Department of

# Mathematics

Seminar Calendar
for Geometry, Groups and Dynamics/GEAR events the next 12 months of Friday, January 1, 2016.

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events for the
events containing

Questions regarding events or the calendar should be directed to Tori Corkery.
    December 2015           January 2016          February 2016
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Thursday, January 21, 2016

12:00 pm in 243 Altgeld Hall,Thursday, January 21, 2016

#### Limits of Teichmüller geodesics

###### Chris Leininger (UIUC Math)

Abstract: I will discuss joint work with Brock, Modami, and Rafi in which we describe local combinatorial constructions for quasi-geodesics in the curve complex which limit to non-uniquely ergodic laminations. We have good control on the intersection numbers for these sequences, and in particular, we are able to analyze the limiting behavior in the Thurston boundary of Teichmüller space of Teichmüller rays defined by these laminations. These limits exhibit new and unexpected phenomena.

Tuesday, January 26, 2016

12:00 pm in Altgeld Hall 243,Tuesday, January 26, 2016

#### Slowly converging Pseudo-Anosovs

###### Mark Bell (UIUC Math)

Abstract: A classic property of pseudo-Anosov mapping classes is that they act on PML with north-south dynamics. This means that for such a mapping class laminations converge exponentially towards its stable lamination under iteration. We will discuss a new result which shows that (whenever the surface is sufficiently complex) there are pseudo-Anosovs where this convergence is exponential but arbitrarily slow, that is, with base arbitrarily close to one. This work is joint with Saul Schleimer.

Thursday, January 28, 2016

12:00 pm in Altgeld Hall 243,Thursday, January 28, 2016

#### Extending the $\log (2k-1)$-Theorem

###### Rosemary Guzman (UIUC Math)

Abstract: In this talk, I discuss current work that expands the scope of the $\log (2k-1)$-Theorem of Anderson, Canary, Culler and Shalen. This was a seminal result in that it articulated a relationship between a set of $k$ freely-generating isometries of hyperbolic 3-space and how they interacted with points in hyperbolic 3-space; namely, under certain conditions, at least one of the given isometries must move a point $P$ by a distance $\ge \log(2k-1)$. The result lay the foundation for future novel geometric-topological results. Here I discuss an expansion of the theorem, wherein we consider sets of length-$n$ words contained in a rank-2 free group $\Xi$ on 2 letters (one can consider $\ge 2$ letters via the same methods), and present a generalized version that restricts how these isometries displace points in hyperbolic 3-space. This has application to classifying certain hyperbolic 3-manifolds in that the volume of the resulting manifold $M$ gotten by quotient of hyperbolic 3-space with $\Xi$, is expected to have a bounded volume which is improved from known volume bounds.

Thursday, February 4, 2016

12:00 pm in Altgeld Hall 243,Thursday, February 4, 2016

#### Flattening equations for Chromatic Polynomials

###### Frank Bernhart (Rochester Institute of Technology)

Abstract: In the 1940s a long treatment of “planar” chromatic polynomials by G.D. Birkhoff and D.C. Lewis led to the discovery of mysterious identities concerning “chromials” obtained by fixing the configuration and varying the boundary coloring pattern. In the 1970s W. T. Tutte at the University of Waterloo (Canada) fruitfully suggested a distinction between “planar” boundary colorings and, the others. Joint research, never fully published, showed that chromials with planar boumndary patterns give a linear basis, and dimension numbers form a sequence related to Motzkin and Catalan numbers, which I had previously called Riordan numbers. Given a circular arrangement of vertices, planar partitions may be defined. The unrestricted count is Catalan. To get the numbers R(n) make the additional restriction that vertices adjacent on the circle cannot be in the same part.

Tuesday, February 9, 2016

12:00 pm in Altgeld Hall 243,Tuesday, February 9, 2016

#### L^2-torsion of free-by-cyclic groups

###### Matthew Clay (University of Arkansas)

Abstract: I will provide an upper bound on the $L^2$-torsion of a free-by-cyclic group, $-p^{(2)}(G_\Phi)$, in terms of a relative train-track representative for $\Phi$ in Aut(F). This result shares features with a theorem of Luck-Schick computing the $L^2$-torsion of the fundamental group of a 3-manifold that fibers over the circle in that it shows that the $L^2$-torsion is determined by the exponential dynamics of the monodromy. In light of the result of Luck-Schick, a special case of this bound is analogous to the bound on the volume of a 3-manifold that fibers over the circle with pseudo-Anosov monodromy by the normalized entropy recently demonstrated by Kojima-McShane.

Thursday, February 11, 2016

12:00 pm in Altgeld Hall 243,Thursday, February 11, 2016

#### Characterization of Aleksandrov Spaces of Curvature Bounded Above by Means of the Metric Cauchy-Schwarz Inequality.

###### David Berg (Illinois Math)

Abstract: Joint work of I.D.Berg and I.G.Nikolaev. We employ the previously introduced notion of the K-quadrilateral cosine,the cosine under parallel transport in model K-space,denoted by cosqK. In K-space, modulus cosqK bounded by 1 is equivalent to the Cauchy-Schwarz inequality for tangent vectors under parallel transport. Our principal result states that a geodesic space(of diameter bounded by half the hemisphere diameter for positive K) is a catK space if and only if cosqK is bounded by 1. If, in addition, 1 is actually achieved for two directed non-collinear segments, the geodesic span of the two segments is isometric to a section of K-plane. The diameter restriction is significant. This talk will be devoted to illustrating and explaining these results.If there is time, I will give the ideas of the proofs, veiling the difficult computations that arise,especially in the case of nonzero K, in a decent obscurity.

Tuesday, February 16, 2016

12:00 pm in Altgeld Hall 243,Tuesday, February 16, 2016

#### Higgs bundles and flexibility of surface group representations

Abstract: Among their many virtues, Higgs bundles reveal features of surface group representations that can be difficult to detect by other means. This will be illustrated in situations where the image of a representation into a group G lies in a subgroup of G, with special attention given to the case where G is a non-compact real form such as U(p,q) or Sp(2n,R).

Tuesday, February 23, 2016

12:00 pm in Altgeld Hall 243,Tuesday, February 23, 2016

#### Cheeger constants of arithmetic hyperbolic surfaces

###### Grant Lakeland (Eastern Illinois University)

Abstract: Given a Riemannian manifold M, the Cheeger constant, h(M), is a geometric invariant which measures the extent to which M has “bottlenecks” - roughly speaking, these are low volume, separating, codimension one submanifolds. We implement an algorithm of Benson to explicitly compute the Cheeger constant for a collection of arithmetic hyperbolic surfaces. The results have connections to arithmetic reflection groups, and to the relationship between the arithmetic and geometry of Fuchsian groups. This is joint work with Brian Benson.

Thursday, February 25, 2016

12:00 pm in Altgeld Hall 243,Thursday, February 25, 2016

#### Characterization of Aleksandrov Spaces of Curvature Bounded Above by Means of the Metric Cauchy-Schwarz Inequality; part II

###### David Berg (UIUC Math)

Abstract: Joint work of I.D.Berg and I.G.Nikolaev. We employ the previously introduced notion of the K-quadrilateral cosine,the cosine under parallel transport in model K-space,denoted by cosqK. In K-space, modulus cosqK bounded by 1 is equivalent to the Cauchy-Schwarz inequality for tangent vectors under parallel transport. Our principal result states that a geodesic space(of diameter bounded by half the hemisphere diameter for positive K) is a catK space if and only if cosqK is bounded by 1. If, in addition, 1 is actually achieved for two directed non-collinear segments, the geodesic span of the two segments is isometric to a section of K-plane. The diameter restriction is significant. This talk will be devoted to illustrating and explaining these results.If there is time, I will give the ideas of the proofs, veiling the difficult computations that arise,especially in the case of nonzero K, in a decent obscurity.

Tuesday, March 1, 2016

12:00 pm in Altgeld Hall 243,Tuesday, March 1, 2016

#### Some symplectic geometry in the 3-d Einstein Universe

###### Virginie Charette (University of Sherbrooke)

Abstract: The 3-dimensional Einstein Universe is the conformal compactification of 3-dimensional affine Minkowski space. Interestingly, it can also be interpreted as the space of Lagrangian planes in a 4-dimensional symplectic vector space. Using a suitable dictionary, we will discuss how certain facts concerning the Einstein Universe can be nicely stated in this language.

Thursday, March 3, 2016

12:00 pm in Altgeld Hall 243,Thursday, March 3, 2016

#### Equations in free associative algebras and group algebras of hyperbolic groups.

###### Olga Kharlampovich (Hunter College of CUNY)

Abstract: We will talk about Diophantine problems for groups and rings and show that equations are undecidable in a free associative algebra and in a group algebra of a free group (or torsion free hyperbolic group) over a field of characteristic zero. This is a surprising result that shows that answers to first-order questions in a free non-abelian group and in its group algebra are completely different. (Joint results with A. Miasnikov)

Tuesday, March 8, 2016

12:00 pm in Altgeld Hall 243,Tuesday, March 8, 2016

#### Mapping class group and right-angled Artin group actions on the circle

###### Thomas Koberda (University of Virginia)

Abstract: I will discuss virtual mapping class group actions on compact one-manifolds. The main result will be that there exists no faithful C^2 action of a finite index subgroup of the mapping class group on the circle, which generalizes results of Farb-Franks and establishes a higher rank phenomenon for the mapping class group, mirroring a result of Ghys and Burger-Monod. This talk will represent joint work with H. Baik and S. Kim.

Tuesday, March 15, 2016

12:00 pm in Altgeld Hall 243,Tuesday, March 15, 2016

#### SO(n,n+1) surface group representations

###### Brian Collier (UIUC Math)

Abstract: In this talk I will discuss a parameterization of n(2g-2) connected components of the SO(n,n+1) character variety of a closed surface of genus g. We will see how this parameterization generalizes both Hitchin's parameterization of the Hitchin component as a vector space of holomorphic differentials of degree 2,4,...,2n and Hitchin's parameterization of the nonzero Toledo invariant components of the PSL(2,R)=SO(1,2) Higgs bundle moduli space by holomorphic quadratic differentials twisted by an effective divisor.

Thursday, March 17, 2016

12:00 pm in 243 Altgeld Hall,Thursday, March 17, 2016

#### Heegaard Floer homology and 3-manifold mutation

###### Corrin Clarkson (Indiana University)

Abstract: The mutation operation is easy to define: cut a 3-manifold along an embedded surfaces and gluing it back together. It is also familiar; special cases include Dehn surgery and the construction of mapping tori. In many cases, the effects of mutation are significant and easily detected, but some gluings are more subtle. One such subtle gluing map is the genus-2 hyperelliptic involution. This is the only higher genus mapping class whose mutations might preserve the total rank of Heegaard Floer homology. I will show that all other gluings can change the total rank of HF-hat and give and overview of our understanding of the exceptional case: mutating by the genus two hyper elliptic involution.

Tuesday, March 29, 2016

12:00 pm in Altgeld Hall 243,Tuesday, March 29, 2016

#### On /\-positioning of arcs between parallel support planes

###### Yevgenya Movshovich (EIU)

Abstract: The following result for an arc, called by J. Wetzel, the /\-property, was proved in Theorem 5.1 of "Besicovitch triangles cover unit arcs", Geom. Dedicata, vol. 123, (2006), ] by P. Coulton and Y. Movshovich: Any simple plane polygonal finite arc g has two parallel support lines and three parameters r < t < u; so that g(t) lies on one line, while g(r) and g(u) lie on the other. When showing that a convex set contains all unit arcs, the /\-property allow us to study only 3 and 4-segment arcs, shaped as letters S and W or a staple. There were two announcements on extending the result of Theorem 5.1 from polygonal to simple arcs: one by Y. M. (Geometry Seminar, UIUC, 2009) and the other by R. Alexander, J. E. Wetzel, W. Wichiramala in their recently submitted paper "The /\-property of a simple arc". In this talk we prove Theorem 5.1 omitting all three requirements on a rectifiable arc: polygonal, simple and plane.

Tuesday, April 5, 2016

12:00 pm in Altgeld Hall 243,Tuesday, April 5, 2016

#### Homology and volume for hyperbolic 3-orbifolds, and enumeration of arithmetic groups

###### Peter Shalen (University of Illinois at Chicago)

Abstract: A theorem of Borel's asserts that for any positive real number $V$, there are at most finitely many arithmetic lattices in ${\rm PSL}_2({\mathbb C})$ of covolume at most $V$, or equivalently at most finitely many arithmetic hyperbolic $3$-orbifolds of volume at most $V$. Determining all of these for a given $V$ is algorithmically possible for a given $V$ thanks to work by Chinburg and Friedman, but appears to be impractical except for very small values of $V$, say $V=0.41$. (The smallest covolume of a hyperbolic $3$-orbifold is about $0.39$.) It turns out that the difficulty in the computation for a larger value of $V$ can be dealt with if one can find a good bound on $\dim H_1(O,\mathbb Z/2 \mathbb Z)$, where $O$ is a hyperbolic $3$-orbifold of volume at most $V$. In the case of a hyperbolic $3$-manifold $M$, not necessarily arithmetic, joint work of mine with Marc Culler and others gives good bounds on the dimension of $H_1(M,\mathbb Z/2 \mathbb Z)$ in the presence of a suitable bound on the volume of $M$. In this talk I will discuss some analogous results for hyperbolic $3$-orbifolds, and the prospects for applying results of this kind to the enumeration of arithmetic lattices. A feature of the work that I find intriguing is that while it builds on my geometric work with Culler, the new ingredients involve primarily purely topological arguments about manifolds---the underlying spaces of the orbifolds in question---and have a classical, combinatorial flavor. At this point it appears that I can prove the following statement: If $\Omega$ is a hyperbolic 3-orbifold of volume at most $1.72$, having a link as singular set and containing no embedded turnovers, then $$\dim H_1(\Omega;\mathbb Z_2)\le 1+ \max\bigg(3,7\bigg\lfloor\frac{10}{3}{\rm vol}(\Omega)\bigg\rfloor\bigg)+ \max\bigg(3, 7\bigg\lfloor\frac{5}{3}{\rm vol}(\Omega)\bigg\rfloor\bigg).$$ In particular, $\dim H_1(\Omega;\mathbb Z_2)\le50$. Various stronger bounds on $\dim H_1(\Omega;\mathbb Z_2)$ follow from stronger bounds on the volume of $\Omega$. The restriction on turnovers is not an obstruction to applying the results to the enumeration of arithmetic groups. The assumption that the singular set is a link is more serious, but as it is used only in a mild way in this work, the methods seem promising for the prospective application.

Thursday, April 7, 2016

12:00 pm in 243 Altgeld Hall,Thursday, April 7, 2016

#### Vortex Moduli Spaces and the Space of Holomorphic Maps

###### Chih-Chung Liu (National Cheng Kung University, Taiwan)

Abstract: The space of holomorphic maps from a closed Riemann surface to complex projective space is of great physical interest. This space naturally corresponds to the space of (gauge classes of) solutions to the vortex equations on the surface. In this talk I will present my past work on strengthening the correspondence so that it is an isometry at some asymptotic region. Such a result encourages us to prove Baptista’s conjectural formula on the L^2 volume of the space of maps, generalizing Speight’s earlier result on the case for Riemann spheres.

Tuesday, April 12, 2016

12:00 pm in Altgeld Hall 243,Tuesday, April 12, 2016

#### Subset currents on surface groups

###### Dounnu Sasaki (Waseda University, Tokyo)

Abstract: Subset currents on free groups and on surface groups, which were introduced by I. Kapovich and T. Nagnibeda in 2013, are the generalization of geodesic currents on surface groups originally studied by Bonahon in late 1980s. The space of geodesic currents on the fundamental group $G$ of a closed hyperbolic surface $\Sigma$ can be thought of as a measure-theoretic completion of the set of all conjugacy classes in $G$. Similarly, subset currents are measure-theoretic generalizations of conjugacy classes of finitely generated subgroups of $G$. Geodesic currents have been successfully used in the study of the Teichm\”uller space and the mapping class group of $\Sigma$. In this talk I will discuss some new results on subset currents on surface groups. To every nontrivial finitely generated subgroup $H\le G=\pi_1(\Sigma)$ we associate a "counting subset current" $\eta_H$. We prove that the set of all scalar multiples of counting currents is dense in the space of all subset currents on $\Sigma$, generalizing a result of Kapovich and Nagnibeda in the free group case. We extend Bonahon's "geometric intersection number" between geodesic currents to an intersection number between subset currents. We also construct a continuous linear "Euler characteristic" functional on the space of subset currents on $\Sigma$ such that, when evaluated on a counting current $\eta_H$, this functional computes the Euler characteristic of the surface corresponding to the core of the cover of $\Sigma$ corresponding to $H$.

Thursday, April 14, 2016

12:00 pm in Altgeld Hall 243,Thursday, April 14, 2016

#### (Hyper-)Kähler geometry of character varieties

###### Brice Loustau (Rutgers - Newark)

Abstract: I will present a construction due to Andy Sanders of a (hyper-)Kähler metric in the character variety of a closed surface group which generalizes the Weil-Petersson metric on Teichmüller space as well as Hitchin's metric in the moduli space of Higgs bundles.

Tuesday, April 19, 2016

12:00 pm in Altgeld Hall 243,Tuesday, April 19, 2016

#### Separated nets in the Heisenberg group

###### Anton Lukyanenko (University of Michigan)

Abstract: A co-compact lattice is a standard example of a separated net, but other nets also arise in applications (most famously the quasi-crystals in chemistry), and one would like to know whether they are simply perturbations of lattices. In Euclidean space, a criterion of Laczkovich allows one to easily make nets that are not a bounded-distance perturbation of any lattice (not BD rectifiable), and an intricate construction due to McMullen and Burago-Kleiner provides a net that is not even bi-Lipschitz to any lattice (not BL rectifiable). We study nets and quasi-crystals in the Heisenberg group and more generally (rational) Carnot groups. Lattices in these groups are quite tame, and by a theorem of Malcev may even be viewed as the integer points in appropriate coordinates. We show that a generic net need not be well-behaved: in addition to nets that are not BD or BL rectifiable, there exist BD-rectifiable exotic nets'' that are neither coarsely dense nor uniformly discrete in Malcev coordinates. On the other hand, in applications, a natural construction of quasi-crystals yields easily-understood nets whose BD rectifiability is based on a certain Diophantine condition, showing that almost every Heisenberg quasi-crystal is a BD perturbation of a lattice.

Thursday, April 21, 2016

12:00 pm in Altgeld Hall 243,Thursday, April 21, 2016

#### Finitely generated groups with co-c.e. word problem (d'apres Morozov)

###### Paul Schupp (UIUC Math)

Abstract: Let $\mathcal{C}$ be the group of all computable permutations of the natural numbers. The general question is: What can one say about finitely generated subgroups of $\mathcal{C}$? While most groups studied in geometric group theory have computably enumerable word problems, one sees immediately that a finitely generated subgroup of $\mathcal{C}$ must have co-c.e. word problem, that is, the set of words equal to the identity in $G$ is the complement of a computably enumerable set. Andrey Morozov proved two important theorems about finitely generated subgroups of $\mathcal{C}$. We will discuss these theorems and interesting connections of the basic question to other groups.

Tuesday, April 26, 2016

12:00 pm in Altgeld Hall 243,Tuesday, April 26, 2016

#### The geometry of hyperbolic free group extensions

###### Spencer Dowdall (Vanderbilt University)

Abstract: Each subgroup $\Gamma$ of the outer automorphism group of a free group naturally gives rise to a group extension $E_\Gamma$ of the free group. In joint work with Sam Taylor, we have given geometric conditions on $\Gamma$ that imply the extension is a hyperbolic group. After describing these conditions, I will present recent results about the fine geometric structure of these group extensions. These results include a Scott--Swarup type theorem proving quasiconvexity of infinite-index subgroups of the free group and a width theorem that characterizes these hyperbolic extensions in terms of the axes of primitive elements of the free group.

Thursday, April 28, 2016

12:00 pm in Altgeld Hall 243,Thursday, April 28, 2016

#### Induced group actions

###### Denis Osin (Vanderbilt University)

Abstract: I will discuss the following natural extension problem for group actions: Given a group G, a subgroup H\le G, and an action of H on a metric space S, when is it possible to extend it to an action of the whole group G? When does such an extension preserve interesting properties of the original action of H? I will explain how to formalize this problem and will present a construction of the induced action which behaves well when G is hyperbolic relative to H or, more generally, H is hyperbolically embedded in G. We will also discuss some applications. This talk is based on my work in progress with C. Abbott and S. Balasubramanya.

Tuesday, May 3, 2016

12:00 pm in Altgeld Hall 243,Tuesday, May 3, 2016

#### Hitchin representations in Sp(4,R) and conformally flat space-times

###### Jérémy Toulisse (University of Southern California)

Abstract: Given a closed oriented surface S of genus g>1 and using the isomorphism between Sp(4,R) and SO(2,3), we can associate to a representation from the fundamental group of S to Sp(4,R) an action on the Einstein universe (which is the model space for 3-dimensional conformally flat Lorentz structure). In this talk, we will explain how to realize Hitchin representations in Sp(4,R) as holonomy of conformally flat Lorentz structures on the unit tangent bundle of S. This is a joint work with Nicolas Tholozan.

Tuesday, August 23, 2016

12:00 pm in 243 Altgeld Hall,Tuesday, August 23, 2016

#### Organizational meeting

Abstract: We will hold a brief organizational meeting, then head directly to lunch.

Thursday, September 1, 2016

12:00 pm in Altgeld Hall 243,Thursday, September 1, 2016

#### Gauge theory for webs and foams

###### Tomasz Mrowka (Massachusetts Institute of Technology)

Abstract: I will discuss how to use Instanton Floer theory ideas to construct invariants of Webs (knotted trivalent graphs) in the three manifolds. These invariants are functorial for Foam cobordisms. The talk will the discuss some properties of these invariants and some computational tools. A particular variant of this story might be applicable lead to a non-computer aided proof of the four color theorem.

Thursday, September 8, 2016

12:00 pm in 243 Altgeld Hall,Thursday, September 8, 2016

#### Floer homology, group orders, and taut foliations of hyperbolic 3-manifolds

###### Nathan Dunfield (U of I Math)

Abstract: A bold conjecture of Boyer-Gorden-Watson and others posit that for any irreducible rational homology 3-sphere M the following three conditions are equivalent: (1) the fundamental group of M is left-orderable, (2) M has non-minimal Heegaard Floer homology, and (3) M admits a co-orientable taut foliation. Very recently, this conjecture was established for all graph manifolds by the combined work of Boyer-Clay and Hanselman-Rasmussen-Rasmussen-Watson. I will discuss a computational survey of these properties involving several hundred thousand hyperbolic 3-manifolds.

Tuesday, September 20, 2016

12:00 pm in 243 Altgeld Hall,Tuesday, September 20, 2016

#### Dynamics of free group automorphisms and a subgroup alternative for Out(F_N)

###### Caglar Uyanik (Illinois Math)

Abstract: The study of outer automorphism group of a free group is closely related to the study of mapping class groups of hyperbolic surfaces. In this talk, I will draw analogies between these two groups, and deduce several structural results about subgroups of Out(F_N) by proving several north-south dynamics type results. Part of this talk is based on joint work with Matt Clay.

Tuesday, October 11, 2016

12:00 pm in 243 Altgeld Hall,Tuesday, October 11, 2016

#### Hypersurfaces with Central Convex Cross-Sections

###### Alper Gur (Indiana U Math)

Abstract: The compact transverse cross-sections of a cylinder over a central ovaloid in $$\bf R^n$$, $$n \geq 3$$, with hyperplanes are central ovaloids. A similar result holds for quadrics (level sets of quadratic polynomials in $$\bf R^n$$, $$n \geq 3$$ ). Their compact transverse cross-sections with hyperplanes are ellipsoids, which are central ovaloids.
$$\quad$$In $$\bf R^3$$, Blaschke, Brunn, and Olovjanischnikoff found results for compact convex surfaces that motivated B. Solomon to prove that these two kinds of examples provide the only complete, connected, smooth surfaces in $$\bf R^3$$, whose ovaloid crosssections are central. We generalize that result to all higher dimensions, proving: If $$M^{n-1} \subseteq \bf R^n$$, $$n \geq 4$$, is a complete, connected, smooth hypersurface, which intersects at least one hyperplane transversally along an ovaloid, and every such ovaloid on $$M$$ is central, then $$M$$ is either a cylinder over a central ovaloid or a quadric.

Thursday, November 10, 2016

12:00 pm in Altgeld Hall 243,Thursday, November 10, 2016

#### TALK CANCELLED DUE TO TRAVEL COMPLICATIONS

###### Ursula Hamenstädt (University of Bonn)

Tuesday, November 15, 2016

12:00 pm in 243 Altgeld Hall,Tuesday, November 15, 2016

#### Polynomial-time Nielsen--Thurston classification

###### Mark Bell (Illinois Math)

Abstract: We will discuss a new polynomial-time algorithm, joint with Richard Webb, for computing the Nielsen--Thurston type of a mapping class. The procedure works by considering the maps action on the curve graph. To be able to compute this action, we need to be able to construct geodesics through the curve graph. However, this graph is locally infinite and so standard pathfinding algorithms struggle. We will discuss a new refinement of the techniques of Leasure, Shackleton, Watanabe and Webb for overcoming this local infiniteness that allows such geodesics to be found in polynomial time.

Thursday, November 17, 2016

12:00 pm in Altgeld Hall 243,Thursday, November 17, 2016

#### The horofunction boundary of the lamplighter group

###### Gregory Kelsey (Bellarmine University)

Abstract: The horofunction boundary of a metric space is defined by embedding it in the space of continuous functions. This boundary frequently behaves more nicely than the visual boundary for spaces that are neither hyperbolic nor CAT(0). The lamplighter group is an amenable group with exponential growth that is neither hyperbolic nor CAT(0) and has provided many interesting examples and counterexamples in group theory. In joint work with Keith Jones, we fully describe the horofunction boundary of the lamplighter group with the word metric arising from its finite state automaton generating set.

Tuesday, November 29, 2016

12:00 pm in 243 Altgeld Hall,Tuesday, November 29, 2016

#### The Symplectic Geometry of Polygon Space and How to Use It

###### Clayton Shonkwiler (Colorado State University)

Abstract: In statistical physics, the basic (and highly idealized) model of a ring polymer is a closed random flight in 3-space with equal-length steps, often called a random equilateral polygon. In this talk, I will describe the moduli space of random equilateral polygons, giving a sense of how this fits into a larger symplectic and algebraic geometric story. In particular, the space of equilateral n-gons turns out to (almost) be a toric symplectic manifold, yielding a (nearly) global coordinate system. These coordinates are powerful tools both for proving theorems and for developing numerical techniques, some of which I will describe, including a very fast and surprisingly simple algorithm for directly sampling random polygons recently developed with Jason Cantarella (University of Georgia), Bertrand Duplantier (CEA/Saclay), and Erica Uehara (Ochanomizu University).