MPS Conference on Singularities: Geometric, Topological, and Analytic Aspects

Much of mathematics takes place in smooth settings, but the study of smooth objects often leads us to the study of singular ones. This workshop will bring together experts from many of the different areas of mathematics where singularities occur (topology, algebraic geometry, analysis, etc.), with the hope that ideas developed in one area will shed new light elsewhere. The themes of the workshop will include intersection homology and L_2 cohomology, stratified spaces, resolution and deformation of singularities, and analysis on nonsmooth spaces.

Julius Shaneson
University of Pennsylvania

Singularities, Old and New  

Singularities (i.e., points at which a space is not locally homogeneous or where a map is not a locally smoothable immersion or a locally trivial projection) are central to lots of mathematics, such as topology, algebra and mathematical physics. Singularities tend to force themselves into the picture no matter how much one may try to avoid them. Some of the more classical phenomena and applications will be discussed, including some of the fundamental results of Cappell and others, and then some recent developments related to physics and the geometry of threefolds.

Steve Ferry
Rutgers University

Some New Manifold-like Spaces

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In his 1994 ICM talk, Shmuel Weinberger, inspired by work of Edwards, Quinn, Chapman-F., Cannon, Bryant-F.-Mio and himself, conjectured the existence of a new collection of spaces with many of the properties of topological manifolds. The authors have constructed spaces in dimensions n≥6 satisfying many parts of Weinberger’s conjecture. Our spaces are finite dimensional and locally contractible. They have the local and global separation properties of topological manifolds, satisfying Alexander duality both locally and globally. More technically, they are integral ENR homology manifolds. They are homogeneous, meaning that, for every x and y in a component of one of these spaces, there is a homeomorphism carrying x to y. These spaces are detected by a 0th Pontryagin class, or Quinn index of the form 8k+1, spaces with 0th Pontryagin number equal to 1 being classical topological manifolds. There are spaces of every possible Quinn index in the homotopy type of any closed, simply connected manifold. The situation for nonsimply connected manifolds is more complicated. In particular, none of our new spaces can have the homotopy type of a torus. In high dimensions, the h- and s-cobordism theorems hold for these topologically exotic manifolds. These exotic manifolds complete Siebenmann periodicity in the topological category and remove a discrepancy between the geometric and algebraic classifications of topological manifolds. This work is joint with J. Bryant.

Shoji Yokura
Kagoshima University

Motivic Hirzebruch Classes of Singular Varieties and Some Homological Congruence Formulae

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In this talk, Yokura will give a quick review of the so-called ‘motivic Hirzebruch’ class of singular varieties and discuss some homological congruence formulae for this class as a generalization or an extension of preceding results of congruence formulae for signature and Hirzebruch chi-y genera, and some related topics.

Jörg Schürmann
Universität Münster

(Equivariant) Characteristic Classes of Singular Toric Varieties

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In this survey talk, we discuss four singular toric varieties different (equivariant) characteristic classes like Chern, Todd, L and Hirzebruch classes, with applications to (weighted) lattice points counting and Euler-MacLaurin type formulae for lattice polytopes.

Laurentiu Maxim
University of Wisconsin, Madison

Perverse Sheaves on Semi-abelian Varieties: Structure and Applications

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Maxim will present a new characterization of perverse sheaves on complex semi-abelian varieties in terms of their cohomology jump loci, generalizing results of Gabber-Loeser and Schnell. He will also discuss propagation properties and codimension lower bounds for the cohomology jump loci of perverse sheaves. As concrete applications, Maxim will mention: (a) generic vanishing for perverse sheaves on semi-abelian varieties; (b) homological duality properties of complex algebraic manifolds, via abelian duality; and (c) new topological characterizations of (semi-)abelian varieties. This is joint work with Y. Liu and B. Wang.

Regina Rotman 
University of Toronto

Periodic Geodesics and Geodesic Nets on Riemannian Manifolds

Geodesic nets are singular objects that are homological equivalents of periodic geodesics. They turn out to be useful for problems about closed geodesics. For example, I have recently proved the existence of ‘wide’ geodesic loops (with an angle θ that is arbitrarily close to π) on a closed Riemannian manifold Mⁿof volume v, where the length of the loop can be majorized in terms of v, n, and the difference between πand θ. Geodesic nets can also be used to prove the existence of periodic geodesics on complete, non-compact Riemannian manifolds satisfying an additional, easy-to-state geometric assumption.

Leslie Saper
Duke University

Weights and Singularities

Various notions of weight play a role in the study of singularities. They arise from different fields: characteristic p algebraic geometry, representation theory, metrical decay rates, mixed Hodge theory. We will discuss the relations between these notions and their application to the study of singularities. If there is time, we will discuss ongoing work on a new theory coming from CR geometry.

Paolo Aluffi
Florid State University

Chern Classes of Embeddable Schemes

We prove a formula computing the Chern-Schwartz-MacPherson (CSM) class of an arbitrary sub-scheme of a nonsingular variety in terms of the Segre class of an associated scheme. This formula generalizes an old result expressing the CSM class of a hypersurface in terms of the Segre class of its singularity sub-scheme. For local complete intersections, the result yields a new expression for the Milnor class.

Jon Woolf
University of Liverpool

Applications of Balmer-Witt Groups to Stratified Spaces

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Balmer-Witt groups of constructible derived categories are a natural home for signatures and related invariants of stratified and singular spaces. Woolf will give a survey of the theory and some of its applications.

John Francis
Northwestern University

Moduli Spaces of Stratifications and Factorization Homology

The alpha form of factorization homology after Beilinson and Drinfeld is based on the topology of Ran spaces. Here Ran(X) is the moduli space of finite subsets of X(introduced by Borsuk-Ulam), topologized so that points can collide. This alpha factorization homology takes as input a manifold or variety X, together with a suitable algebraic coefficient system A, and it outputs the cosheaf homology of Ran(X) with coefficients defined by A. Factorization homology simultaneously generalizes singular homology, Hochschild homology and conformal blocks or observables in conformal field theory. He will discuss applications of this alpha form of factorization homology in the study of mapping spaces in algebraic topology, bundles on algebraic curves and perturbative quantum field theory. Francis will then describe a beta form of factorization homology, where one replaces Ran(X) with a moduli space of stratifications of X, designed to overcome certain strict limitations of the alpha form. A main result, joint with Ayala and Rozenblyum, is that an (∞,n)-category defines a cosheaf on the moduli space of vari-framed stratifications. A theorem-in-progress is that an (∞,n)-category with adjoints defines a cosheaf on the moduli space of solidly n-framed stratifications. An immediate consequence is a proof of the cobordism hypothesis (after Baez-Dolan, Costello, Hopkins-Lurie, and Lurie) exactly in the manner of Pontryagin-Thom theory. This is joint work with David Ayala.

\(H^*_{\overline{p}}(X;\mathbb{Z})\)

Martintxo Saralegi-Aranguren
Université d’Artois

Two Intersection Cohomologies  

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Intersection cohomology has been introduced for pseudomanifolds and its behavior is mainly derived from sheaf theory. In this talk, we show how cochain complexes can also be used in an efficient way.

We present two cohomologies. The first one comes from the linear dual of the intersection chain complex; we denote it \(H^*_{\overline{p}}(X;\mathbb{Z})\). The second one arises from a simplicial blow-up; we denote it \(\mathscr H^*_{\ov{p}}(X;\mathbb{Z})\).

A simple calculation on the cone of a manifold shows that they are, in general, different. Also, the first one verifies a universal coefficient formula by construction, while the second one is related to intersection homology by a cap product, \(\mathscr H^*_{\ov{p}}(X;\mathbb{Z})\cong H_{n-*}^{\ov{p}}(X;\mathbb{Z})\), exactly as for manifolds. With these two properties, the two cohomologies intertwine through a nonsingular pairing \(\mathscr H_{*}^{\ov{p}}(X;\mathbb{Z})\cong {\rm Hom\,}(H^*_{\ov{p}}(X;\mathbb{Z}),I^*_{\mathbb{Z}})\).

By using this kind of Lefschetz duality, we specify existence and defects of duality in intersection homology. If part of these results is already known, the originality comes from the use of cup and cap products to express them, as in the case of manifolds.

Jim Davis
Indiana University

A Remark on Nielsen Realization

Jakob Nielsen asked if a finite subgroup of outer automorphisms of the fundamental group of a compact surface can be realized by a group action. This was proved by Steve Kerckho in 1980. It is an open question whether this same statement is true for any compact, negatively curved manifold. Surgery theory, a geometric construction, Cappell UNil groups and the Farrell-Jones conjecture gives positive results in a special case. This is the first positive process in Nielsen realization in twenty years. We will also discuss connections with the existence part of the equivariant Borel conjecture.

Min Yan
Hong Kong University of Science and Technology

Converse of Smith Theory

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Suppose \(G\) is a finite group, and \(f\colon F\to Y\) is a map between finite \(CW\)-complexes. Is it possible to extend \(F\) to a finite \(G\)-CW complex \(X\) satisfying \(X^G=F\), and extend \(f\) to a \(G\)-map \(g\colon X \to Y\) (\(G\) acts trivially on \(Y\)), such that \(g\) is a homotopy equivalence after forgetting the \(G\)-action?
In case \(Y\) is a single point, the problem becomes whether a given finite \(CW\)-complex \(F\) is the fixed point of a \(G\)-action on a finite contractible \(CW\)-complex. In 1942, P.A. Smith showed that the fixed point of a \(p\)-group action on a finite \({\mathbb Z}_p\)-acyclic complex is still \({\mathbb Z}_p\)-acyclic. In 1971, Lowell Jones proved a converse, that any \({\mathbb Z}_n\)-acyclic finite \(CW\)-complex is the fixed point of a semi-free \({\mathbb Z}_n\)-action on a finite contractible \(CW\)-complex. In 1975, Robert Oliver proved that, for a given finite group \(G\) of not prime power order, whether a finite \(CW\)-complex \(F\) is the fixed point of a general \(G\)-action on a finite contractible \(CW\)-complex is determined by the Euler characteristic of \(F\).

For our general problem, we find that Oliver’s theory on fixed point of general group actions largely remain true. Moreover, Jones’s theory on fixed point of semi-free group actions can also be extended, with an obstruction in \(\tilde{K}_0\). We also calculate some examples of semi-free actions.

This is joint work with Sylvain Cappell of New York University and Shmuel Weinberger of University of Chicago.

Ed Bierstone
University of Toronto

Geometry of Quasianalytic Classes

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Quasianalytic classes are classes of infinitely differentiable functions that enjoy the analytic continuation property of holomorphic functions. They are the objects of classical studies in real analysis (e.g., the Denjoy-Carleman theorem), and the last 20 years have seen the development of remarkable relationships with algebraic geometry (resolution of singularities) and model theory (o-minimal structures). Bierstone will talk about these developments, recent results on ‘quasianalytic continuation’ and the solutions of quasianalytic equations and open problems.

Tam Nguyen-Phan
Max Planck Institute for Mathematics

A Geometric Analogue of the Rational Tits Building in Non-positive Curvature

Locally symmetric manifolds of noncompact type form an interesting class of noncompact, non-positively curved manifolds. For example, they have a compactification, due to Borel-Serre, that is manifolds with corners. Associated to a locally symmetric space M is the so-called ‘rational Tits building,’ which can be thought of abstractly or as a subset of the visual boundary of the universal cover of M. This arithmetically constructed object controls the topology of the end of M since it happens to be the same as the nerve of the boundary strata of the universal cover of the Borel-Serre compactification of M. The rational Tits building of a locally symmetric space is always homotopically a wedge of spheres of dimension (q-1), where q is the Q-rank of the locally symmetric space. In general, q is less than or equal to half the dimension of the locally symmetric space. We show that this is not an arithmetic coincidence but a consequence of nonpositive curvature alone. Inspired by the Borel-Serre compactification, we build an analogue of the rational Tits building, viewed as a subset of the visual boundary at infinity, for general noncompact, finite volume, complete, bounded nonpositively curved n-manifolds with no arbitrarily small geodesic loops (the latter condition is so that M is tame but also holds for all locally symmetric spaces). We use this to show that any polyhedron, in the thin part (i.e., the end) of M that lifts to the universal cover can be homotoped within the thin part of Mto one with dimension less than or equal to (n/2 – 1). Loosely speaking, this says that any topological feature that survives from being pushed to infinity must be in dimension less than n/2. Nguyen-Phan will describe how this is done. This is joint work with Grigori Avramidi.

This talk is about non-positively curved geometry. No knowledge of Tits buildings is required (or will be given).

Pierre Albin
University of Illinois at Urbana-Champaign

Mapping Stratified Surgery to Analysis

In an influential series of papers, Higson and Roe related the K-theoretic higher index of the signature operator on an oriented closed manifold with the surgery long exact sequence of that manifold. Following up on work with Eric Leichtnam, Rafe Mazzeo and Paolo Piazza, where we studied the higher signatures of stratified spaces, Albin will report on joint work with Piazza relating these higher signatures with the Browder-Quinn surgery long exact sequence of a stratified space.

Fedor Manin

Ohio State University

Lipschitz Homotopy and Shadows of Homomorphisms  

In the 1970s and again in the 1990s, Gromov gave a number of results and conjectures about the Morse landscape of mapping spaces between compact manifolds and finite complexes with respect to the Lipschitz functional. These were motivated by the observation that Sullivan’s algebraicization of such maps via rational homotopy theory, stated in the right way, preserves some geometric as well as topological information. New results confirm this intuition by showing that this process is almost reversible: the images of genuine maps between spaces are coarsely dense in a much larger space of DGA homomorphisms. This allows us to make great progress toward resolving Gromov’s questions.

An example (near-sharp) theorem: if X and Y are finite complexes with Y simply connected, then every nullhomotopic L-Lipschitz map X → Y has a C(X,Y)L²-Lipschitz null-homotopy.

Jeff Cheeger
New York University 

Quantitative Differentiation and Generalized Differentiation

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Assaf Naor
Princeton University

  Coarse (non)Universality of Alexandrov Spaces

We will show that there exists a metric space that does not admit a coarse embedding into any Alexandrov space of nonpositive curvature, thus answering a question of Gromov (1993). In contrast, any metric space embeds coarsely into an Alexandrov space of nonnegative curvature. Based on joint works with Andoni and Neiman, and Eskenazis and Mendel.

Markus Banagl 
Universität Heidelberg

Sylvain Cappell — Opera Selecta

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We will discuss selected works of Sylvain Cappell, representing different aspects

of his wide-ranging mathematical interests. In some instances, we hope to present not only the results, but also to sketch the beautiful methods used.

Carmen Rovi
Indiana University

The Reinterpretation of Davis-Lueck Equivariant Homology in Terms of L-theory

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The K-theory Kn(ℤG) and quadratic L-theory Ln(ℤG) functors provide information about the algebraic and geometric topology of a smooth manifold X with fundamental group G=π1(X,x0). Both K– and L-theory are difficult to compute in general and assembly maps give important information about these functors. Ranicki developed a combinatorial version of assembly by describing L-theory over additive bordism categories indexed over simplicial complexes. The chain duality defined for such categories also has an interpretation as a Verdier duality.

In this talk, Rovi will present current work with Jim Davis where they define an equivariant version of Ranicki’s local/global assembly map and identify their local/global assembly map with the map on equivariant homology defined by Davis and Lueck. Furthermore, Rovi will discuss some applications. In particular, it is a folklore statement that the L-theoretic Farrell-Jones conjecture holds for G=H⋊αℤassuming that it holds for the group H. Nonetheless, a satisfactory proof of this often-used result has never been given. Rovi will give insight into how they use their investigation of the equivariant assembly maps to prove this result.

Diarmuid Crowley
University of Melbourne

On the Topological Period-Index Conjecture for Singular Varieties

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The (Algebraic) Period-Index Conjecture (APIC) of Colliot-Thélène provides a formula relating the order of an element \(\alpha\) of the Brauer group of an algebraic variety \(V\), the minimal rank of a bundle representing \(\alpha\) and the dimension of \(V\).

The Topological Period-Index Conjecture (TPIC) is a topological statement analogous to the APIC. It which was identified by Antieau and Williams who proved that the TPIC fails in general and also that the APIC implies the TPIC for smooth varieties \(V\).

In this talk, Crowley will report on joint work with Mark Grant, where we verify the TPIC for (real) 6-dimensional spin^c-manifolds. Crowley will discuss generalizations of this proof for the topological spaces underlying singular complex varieties of (real) dimension 6.

Kent Orr
University of Indiana

New Perspectives on an Old Problem of Milnor

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In the mid-1950s, Milnor introduced his link invariants, a vast and profound extension of the classical linking number. His examples, results and a seminal list of problems have driven decades of research. One of Milnor’s original questions remains unresolved. How can one extract a version of Milnor’s invariants from the transfinite lower central series? We consider this problem anew and present a solution in a broader setting. We include detailed computations for a key and illustrative example, as well as a brief outline of past progress on Milnor’s invariants. This is joint work with Jae Choon Cha.