Geometry/Topology Seminar

Winter 2014

Thursdays (and sometimes Tuesdays) 3-4pm, in Eckhart 308

- Thursday January 9 at 3-4pm in Eck 308
- David Aulicino, University of Chicago
- Higher Rank Orbit Closures in H^odd(4)
- Abstract: The moduli space of genus 3 translation surfaces with a single zero has two connected components. We show that in the odd connected component H^odd(4) the only GL⁺(2,R) orbit closures are closed orbits, the Prym locus \tilde{Q}(3,-1³), and H^odd(4). This is joint work with Duc-Manh Nguyen and Alex Wright.
- Monday January 13 at 4-5pm in Eck 308
- Lee Mosher, Rutgers University
- The geometry, topology, and dynamics of free groups (I)
- Abstract: Over a long span of time, a geometric understanding of free groups has emerged from the study of some very basic questions, such as: How do you tell when a set of elements of a free group forms a basis up to conjugacy (Whitehead's algorithm)? What are the possible values of the cyclically reduced word norm growth for an automorphism of F_n? What are the possible ranks of fixed subgroups of automorphisms of F_n (the Scott Conjecture, now a theorem of Bestina and Handel)? Motivated by these questions and others, in these lectures we shall survey the geometric methods which over the last 25 years have led to a quite revolution in the study of free groups. Our survey will include various deformation spaces of geometric/topological structures, such as the outer space of the free group (introduced by Culler and Vogtmann), the sphere complex aka the free splitting complex (Hatcher), and the free factor complex (Hatcher and Vogtmann). We also survey the dynamical theory of Out(F_n), the outer automorphism group, which developed along the lines of Thurston's dynamical theory for mapping class gorups, using tools of relative train track maps (Bestvina and Handel), attracting laminations (Bestvina, Feighn, and Handel), and R-trees (Culler, Morgan, Levitt, Lustig, and others).
- Tuesday January 14 at 3-4pm in Eck 308
- Lee Mosher, Rutgers University
- The geometry, topology, and dynamics of free groups (II)
- Abstract: Over a long span of time, a geometric understanding of free groups has emerged from the study of some very basic questions, such as: How do you tell when a set of elements of a free group forms a basis up to conjugacy (Whitehead's algorithm)? What are the possible values of the cyclically reduced word norm growth for an automorphism of F_n? What are the possible ranks of fixed subgroups of automorphisms of F_n (the Scott Conjecture, now a theorem of Bestina and Handel)? Motivated by these questions and others, in these lectures we shall survey the geometric methods which over the last 25 years have led to a quite revolution in the study of free groups. Our survey will include various deformation spaces of geometric/topological structures, such as the outer space of the free group (introduced by Culler and Vogtmann), the sphere complex aka the free splitting complex (Hatcher), and the free factor complex (Hatcher and Vogtmann). We also survey the dynamical theory of Out(F_n), the outer automorphism group, which developed along the lines of Thurston's dynamical theory for mapping class gorups, using tools of relative train track maps (Bestvina and Handel), attracting laminations (Bestvina, Feighn, and Handel), and R-trees (Culler, Morgan, Levitt, Lustig, and others).
- Wednesday January 15 at 2:30-4pm in Eck 308
- Lee Mosher, Rutgers University
- The geometry, topology, and dynamics of free groups (III)
- Abstract: Over a long span of time, a geometric understanding of free groups has emerged from the study of some very basic questions, such as: How do you tell when a set of elements of a free group forms a basis up to conjugacy (Whitehead's algorithm)? What are the possible values of the cyclically reduced word norm growth for an automorphism of F_n? What are the possible ranks of fixed subgroups of automorphisms of F_n (the Scott Conjecture, now a theorem of Bestina and Handel)? Motivated by these questions and others, in these lectures we shall survey the geometric methods which over the last 25 years have led to a quite revolution in the study of free groups. Our survey will include various deformation spaces of geometric/topological structures, such as the outer space of the free group (introduced by Culler and Vogtmann), the sphere complex aka the free splitting complex (Hatcher), and the free factor complex (Hatcher and Vogtmann). We also survey the dynamical theory of Out(F_n), the outer automorphism group, which developed along the lines of Thurston's dynamical theory for mapping class gorups, using tools of relative train track maps (Bestvina and Handel), attracting laminations (Bestvina, Feighn, and Handel), and R-trees (Culler, Morgan, Levitt, Lustig, and others).
- Tuesday January 21 at 3-4:30pm in Eck 206
- Curt McMullen, Harvard
- Entropy: graphs, surfaces, knots and number fields
- Abstract: (joint with dynamics seminar)
- Wednesday January 22 at 2:30-4pm in Eck 206
- Curt McMullen, Harvard
- Entropy: graphs, surfaces, knots and number fields
- Abstract: (joint with dynamics seminar)
- Thursday January 23 at 3-4:30pm in Ry 277
- Curt McMullen, Harvard
- Entropy: graphs, surfaces, knots and number fields
- Abstract: (joint with dynamics seminar)
- Tuesday February 18 at 3-4pm in Eck 206
- Maia Fraser, University of Toronto
- Bi-invariant metrics on contactomorphism groups
- Abstract: In contrast to the situation for the group of Hamiltonian symplectomorphisms of a symplectic manifold where Hofer's metric has been well studied since the early 90's, non-trivial bi-invariant metrics on contactomorphism groups have only recently been developed: Sandon 2010, Zapolsky 2012, Colin-Sandon 2012, Albers-Merry 2013 propose several. I will describe joint work with Polterovich and Rosen where we give yet another construction. It applies in contact manifolds which admit a 1-periodic Reeb flow and are orderable in the sense of Eliashberg and Polterovich. Stable unboundedness of the norm can be deduced in various settings including certain prequantization spaces (where the result follows from basic results on symplectic intersections) and the standard projective spaces RP²ⁿ⁺¹ (using Givental's non-linear Maslov index).
- Wednesday February 26 at 2:30-4pm in Eck 206
- Frank Calegari, Northwestern
- The stable homology of congruence subgroups (Part 1)
- Abstract: (joint with Algebraic Topology and with Number Theory Seminar) Let F be a number field. A stability result due to Charney and Maazen says that the homology groups H_d(SL_N(O_F),Z) (for d fixed) are independent of N for N sufficiently large. The resulting stable cohomology groups are intimately related to the algebraic K-theory of O_F. In these talks, we shall explore the homology of the p-power congruence subgroups of SL_N(O_F) in fixed degree d as N becomes large. We show that the resulting homology groups consist of two parts: an “unstable” part which depends only on local behavior concerning how the prime p splits in F, and a “stable” part which contains global information concerning p-adic regulator maps. Our argument consists of two parts. The first part (which is joint work with Matthew Emerton) explains how to modify the homology of congruence subgroups in a suitable way (using completed homology) to obtain groups which are literally stable for large N. We prove this by combining recent results in representation stability with the representation theory of p-adic groups. The second part consists of relating these stable homology groups to K-theory using ideas from algebraic topology. We give some applications of our results. The first concerns the computation of H₂ of the p-congruence subgroup of SL_N(Z) for sufficiently large N. The second concerns describing the extent to which the stable classes in characteristic zero constructed by Borel become more and more divisible by p when one passes to higher and higher p-power level.
- Thursday February 27 at 3-4pm in Eck 308
- Michael Farber, Warwick
- Fundamental groups of large random spaces
- Abstract: I will discuss several probabilistic models producing random simplicial complexes and will describe properties of their fundamental groups (torsion and cohomological dimension). Besides, I will examine the probabilistic validity of the Whitehead conjecture for aspherical 2-dimensional subcomplexes. Joint work with A. Costa.
- Friday February 28 at 2:30-4pm in Eck 206
- Frank Calegari, Northwestern
- The stable homology of congruence subgroups (Part 2)
- Abstract: (joint with Algebraic Topology and with Number Theory Seminar) Let F be a number field. A stability result due to Charney and Maazen says that the homology groups H_d(SL_N(O_F),Z) (for d fixed) are independent of N for N sufficiently large. The resulting stable cohomology groups are intimately related to the algebraic K-theory of O_F. In these talks, we shall explore the homology of the p-power congruence subgroups of SL_N(O_F) in fixed degree d as N becomes large. We show that the resulting homology groups consist of two parts: an “unstable” part which depends only on local behavior concerning how the prime p splits in F, and a “stable” part which contains global information concerning p-adic regulator maps. Our argument consists of two parts. The first part (which is joint work with Matthew Emerton) explains how to modify the homology of congruence subgroups in a suitable way (using completed homology) to obtain groups which are literally stable for large N. We prove this by combining recent results in representation stability with the representation theory of p-adic groups. The second part consists of relating these stable homology groups to K-theory using ideas from algebraic topology. We give some applications of our results. The first concerns the computation of H₂ of the p-congruence subgroup of SL_N(Z) for sufficiently large N. The second concerns describing the extent to which the stable classes in characteristic zero constructed by Borel become more and more divisible by p when one passes to higher and higher p-power level.
- Thursday March 6 at 3-4pm in Eck 308
- Anders Sodergern, IAS
- The distribution of angles in hyperbolic lattices
- Abstract: We consider certain questions related to the hyperbolic lattice point problem in general dimension n≥2. More precisely, we describe the distribution of angles between geodesic rays associated with the points in an orbit of a hyperbolic lattice. Our main interest is in properties of the corresponding pair correlation density function and the starting point of our discussion will be the recent results of Kelmer and Kontorovich in dimension 2. This is joint work with Morten Risager.
- Tuesday March 18 at 3-4pm in E308
- Dima Burago, Penn State
- Just so stories (Kipling)
- Abstract: This is not a usual type of a seminar talk, though I have given several talks with almost the same title lately. Still the talks are different. I made transparencies for more than 20 topics, two to three slides per topic. Necessary definitions, formulations of key results, hints towards proofs, open problems. I chose about 8 topics per talk. The choice depends on the audience, the weather, what I had for breakfast and such. The only thing that unites the topics is that they have been of interest to me in past number of years. They all are related to geometry, PDEs, dynamics, geometric group theory and such. If the weather cooperates, the breakfast goes well and there are no terrible traffic jams, I probably would do something around there topics: -- Uniform approximations of metric spaces by graphs of bounded geometry; -- Graph approximations of the Beltrami-Laplace on Riemannian manifolds; (including a Marriage Lemma for measures); -- Local generation of metric entropy and surjectivity of the boundary distance function near simple Finsler metrics; -- Why curvature in Finsler Geometry cannot be responsible for basic geometric phenomena (no rotten tomatoes, please!); -- On conjugation invariant metrics of unbounded diameter on groups of geometric origin; -- Area minimizers in normed spaces (including Busemann's Conjecture in D=2 and a formula for the area of a convex polygon in the Euclidean plane); -- Area spaces: now filling area determines metric; -- Tomography: imaging vs. volume minimality. -- Maybe something completely different.
- Thursday March 20 at 2-3pm in Ry 352
- Misha Belilopetsky, IMPA
- 2-systoles and quantum codes
- Abstract: I will review a recent paper of Guth and Lubotzky and discuss some related problems.
- Thursday March 20 at 3-4pm in Eck 308
- Jing Tao, University of Oklahoma
- Evolving structures on graphs and uniform growth rate of some families of groups
- Abstract: Thurston asserted during his birthday conference that the growth rate of mapping class group is independent of genus. In this talk, I will explain what this means and sketch a proof of this theorem, following Sleator-Tarjan-Thurston.
- Thursday April 3 at 3-4pm in Eck 308
- Stergios Antonakoudis, Harvard University
- The intrinsic geometry of Teichmuller space and symmetric domains
- Abstract: In this talk we will look at complex domains in their intrinsic Kobayashi metrics with a goal to contribute to the following important question: To what extent does the geometry of Teichmuller space resemble the geometry of a bounded symmetric domain?
- Monday April 7 at 2:30-3:30 pm in Ry358
- Satyan Devadoss, Stanford University
- Phylogenetic networks and the real moduli space of curves
- Abstract: Infused with visual imagery, this talk is motivated by the the moduli spaces of punctured Riemann spheres. In the 1970s, Grothendieck, Deligne, and Mumford constructed their compactification using Geometric Invariant Theory. In the 1990s, Gromov and Witten utilized them as invariants arising from string field theory and quantum cohomology. We consider real points of these spaces, which have elegant geometric and combinatorial properties, being compact hyperbolic manifolds with a beautiful tessellation by convex polytopes. In recent years, they have gained importance in their own right, appearing in areas such as representation theory, geometric group theory, tropical geometry, and lately reinterpreted as spaces of phylogenetic networks. In particular, they resolve the singularities of tree spaces studied by Billera, Holmes, and Vogtmann.
- Tuesday April 8 at 3-4 pm in E308
- Jonathan Hillman, Sydney
- Poincare duality in low dimensions (1)
- Abstract: The goal of this minicourse is to show how Poincare duality imposes constraints on and interactions between the two most basic invariants of algebraic topology, namely, the Euler characteristic \chi(M) and the fundamental group \pi₁(M), for M a manifold (or a Poincare duality complex) of dimension 3 or 4. (In lower dimensions these invariants determine each other, while in higher dimensions they are decoupled.) We shall concentrate on recent work and open questions. In dimension 3 there is a simple system of homotopy invariants, and every orientable PD₃-complex is a connected sum of indecomposables. These are either aspherical or have virtually free fundamental group. The latter class is largely known, and the main question here is whether every aspherical PD₃-complex is homotopy equivalent to a closed 3-manifold. In dimension 4 the situation is less clear, but aspherical PD₄-complexes may be characterised in terms of \pi and \chi, while the fundamental group, Stiefel-Whitney classes and the homotopy intersection pairing determine the homotopy type in many cases with c.d.\pi≤2.
- Thursday April 10 at 3-4 pm in E308
- Jonathan Hillman, Sydney
- Poincare duality in low dimensions (2)
- Abstract: The goal of this minicourse is to show how Poincare duality imposes constraints on and interactions between the two most basic invariants of algebraic topology, namely, the Euler characteristic \chi(M) and the fundamental group \pi₁(M), for M a manifold (or a Poincare duality complex) of dimension 3 or 4. (In lower dimensions these invariants determine each other, while in higher dimensions they are decoupled.) We shall concentrate on recent work and open questions. In dimension 3 there is a simple system of homotopy invariants, and every orientable PD₃-complex is a connected sum of indecomposables. These are either aspherical or have virtually free fundamental group. The latter class is largely known, and the main question here is whether every aspherical PD₃-complex is homotopy equivalent to a closed 3-manifold. In dimension 4 the situation is less clear, but aspherical PD₄-complexes may be characterised in terms of \pi and \chi, while the fundamental group, Stiefel-Whitney classes and the homotopy intersection pairing determine the homotopy type in many cases with c.d.\pi≤2.
- Tuesday April 15 at 3-4 pm in E203
- Jonathan Hillman, Sydney
- Poincare duality in low dimensions (3)
- Abstract: The goal of this minicourse is to show how Poincare duality imposes constraints on and interactions between the two most basic invariants of algebraic topology, namely, the Euler characteristic \chi(M) and the fundamental group \pi₁(M), for M a manifold (or a Poincare duality complex) of dimension 3 or 4. (In lower dimensions these invariants determine each other, while in higher dimensions they are decoupled.) We shall concentrate on recent work and open questions. In dimension 3 there is a simple system of homotopy invariants, and every orientable PD₃-complex is a connected sum of indecomposables. These are either aspherical or have virtually free fundamental group. The latter class is largely known, and the main question here is whether every aspherical PD₃-complex is homotopy equivalent to a closed 3-manifold. In dimension 4 the situation is less clear, but aspherical PD₄-complexes may be characterised in terms of \pi and \chi, while the fundamental group, Stiefel-Whitney classes and the homotopy intersection pairing determine the homotopy type in many cases with c.d.\pi≤2.
- Thursday April 17 at 3-4 pm in E308
- Jonathan Hillman, Sydney
- Poincare duality in low dimensions (4)
- Abstract: The goal of this minicourse is to show how Poincare duality imposes constraints on and interactions between the two most basic invariants of algebraic topology, namely, the Euler characteristic \chi(M) and the fundamental group \pi₁(M), for M a manifold (or a Poincare duality complex) of dimension 3 or 4. (In lower dimensions these invariants determine each other, while in higher dimensions they are decoupled.) We shall concentrate on recent work and open questions. In dimension 3 there is a simple system of homotopy invariants, and every orientable PD₃-complex is a connected sum of indecomposables. These are either aspherical or have virtually free fundamental group. The latter class is largely known, and the main question here is whether every aspherical PD₃-complex is homotopy equivalent to a closed 3-manifold. In dimension 4 the situation is less clear, but aspherical PD₄-complexes may be characterised in terms of \pi and \chi, while the fundamental group, Stiefel-Whitney classes and the homotopy intersection pairing determine the homotopy type in many cases with c.d.\pi≤2.
- Tuesday April 22 at 3-4pm in Eck 308
- Weiwei Wu, Michigan State University
- Symplectic mapping class groups in dimension four, and their relations with Lagrangians
- Abstract: This talk will survey some recent progress and open problems on understanding symplectic mapping class groups in dimension four. In particular, we explain their interactions with Lagrangians in both directions: on the one hand, we show some evidence that, in nice situations, Lagrangian spheres creates Dehn twists that generates the symplectic mapping class groups; on the other hand, the understandings on symplectic mapping class groups can be used to resolve problems on isotopy classes of Lagrangian or symplectic objects, not limited to spheres. The bridge between the two problems is provided by the so-called ball-swapping construction.
- Thursday May 1 at 3-4pm in Eck 308
- Burglind Joricke, Indiana
- Braids, Conformal Module and Entropy
- Abstract: We will discuss two invariants of conjugacy classes of braids. The first invariant is the conformal module which occurred in connection with the interest in the 13. Hilbert problem. The second is a popular dynamical invariant, the entropy. It occurred in connection with Thurston's theory of surface homeomorphisms. It turns out that these invariants are related: They are inverse proportional. This allows to use known results on entropy for applications to the concept of conformal module, in particular to give a conceptional proof of a previous theorem. We will also give an application of the concept of conformal module to the problem of isotopy of continuous objects involving braids to the respective holomorphic objects.
- Thursday May 8 at 3-4pm in Eck 308
- Rich Schwartz, Brown
- The projective heat map on pentagons
- Abstract: I'll describe several geometrically defined and projectively natural iterations on polygons. One of them, the pentagram map, is now known to be a discrete completely integrable system. The other one, which I call the projective heat map, is sort of a marriage of the pentagram map and heat flow. I'll sketch a computer-assisted (but rigorous) analysis of how the projective heat map acts on the moduli space of projective equivalence classes of pentagons.
- Friday May 9 at 4-5pm in Ry 251
- Ian Agol, University of California at Berkeley
- The virtual Haken conjecture (Namboodiri lecture 1)
- Abstract: Waldhausen conjectured in 1968 that every aspherical 3-manifold has a finite-sheeted cover which is Haken (contains an embedded essential surface). Thurston conjectured that hyperbolic 3-manifolds have a finite-sheeted cover which fibers over the circle. The first lecture will be an overview of 3-manifold topology in order to explain the meaning of Waldhausen's virtual Haken conjecture and Thurston's virtual fibering conjecture, and how they relate to other problems in 3-manifold theory. The second lecture will give some background on geometric group theory, including the topics of hyperbolic groups and CAT(0) cube complexes after Gromov, and explain how the above conjectures may be reduced to a conjecture of Dani Wise in geometric group theory. The third lecture will discuss the proof of Wise's conjecture, that cubulated hyperbolic groups are virtually special, and some discussion of a new proof of Wise's Malnormal Special Quotient Theorem. Part of these results are joint work with Daniel Groves and Jason Manning.
- Monday May 12 at 4-5pm in Ry 251
- Ian Agol, University of California at Berkeley
- The virtual Haken conjecture (Namboodiri lecture 2)
- Abstract: Waldhausen conjectured in 1968 that every aspherical 3-manifold has a finite-sheeted cover which is Haken (contains an embedded essential surface). Thurston conjectured that hyperbolic 3-manifolds have a finite-sheeted cover which fibers over the circle. The first lecture will be an overview of 3-manifold topology in order to explain the meaning of Waldhausen's virtual Haken conjecture and Thurston's virtual fibering conjecture, and how they relate to other problems in 3-manifold theory. The second lecture will give some background on geometric group theory, including the topics of hyperbolic groups and CAT(0) cube complexes after Gromov, and explain how the above conjectures may be reduced to a conjecture of Dani Wise in geometric group theory. The third lecture will discuss the proof of Wise's conjecture, that cubulated hyperbolic groups are virtually special, and some discussion of a new proof of Wise's Malnormal Special Quotient Theorem. Part of these results are joint work with Daniel Groves and Jason Manning.
- Tuesday May 13 at 3-4pm in Eck 308
- Jorgen Ellegaard Andersen, QGM Aarhus
- The Witten-Reshetikhin-Turaev quantum representations of mapping class groups
- Abstract: As part of the Witten-Reshetikhin-Turaev topological quantum field theory one has the corresponding quantum representations of the mapping class groups. They can be constructed both by purely combinatorial means and by applying geometric quantization to the moduli spaces of flat connections on a two dimensional surface. We shall review this geometric construction of these representations and turn to a number of applications of these representations to the study of mapping class groups.
- Tuesday May 13 at 4:30-5:30pm in Eck 206
- Ian Agol, University of California at Berkeley
- The virtual Haken conjecture (Namboodiri lecture 3)
- Abstract: Waldhausen conjectured in 1968 that every aspherical 3-manifold has a finite-sheeted cover which is Haken (contains an embedded essential surface). Thurston conjectured that hyperbolic 3-manifolds have a finite-sheeted cover which fibers over the circle. The first lecture will be an overview of 3-manifold topology in order to explain the meaning of Waldhausen's virtual Haken conjecture and Thurston's virtual fibering conjecture, and how they relate to other problems in 3-manifold theory. The second lecture will give some background on geometric group theory, including the topics of hyperbolic groups and CAT(0) cube complexes after Gromov, and explain how the above conjectures may be reduced to a conjecture of Dani Wise in geometric group theory. The third lecture will discuss the proof of Wise's conjecture, that cubulated hyperbolic groups are virtually special, and some discussion of a new proof of Wise's Malnormal Special Quotient Theorem. Part of these results are joint work with Daniel Groves and Jason Manning.
- Tuesday May 27 at 3-4pm in Eck 308
- Jon Chaika, University of Utah
- TBA
- Abstract: TBA.
- Thursday May 29 at 3-4pm in Eck 308
- Barry Minemyer, Alfred University
- Isometric Embeddings of Polyhedra
- Abstract: In 1954-56 John Nash solved the isometric embedding problem for Riemannian manifolds. This, along with a result due to Zalgaller, prompted Gromov to ask whether or not Euclidean polyhedra admit isometric embeddings into Euclidean space. Here, a Euclidean polyhedra is just a metric space which admits a locally finite triangulation such that every k-dimensional simplex is affinely isometric to a simplex in Euclidean space E^k. In this talk we will discuss recent results toward answering Gromov's question and, if time permits, we will discuss indefinite metric analogues to the above results.

For questions, contact

Benson Farb, farb at math.uchicago.edu
Alex Eskin, eskin at math.uchicago.edu
Shmuel Weinberger, shmuel at math.uchicago.edu
Howard Masur, masur at math.uchicago.edu
Danny Calegari, dannyc at math.uchicago.edu
Qian Yin, qyin at math.uchicago.edu
Sebastian Hensel, hensel at math.uchicago.edu
Jonathan Chaika, jonchaika at math.uchicago.edu