based on Fourier analysis.

In this talk, Guth will discuss the connection between Fourier analysis and quantitative homotopy theory. How does Fourier analysis help in the proof mentioned above? Guth will also discuss some open problems of this flavor. By a theorem of Manin called the shadowing principle, problems of the type above can be reduced to analytic questions involving differential forms. These analytic questions are not understood in general, and Larry Guth will discuss some of the difficulties in trying to understand them.
 

Alex Lubotzky
Weizmann Institute

Topological Expanders, Locally Testable Codes and (Non-)Sofic Groups

In the talk, which is based on a paper below dedicated to Shmuel Weinberger, Alex Lubotzky will explain why the three, seemingly unrelated, topics in the title are related. Moreover, the known solutions to the first two problems suggest a pattern to solve the third. All notions will
be explained.
 

Assaf Naor
Princeton University

Quantitative Wasserstein Rounding

The main focus of this talk will be to describe recent work (joint with Braverman) on the Lipschitz extension problem that obtains solutions to various natural quantitative questions by thinking about its (known) dual formulation as a question about randomly rounding an ambient metric space to its subset while preserving certain natural guarantees that are measured in terms of transportation cost. Assaf Naor will start by discussing the classical formulation of these old questions as well as some background and earlier results, before passing to examples of how one could reason quantitatively using the dual perspective.
 

Alex Nabutovsky
University of Toronto

Boxing Inequality in Banach Spaces and Related Inequalities
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Let \(M^n\) be a manifold in a finite or infinite-dimensional Banach space \(B\), and \(m\leq n\) a positive number.

Then there exists a pseudomanifold \(W^{n+1}\) in \(B\) such that \(\partial W^{n+1}=M^n\) and the \(m\)-dimensional Hausdorff content \(HC_m(W^{n+1})\) of \(W^{n+1}\) does not exceed \(c(m)HC_m(M^n)\). Recall that \(HC_m(X)\) is defined as the infimum of \(\Sigma_i r_i^m\) over all coverings of \(X\) by metric balls in \(B\), where \(r_i\) denote the radii of these balls.

When \(M^n\) is a closed hypersurface in \(\mathbb{R}^{n+1}\), this result implies that for all \(m\in (0,n]\) \(HC_m(\Omega)\leq c(m)HC_m(M^n)\), where \(\Omega\) denotes the domain bounded by \(M^n\). (The case of \(m=n\) is the well-known boxing inequality first proven by W. Gustin.)

Alex Nabutovsky will discuss further generalizations of this result, and its connections with Urysohn width-volume inequalities first proven by L. Guth. Nabutovsky will present two new \(l^\infty\)-width – volume inequalities and discuss their implications for systolic geometry.

Joint work with Sergey Avvakumov.
 

Leonid Polterovich
Tel Aviv University

Courant, Bézout and Topological Persistence
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Leonid Polterovich discuss generalizations of two classical results, Courant’s nodal domain theorem and Bézout’s theorem, based on ideas of topological data analysis. Joint with Lev Buhovsky, Jordan Payette, Iosif Polterovich, Egor Shelukhin and Vukašin Stojisavljević.
 

Sahana Vasudevan
Institute for Advanced Study

Twisted Torus Embeddings

Consider an unknotted torus embedded in R^3 with the Euclidean metric. Under certain conditions depending only on the topology of the embedding and the induced metric on the torus, one necessarily finds non-nullhomotopic loops on the torus that have much smaller diameter in R^3 than in the induced metric. For example, a standard flat torus embedded Nash-isometrically in R^3 in a twisted way contains a non-nullhomotopic loop with small R^3-diameter. This result is related to questions about distortion of knots.
 

Guoliang Yu
Texas A&M University

An Index Theorem for Manifolds with Polyhedral Boundary and Scalar Curvature Rigidity

Guoliang Yu will explain how to develop a new index theorem for Dirac operators on manifolds with polyhedral boundary and use it to prove Gromov’s dihedral extremality conjecture on scalar curvature for polyhedra.

This is joint work with Jinmin Wang and Zhizhang Xie.