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

Markus Banagl, Ph.D.Universität Heidelberg

Greg Friedman, Ph.D.Texas Christian University

Shmuel Weinberger, Ph.D.University of Chicago

Robert Young, Ph.D.New York University
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.
Talks
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 Manifoldlike Spaces
In his 1994 ICM talk, Shmuel Weinberger, inspired by work of Edwards, Quinn, ChapmanF., Cannon, BryantF.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 scobordism 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
In this talk, Yokura will give a quick review of the socalled ‘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 chiy genera, and some related topics.
Jörg Schürmann
Universität Münster
(Equivariant) Characteristic Classes of Singular Toric Varieties
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 EulerMacLaurin type formulae for lattice polytopes.
Laurentiu Maxim
University of Wisconsin, Madison
Perverse Sheaves on Semiabelian Varieties: Structure and Applications
Maxim will present a new characterization of perverse sheaves on complex semiabelian varieties in terms of their cohomology jump loci, generalizing results of GabberLoeser 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 semiabelian 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 M^{n}of 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, noncompact Riemannian manifolds satisfying an additional, easytostate 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 ChernSchwartzMacPherson (CSM) class of an arbitrary subscheme 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 subscheme. For local complete intersections, the result yields a new expression for the Milnor class.
Jon Woolf
University of Liverpool
Applications of BalmerWitt Groups to Stratified Spaces
BalmerWitt 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 BorsukUlam), 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 variframed stratifications. A theoreminprogress is that an (∞,n)category with adjoints defines a cosheaf on the moduli space of solidly nframed stratifications. An immediate consequence is a proof of the cobordism hypothesis (after BaezDolan, Costello, HopkinsLurie, and Lurie) exactly in the manner of PontryaginThom theory. This is joint work with David Ayala.
\(H^*_{\overline{p}}(X;\mathbb{Z})\)
Martintxo SaralegiAranguren
Université d’Artois
Two Intersection Cohomologies
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 blowup; 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 FarrellJones 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
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 semifree \({\mathbb Z}_n\)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 semifree group actions can also be extended, with an obstruction in \(\tilde{K}_0\). We also calculate some examples of semifree 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
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 DenjoyCarleman theorem), and the last 20 years have seen the development of remarkable relationships with algebraic geometry (resolution of singularities) and model theory (ominimal structures). Bierstone will talk about these developments, recent results on ‘quasianalytic continuation’ and the solutions of quasianalytic equations and open problems.
Tam NguyenPhan
Max Planck Institute for Mathematics
A Geometric Analogue of the Rational Tits Building in Nonpositive Curvature
Locally symmetric manifolds of noncompact type form an interesting class of noncompact, nonpositively curved manifolds. For example, they have a compactification, due to BorelSerre, that is manifolds with corners. Associated to a locally symmetric space M is the socalled ‘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 BorelSerre compactification of M. The rational Tits building of a locally symmetric space is always homotopically a wedge of spheres of dimension (q1), where q is the Qrank 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 BorelSerre 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 nmanifolds 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. NguyenPhan will describe how this is done. This is joint work with Grigori Avramidi.
This talk is about nonpositively curved geometry. No knowledge of Tits buildings is required (or will be given).
Pierre Albin
University of Illinois at UrbanaChampaign
Mapping Stratified Surgery to Analysis
In an influential series of papers, Higson and Roe related the Ktheoretic 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 BrowderQuinn 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 (nearsharp) theorem: if X and Y are finite complexes with Y simply connected, then every nullhomotopic LLipschitz map X → Y has a C(X,Y)L^{2}Lipschitz nullhomotopy.
Jeff Cheeger
New York University
Quantitative Differentiation and Generalized Differentiation
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
We will discuss selected works of Sylvain Cappell, representing different aspects
of his wideranging 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 DavisLueck Equivariant Homology in Terms of Ltheory
The Ktheory Kn(ℤG) and quadratic Ltheory 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 Ltheory are difficult to compute in general and assembly maps give important information about these functors. Ranicki developed a combinatorial version of assembly by describing Ltheory 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 Ltheoretic FarrellJones conjecture holds for G=H⋊αℤassuming that it holds for the group H. Nonetheless, a satisfactory proof of this oftenused 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 PeriodIndex Conjecture for Singular Varieties
The (Algebraic) PeriodIndex Conjecture (APIC) of ColliotThélène 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 PeriodIndex 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) 6dimensional 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
In the mid1950s, 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.

Abstracts
\( \newcommand{\ov}[1]{\overline #1} \)
Pierre Albin
University of Illinois at UrbanaChampaignMapping Stratified Surgery to Analysis
In an influential series of papers, Higson and Roe related the Ktheoretic 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 BrowderQuinn surgery long exact sequence of a stratified space.
Paolo Aluffi
Florid State UniversityChern Classes of Embeddable Schemes
We prove a formula computing the ChernSchwartzMacPherson (CSM) class of an arbitrary subscheme 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 subscheme. For local complete intersections, the result yields a new expression for the Milnor class.
Markus Banagl
Universität HeidelbergSylvain Cappell — Opera Selecta
We will discuss selected works of Sylvain Cappell, representing different aspects
of his wideranging mathematical interests. In some instances, we hope to present not only the results, but also to sketch the beautiful methods used.
Ed Bierstone
University of TorontoGeometry of Quasianalytic Classes
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 DenjoyCarleman theorem), and the last 20 years have seen the development of remarkable relationships with algebraic geometry (resolution of singularities) and model theory (ominimal structures). Bierstone will talk about these developments, recent results on ‘quasianalytic continuation’ and the solutions of quasianalytic equations and open problems.
Jeff Cheeger
New York UniversityTitle TBA
Diarmuid Crowley
University of MelbourneOn the Topological PeriodIndex Conjecture for Singular Varieties
The (Algebraic) PeriodIndex Conjecture (APIC) of ColliotThélène 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 PeriodIndex 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) 6dimensional spin^{c}manifolds. Crowley will discuss generalizations of this proof for the topological spaces underlying singular complex varieties of (real) dimension 6.
Jim Davis
Indiana UniversityA 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 FarrellJones 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.
Steve Ferry
Rutgers UniversitySome New Manifoldlike Spaces
In his 1994 ICM talk, Shmuel Weinberger, inspired by work of Edwards, Quinn, ChapmanF., Cannon, BryantF.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 \(n \ge 6\) 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 scobordism 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.
John Francis
Northwestern UniversityModuli 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 BorsukUlam), 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 variframed stratifications. A theoreminprogress is that an (∞,n)category with adjoints defines a cosheaf on the moduli space of solidly nframed stratifications. An immediate consequence is a proof of the cobordism hypothesis (after BaezDolan, Costello, HopkinsLurie, and Lurie) exactly in the manner of PontryaginThom theory. This is joint work with David Ayala.
Fedor Manin
Ohio State UniversityLipschitz 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 (nearsharp) theorem: if X and Y are finite complexes with Y simply connected, then every nullhomotopic LLipschitz map X → Y has a C(X,Y)L^{2}Lipschitz nullhomotopy.
Laurentiu Maxim
University of Wisconsin, MadisonPerverse Sheaves on Semiabelian Varieties: Structure and Applications
Maxim will present a new characterization of perverse sheaves on complex semiabelian varieties in terms of their cohomology jump loci, generalizing results of GabberLoeser 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 semiabelian 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.
Assaf Naor
Princeton UniversityCoarse (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 Alexandorv space of nonnegative curvature. Based on joint works with Andoni and Neiman, and Eskenazis and Mendel.
Tam NguyenPhan
Max Planck Institute for MathematicsA Geometric Analogue of the Rational Tits Building in Nonpositive Curvature
Locally symmetric manifolds of noncompact type form an interesting class of noncompact, nonpositively curved manifolds. For example, they have a compactification, due to BorelSerre, that is manifolds with corners. Associated to a locally symmetric space M is the socalled ‘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 BorelSerre compactification of M. The rational Tits building of a locally symmetric space is always homotopically a wedge of spheres of dimension (q1), where q is the Qrank 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 BorelSerre 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 nmanifolds 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 M to 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. NguyenPhan will describe how this is done. This is joint work with Grigori Avramidi.
This talk is about nonpositively curved geometry. No knowledge of Tits buildings is required (or will be given).
Kent Orr
University of IndianaNew Perspectives on an Old Problem of Milnor
In the mid1950s, 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.
Regina Rotman
University of TorontoPeriodic 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 M^{n}of volume v, where the length of the loop can be majorized in terms of v, n, and the difference between π and θ. Geodesic nets can also be used to prove the existence of periodic geodesics on complete, noncompact Riemannian manifolds satisfying an additional, easytostate geometric assumption.
Carmen Rovi
Indiana UniversityThe Reinterpretation of DavisLueck Equivariant Homology in Terms of Ltheory
The Ktheory \(K_n(\mathbb{Z} G)\) and quadratic Ltheory \(L_n(\mathbb{Z} G)\) functors provide information about the algebraic and geometric topology of a smooth manifold X with fundamental group \(G= \pi_1(X, x_0).\) Both K– and Ltheory are difficult to compute in general and assembly maps give important information about these functors. Ranicki developed a combinatorial version of assembly by describing Ltheory 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 Ltheoretic FarrellJones conjecture holds for \(G= H ⋊_{\alpha} \mathbb{Z}\) assuming that it holds for the group H. Nonetheless, a satisfactory proof of this oftenused result has never been given. Rovi will give insight into how they use their investigation of the equivariant assembly maps to prove this result.
Leslie Saper
Duke UniversityWeights 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.
Martintxo SaralegiAranguren
Université d’ArtoisTwo Intersection Cohomologies
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 \(H^*_{\ov{p}}(X;\mathbb{Z})\). The second one arises from a simplicial blowup; 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.
Jörg Schürmann
Universität Münster(Equivariant) Characteristic Classes of Singular Toric Varieties
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 EulerMacLaurin type formulae for lattice polytopes.
Julius Shaneson
University of PennsylvaniaSingularities, 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.
Jon Woolf
University of LiverpoolApplications of BalmerWitt Groups to Stratified Spaces
BalmerWitt 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.
Min Yan
Hong Kong University of Science and TechnologyConverse of Smith Theory
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 semifree \({\mathbb Z}_n\)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 semifree group actions can also be extended, with an obstruction in \(\tilde{K}_0\). We also calculate some examples of semifree actions.
This is joint work with Sylvain Cappell of New York University and Shmuel Weinberger of University of Chicago.
Shoji Yokura
Kagoshima UniversityMotivic Hirzebruch Classes of Singular Varieties and Some Homological Congruence Formulae
In this talk, Yokura will give a quick review of the socalled ‘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 chiy genera, and some related topics.