Department of

# Mathematics

Seminar Calendar
for GEAR events the year of Monday, January 26, 2015.

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events for the
events containing

Questions regarding events or the calendar should be directed to Tori Corkery.
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Tuesday, January 20, 2015

1:00 pm in Altgeld Hall 243,Tuesday, January 20, 2015

#### Thin Monodromy Groups

###### Elena Fuchs (UIUC Math)

Abstract: In recent years, it has become interesting from a number-theoretic point of view to be able to determine whether a finitely generated subgroup of $GL_n(\mathbb Z)$ is a so-called thin group. In general, little is known as to how to approach this question. In this talk we discuss this question in the case of hypergeometric monodromy groups, which were studied in detail by Beukers and Heckman in 1989. We will convey what is known, explain some of the difficulties in answering the thinness question, and show how one can successfully answer it in many cases where the group in question acts on hyperbolic space. This work is joint with Meiri and Sarnak. View talk at http://youtu.be/BJg-UlUr0w8

Thursday, January 22, 2015

1:00 pm in 243 Altgeld Hall,Thursday, January 22, 2015

#### Constructing surface homeomorphisms with given stretching factors

Abstract: Homeomorphisms from a compact surface to itself were classified by Thurston, and he associated to each such map an algebraic integer, called the dilatation - or the stretching factor. The question of which numbers can be realized as the dilatation of a pseudo-Anosov surface homeomorphism has a long history. A well-known necessary condition is that the number must be strictly greater in absolute value than all its Galois conjugates. We give a sufficient conditions for an algebraic number to be a pseudo-Anosov dilatation of a compact surface and describe an explicit construction of the surface and the map when this condition is met. View talk at http://youtu.be/EjBDwx3RvyE

Tuesday, January 27, 2015

1:00 pm in Altgeld Hall 243,Tuesday, January 27, 2015

#### Fully irreducible Automorphisms of the Free Group via Dehn twisting in $\sharp_k(S^2 \times S^1)$

###### Funda Gultepe (UIUC Math)

Abstract: By using a notion of a geometric Dehn twist in $\sharp_k(S^2 \times S^1)$, we prove that when projections of two $Z$-splittings to the free factor complex are far enough from each other in the free factor complex, Dehn twist automorphisms corresponding to the $Z$ splittings generate a free group of rank $2$. Moreover, every element from this free group is either conjugate to a power of one of the Dehn twists or it is a fully irreducible outer automorphism of the free group. View lecture at http://youtu.be/9nSWL4g19lk

Thursday, January 29, 2015

1:00 pm in Altgeld Hall 243,Thursday, January 29, 2015

#### Certifying the Thurston norm via twisted homology

###### Nathan Dunfield (UIUC Math)

Abstract: From the very beginning of 3-manifold topology, a fundamental task has been to find the simplest surface in a given 2-dimensional homology class, e.g. the Seifert genus of a knot in the 3-sphere. The behavior of the minimal topological complexity as the homology class varies is encapsulated in the Thurston norm. In this talk, I will discuss tools for proving that a particular surface has minimal genus. These are generalizations of the classical Alexander polynomial, but are defined using homology with coefficients twisted by some finite-dimensional representation of the fundamental group of the manifold. I will discuss recent work with Ian Agol on situations where using representations coming from hyperbolic geometry suffices to provide such certificates. If time permits, I will sketch how all of this relates to fundamental questions about the computational complexity of finding the Thurston norm. Only basic facts about manifolds and homology will be assumed. View lecture at http://youtu.be/b7a-WXpNLAQ

Tuesday, February 3, 2015

1:00 pm in 243 Altgeld Hall,Tuesday, February 3, 2015

#### Pseudo-Anosov mapping classes not arising from Penner's construction

###### Hyunshik Shin (UIC Math)

Abstract: We show that Galois conjugates of stretch factors of pseudo-Anosov mapping classes arising from Penner’s construction lie off the unit circle. As a consequence, we show that for all but a few exceptional surfaces, there are examples of pseudo-Anosov mapping classes so that no power of them arises from Penner’s construction. This resolves a conjecture of Penner. View talk at http://youtu.be/lL94FzESags

Thursday, February 5, 2015

1:00 pm in Altgeld Hall 243,Thursday, February 5, 2015

#### The primitivity index function for a free group, and untangling closed curves on surfaces

###### Neha Gupta (UIUC Math)

Abstract: A theorem of Scott shows that any closed geodesic on a surface lifts to an embedded loop in a finite cover. Our motivation is to find a worst-case lower bound for the degree of this cover, in terms of the length of the original loop. We establish, via probabilistic methods, lower bounds for certain analogous functions, like the Primitivity Index Function and the Simplicity Index Function, in a free group. These lower bounds, when applied in a suitable way to the surface case, give us some lower bounds for our motivating question. This is joint work with Ilya Kapovich. View talk at http://youtu.be/-RS1SNhj3yA

Tuesday, February 10, 2015

1:00 pm in Altgeld Hall 243,Tuesday, February 10, 2015

#### Around Bochner-Krall problem

###### Boris Shapiro (Stockholm University)

Abstract: A linear differential operator $T=\sum_{i=1}^k Q_i(x) \frac{d^i}{dx^i}$ with polynomials coefficients is called {\it exactly solvable} if the degree of each $Q_i(x)$ is at most $i$ and there exists all least one value of $i$ for which the equality holds. One can easily see that such operators are have one eigenpolynomial in every sufficiently large degree. Already in 1929 S.Bochner asked for which exactly solvable operators the corresponding sequence of eigenpolynomials consists of orthogonal polynomials. This problem was considered in large numbers of publications over many decades. I will present modern results about the root asymptotics for sequences of eigenpolynomials of exactly solvable operators, several conjectures and the relation of the latter root asymptotics to quadratic and higher order differentials in the complex plane. View talk at http://youtu.be/zjag_IS6Gos

Thursday, February 12, 2015

1:00 pm in Altgeld Hall 243,Thursday, February 12, 2015

#### A Liouville theorem on Kaehler Ricci flat metric in $C^n$ with conical singularities along $\{0\}\times C^{n-1}$

###### Xiuxiong Chen (Stony Brook)

Abstract: We prove a new Liouville type theorem which goes back to E. Calabi and Pogorelov. The theorem of Calabi and Pogrelov can be formulated as follows: In C^n, any Kaehler Ricci flat metric which depends on R^n (i.e., admitting standard toric symmetry) must be trivial. We prove that, in C^n, any Kaehler Ricci flat metric which has conical singularity along {0}\times C^{n-1} and is quasi conformal to a standard flat conical metric must be trivial. This has important application in conical Kaehler geometry (such as regularity for conical Kaehler Einstein metrics). Joint with Yuanqi Wang. View talk at http://youtu.be/ejmhtgPY49c

Tuesday, February 17, 2015

1:00 pm in Altgeld Hall 243,Tuesday, February 17, 2015

#### A mapping class group invariant parameterization of maximal $Sp(4,\mathbb{R})$ representations

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

Abstract: Let $S$ be a closed surface of genus at least 2, and consider the moduli space of representations $\rho:\pi_1(S)\rightarrow Sp(4,\mathbb{R}).$ There is an invariant $\tau\in\mathbb{Z},$ called the Toledo invariant, which helps to distinguish connected components. The Toledo invariant satisfies a Milnor-Wood inequality $|\tau|\leq 2g-2.$ Representations with maximal Toledo invariant have many geometrically interesting properties, for instance, they are all discrete and faithful. In this talk, we will give a mapping class group invariant parameterization of all smooth connected components of the maximal $Sp(4,\mathbb{R})$ representations. Our main tool is Higgs bundles. However, to utilize Higgs bundle techniques, one has to fix a conformal structure of the surface $S,$ hence breaking the mapping class group symmetry. To restore the symmetry, we associate a unique preferred' conformal structure to each such representation. This is done by exploiting the relationship between the associated Higgs bundles and minimal surfaces. View talk at http://youtu.be/lhJhEI1FGb8

Thursday, February 19, 2015

1:00 pm in Altgeld Hall 243,Thursday, February 19, 2015

#### The Tits alternative for the automorphism group of a free product

###### Camille Horbez (Rennes)

Abstract: A group $G$ is said to satisfy the Tits alternative if every subgroup of $G$ either contains a nonabelian free subgroup, or is virtually solvable. The talk will aim at presenting a version of this alternative for the automorphism group of a free product of groups. A classical theorem of Grushko states that every finitely generated group $G$ splits as a free product of the form $G_1*...*G_k*F_N$, where $F_N$ is a finitely generated free group, and all $G_i$ are nontrivial, non isomorphic to $Z$, and freely indecomposable. In this situation, I prove that if all groups $G_i$ and $Out(G_i)$ satisfy the Tits alternative, then so does the group $Out(G)$ of outer automorphisms of G. I will present applications to proving the Tits alternative for outer automorphism groups of right-angled Artin groups, or of some classes of relatively hyperbolic groups. I will then present a proof of this theorem, in parallel to a new proof of the Tits alternative for mapping class groups of compact surfaces. The proof relies on a study of the actions of some subgroups of $Out(G)$ on a version of the outer space for free products, and on a hyperbolic simplicial graph. View talk at http://youtu.be/5AP1g7_8MCs

Tuesday, February 24, 2015

1:00 pm in 243 Altgeld Hall,Tuesday, February 24, 2015

#### Planes, Trains & Automorphisms

###### Mark Bell (University of Warwick)

Abstract: We will discuss a new approach using train tracks to solve the conjugacy problem in the automorphism group of a punctured plane. This solution relies on the action of the mapping class group on the space of measured laminations and has several connections to veering triangulations of fibred 3--manifolds. Unlike techniques in braid groups using Garside structures, not only is this algorithm is effective it also generalises to all higher genus surfaces.

Thursday, February 26, 2015

1:00 pm in Altgeld Hall 243,Thursday, February 26, 2015

#### Higher rank geometric structures

###### Andy Zimmer (University of Chicago)

Abstract: Suppose G is a Lie group acting transitively on a manifold X. Then a (G,X)-structure on a manifold M is a collection of coordinate charts whose images are in X and whose transition functions are in G. Some classic examples include the locally symmetric spaces, real projective manifolds, complex projective manifolds, and affine manifolds. In this talk we will consider (G,X)-structures where X is a compact G-space having higher rank'' in the sense that G does not act two-transitively on X. For instance, the complete flag variety of a real vector space with dimension at least two is a higher rank $SL_d(\mathbb{R})$-space. We will show that a higher rank geometric structure on a manifold restricts the fundamental group. In particular, a manifold with Gromov hyperbolic fundamental group cannot have a proper (G,X)-structure when $G=SL_d(\mathbb{R})$ and X is a higher rank compact G-space.

Tuesday, March 3, 2015

1:00 pm in Altgeld Hall 243,Tuesday, March 3, 2015

#### Dense geodesic rays in the quotient of Outer space

###### Catherine Pfaff (Bielefeld)

Abstract: In 1981 Masur proved the existence of a dense Teichmueller geodesic in moduli space. As some form of analogue, we construct dense geodesic rays in certain subcomplexes of the $Out(F_r)$ quotient of outer space. This is joint work in progress with Yael Algom-Kfir.

Thursday, March 5, 2015

1:00 pm in 243 Altgeld Hall,Thursday, March 5, 2015

#### Homology and dynamics of pseudo-Anosovs

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

Abstract: I'll explain a connection between a pseudo-Anosov homeomorphism's stretch factor, and its action on homology. This provides a kind of interpolation between a result of Penner and our prior work with Farb and Margalit, and answers a question of Ellenberg. This is joint work with Agol and Margalit.

Tuesday, March 10, 2015

1:00 pm in Altgeld Hall 243,Tuesday, March 10, 2015

#### How can you have dynamics when all you have is a category?

###### James Pascaleff (UIUC Math)

Abstract: One of the main objects in symplectic topology is the Fukaya Category F(M) of a symplectic manifold M, whose objects are Lagrangian submanifolds. It has recently been advocated (most strongly by P. Seidel), that interesting results in symplectic topology can be obtained by considering "continuous symmetries" of F(M). These symmetries are not related to any kind of group action on the manifold M, but are "hidden" in the category itself. The key (if imprecise) phrase is "dynamics of a vector field on the moduli space of objects in F(M)." The linearization of the vector field around a fixed point gives rise to the notion of an equivariant Lagrangian submanifold, and the closed orbits are also of interest for applications. This talk will not assume prior exposure to the Fukaya category.

Thursday, March 12, 2015

1:00 pm in Altgeld Hall 243,Thursday, March 12, 2015

#### Word Maps and Measure Preservation

###### Doron Puder (IAS Princeton)

Abstract: We establish new characterizations of primitive elements and free factors in free groups, which are based on the distributions they induce on finite groups. More specifically, for every finite group G, a word w in the free group on k generators induces a word map from G^k to G. We say that w is measure preserving with respect to G if given uniform distribution on G^k, the image of this word map distributes uniformly on G. It is easy to see that primitive words (words which belong to some basis of the free group) are measure preserving w.r.t. all finite groups, and several authors have conjectured that the two properties are, in fact, equivalent. In a joint work with O. Parzanchevski, we prove this conjecture.

Tuesday, March 17, 2015

1:00 pm in 243 Altgeld Hall,Tuesday, March 17, 2015

#### Th singular fibers of the Hitchin map

###### Andre Oliveira (University of Trás-os-Montes e Alto Douro, Vila Real, Portugal.)

Abstract: The Hitchin map defines a remarkable fibration on the moduli space of Higgs bundles (closely related with moduli spaces of surface group representations). The generic fibers of this map are well known, but the same is not true for the singular ones. After sketching the main features of the fibration we study these singular fibers (in rank 2) and show that they are connected. This is joint work with Peter Gothen.

Thursday, March 19, 2015

1:00 pm in Altgeld Hall 243,Thursday, March 19, 2015

#### On the Jones subgroup of Thompson's group $F$

###### Mark Sapir (Vanderbilt University)

Abstract: This is a joint work with Gili Golan. Recently Vaughan Jones showed that Thompson's group $F$ encodes in a natural way all knots and links in $\mathbb R^3$, and a certain subgroup $\overrightarrow F$ of $F$ encodes all oriented knots. We answer several questions of Jones about $\overrightarrow F$. In particular we prove that the subgroup $\overrightarrow F$ is generated by $x_0x_1, x_1x_2, x_2x_3$ (where $x_i,i\in \mathbb N$ are the standard generators of $F$) and is isomorphic to $F_3$, the analog of $F$ where all slopes are powers of $3$ and break points are $3$-adic rationals. We also show that $\overrightarrow F$ coincides with its commensurator. Hence the linearization of the permutational representation of $F$ on $F/\overrightarrow F$ is irreducible. Finally we show how to replace $3$ in the above results by an arbitrary $n$, and to construct a series of irreducible representations of $F$ defined in a similar way.

Tuesday, March 31, 2015

1:00 pm in 243 Altgeld Hall,Tuesday, March 31, 2015

#### Approximating codimension one foliations of 3-manifolds

###### Rachel Roberts (Washington University)

Abstract: Eliashberg and Thurston proved that any smooth taut co-oriented foliation of a closed, orientable 3-manifold can be approximated by a pair of contact structures, one positive and one negative. These contact structures are weakly symplectically fillable and universally tight. This result has been used both to establish the fillability of certain contact structures and to prove the nonexistence of taut foliations in certain 3-manifolds. Sometimes this has been done without consideration of the smoothness of the foliations under consideration, resulting in proof gaps. I'll discuss 3-manifolds and 2-plane fields on 3-manifolds. I'll then focus in on the statement of the Elisahberg-Thurston theorem, defining all terms, and give an overview of its proof. As time permits, I will describe how to generalize the Eliashberg-Thurston theorem to continuous foliations. This work is joint with Will Kazez.

Thursday, April 2, 2015

1:00 pm in Altgeld Hall 243,Thursday, April 2, 2015

#### Complexity of finitely generated residually finite groups

###### Alexei Myasnikov (Stevens Institute of Technology)

Abstract: Finitely presented residually finite groups are usually thought of as nice from the algorithmic view-point, in particular, they have decidable word problem. In this talk I will address the following general questions for such groups G: how large could be the Dehn function of G? How large could be the gap between the complexity of the word problem and the Dehn functions of G? What is the time complexity of the classical McKinsey algorithm for the word problem in G (this is the only known uniform algorithm for the word problem in such groups)? How large could be the depth functions in G? The depth function measures how deep one has to go into finite index subgroups to separate a non-trivial element of a given length in G from the identity. These are joint results with O. Kharlampovich and M. Sapir. I will also discuss finitely generated recursively presented residually finite groups with really strange algorithmic properties, so called Dehn monsters. To build them we need Golod-Shafarevich construction and a forcing-type argument from logic. This is based on joint results with D. Osin and B. Khoussainov.

Tuesday, April 7, 2015

1:00 pm in Altgeld Hall 243,Tuesday, April 7, 2015

#### Marked length rigidity for NPC Euclidean cone metrics

###### Christopher J. Leininger (UIUC Math)

Abstract: Otal proved that for negatively curved Riemannian metrics on compact surfaces, the marked length spectrum---the function which assigns the length of the geodesic representative to each homotopy class of curves---determines the metric up to isometry homotopic to the identity. This was extended to nonpositively curved (NPC) Riemannian metrics by Croke-Fathi-Feldman, and to negatively curved cone metrics by Hersonsky-Paulin. In his thesis, Frazier considered the case of NPC Euclidean cone metrics, and showed that the marked length spectrum distinguishes such metrics from the classes above, but was unable to prove that they could be distinguished by such from each other. In joint work with Anja Bankovic, we prove that NPC Euclidean cone metrics are determined by their marked length spectrum. From the proof, we conjecture that they are (almost) determined by a much coarser invariant, namely the support of the associated Liouville current. I'll explain all the terms and sketch the relatively short proof.

Thursday, April 9, 2015

1:00 pm in Altgeld Hall 243,Thursday, April 9, 2015

#### Clairaut's theorem and potential mechanics on metric spaces

###### Richard Bishop (UIUC Math)

Abstract: For surfaces of revolution Clairaut's theorem gives a first integral for geodesics: $r \cos\theta =$ constant, where $r$ is the distance from the axis to the profile curve and $\theta$ is the angle the geodesic makes with the latitude circles. We have generalized this to warped products $W = B\times_fF$ of metric spaces: along any geodesic $\gamma$ in $W$, $f^2v = b$ is constant, where $v$ is the speed of the projection of $\gamma$ to $F$. When $B, F$ are Riemannian manifolds, the geodesic equations have a known form: $$\gamma_B'' = c(1/f^3) {\rm grad} f, \qquad (f^2v)' =0,$$ where $\gamma_B$ is the projection to $B$. This has the interpretation that $\gamma_B$ is a trajectory of the potential function $U = c/2f^2$. The fact that the speed of $\gamma$ is a constant $a = \sqrt{b/c}$ becomes the law of conservation of energy $u^2 + 2U = (b/c)^2$, where $u$ is the speed of $\gamma_B$. Hence, for more general metric spaces $B$, Clairaut's theorem makes it reasonable to interpret the projections of geodesics from a warped product $B\times_fF$ to $B$ as the trajectories of the potential function $U = 1/2f^2 :B \to {\bf R}$. Since we also have shown that these trajectories are independent of the choice of $F$, we can simply take $F$ to be the line or a circle. Joint work with Stephanie Alexander

Tuesday, April 14, 2015

1:00 pm in Altgeld Hall 243,Tuesday, April 14, 2015

#### Stability in mapping class groups and right-angled Artin groups

###### Samuel Taylor (Yale University)

Abstract: A well studied question in surface topology asks whether every purely pseudo-Anosov subgroup of the mapping class group is convex cocompact. This question can be reformulated in a way which references only the geometric structure of the mapping class group using a strong form of quasiconvexity called stability. In joint work with Thomas Koberda and Johanna Mangahas, we recently gave a complete characterization of stable subgroups of right-angled Artin groups (RAAGs), thus answering the RAAG analog of the question above. In particular, we show that any finitely generated subgroup of a RAAG all of whose nontrivial elements have cyclic centralizer is stable and, in particular, quasiconvex. In this talk, I will introduce the general notion of stability, explain its importance in RAAGs, and give some applications of our theorem.

Thursday, April 16, 2015

1:00 pm in Altgeld Hall 243,Thursday, April 16, 2015

#### Boundaries of (some) $Out(F_n)$-complexes

###### Mladen Bestvina (University of Utah)

Abstract: It has recently been shown that Out(F_n) acts on several interesting hyperbolic complexes in analogy with mapping class groups acting on the curve (or arc) complex. In the talk I will try to describe what the boundaries of two of these (free factor and splitting) complexes look like. The answer is modelled on Klarreich's description of the boundary of the curve complex as the space of ending laminations. This is joint work with Reynolds and with Feighn-Reynolds (in progress) respectively.

Tuesday, April 21, 2015

1:00 pm in Altgeld Hall 243,Tuesday, April 21, 2015

#### Homogeneous dynamics and applications to number theory

Abstract: To set the stage for Margulis' Tondeur lectures, I will survey some problems in homogeneous dynamics and their relationship to number theory. I will focus on the Oppenheim and Littlewood conjectures.

Tuesday, April 28, 2015

1:00 pm in Altgeld Hall 243,Tuesday, April 28, 2015

#### An effective asymptotic result for the Lebesgue measure of the sum-level sets for continued fractions

###### Byron Heersink (UIUC Math)

Abstract: For every positive integer $n$, let $C_n$ be the set of real numbers in $[0,1]$ whose continued fraction expansion $[a_1,a_2,...]$ satisfies $a_1+...+a_k=n$ for some $k$. Using results from infinite ergodic theory, Kessebohmer and Stratmann proved that the Lebesgue measure of $C_n$ is asymptotically equivalent to $1/\log_2 n$ as $n$ approaches $\infty$. In this talk, we provide a simplified proof of this result, mostly using basic properties of the transfer operator of the Farey map and Karamata's Tauberian theorem, while avoiding most of the ergodic results in the proof of Kessebohmer and Stratmann. Additionally, we obtain an error term by adapting Freud's effective version of Karamata's theorem to this situation.

Thursday, April 30, 2015

1:00 pm in Altgeld Hall 243,Thursday, April 30, 2015

#### A new continuity flow of Monge-Ampere type, Gromov-Hausdorff limits and Minimal Model program

###### Gabriel La Nave (UIUC Math)

Abstract: I will describe a new continuity equation recently introduced by G.Tian and myself and it's connection with then Minimal Model Program. In particular, in joint work with Tian and Z. Zhang, we show that the Gromov-Hausdorff of our continuity method are homeomorphic to a normal projective variety and that the singularities of the metric are concentrated on a sub variety , under some natural assumptions --which are the natural differential geometric versions of the assumptions in Kawamata's base point free theorem

Thursday, May 7, 2015

1:00 pm in Altgeld Hall 243,Thursday, May 7, 2015

#### Simultaneous dense and nondense orbits

###### Jimmy Tseng (University of Bristol)

Abstract: We consider pairs of maps on (usually) the same phase space and, in particular, examine pairs for which many points have drastically different orbit structures. Our main example is a pair of commuting automorphisms of the d-torus, for which the set of points with dense orbit under one map and nondense orbit under the other has full Hausdorff dimension. Two other examples that we only very briefly mention are two linearly independent elements of the Cartan action on compact higher rank homogeneous spaces and the multiplication-by-n map on the circle and the geodesic flow under the induced map on the circle corresponding to the expanding horospherical subgroup. The last result is an example for which the phase spaces are not the same (because the geodesic flow acts on the space of unimodular lattices) but, nevertheless, it allows us to obtain a counterpart to a classical result of R. Kaufman in Diophantine approximation. This talk is based on my joint work with V. Bergelson and M. Einsiedler and my other recent joint work with R. Shi.

Thursday, August 27, 2015

1:00 pm in 243 Altgeld Hall,Thursday, August 27, 2015

#### A tale of two norms

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

Abstract: Abstract: The first cohomology of a hyperbolic 3-manifold has two natural norms: the Thurston norm, which measure topological complexity of surfaces representing the dual homology class, and the harmonic norm, which is just the L^2 norm on the corresponding space of harmonic 1-forms. Bergeron-Sengun-Venkatesh recently showed that these two norms are closely related, at least when the injectivity radius is bounded below. Their work was motivated by the connection of the harmonic norm to the Ray-Singer analytic torsion and issues of torsion growth in homology of towers of finite covers. After carefully introducing both norms, I will discuss new results that refine and clarify the precise relationship between them; a key tool here will be a third norm based on least-area surfaces. This is joint work with Jeff Brock and will feature some pretty pictures that are joint work with Anil Hirani. View talk at https://youtu.be/29cE5c1F04k

Tuesday, September 1, 2015

12:00 pm in 345 Altgeld Hall,Tuesday, September 1, 2015

#### Recognizing three-manifolds

###### Saul Schleimer (University of Warwick)

Abstract: To the eyes of a topologist manifolds have no local properties: every point has a small neighborhood that looks like euclidean space. Accordingly, as initiated by Poincaré, the classification of manifolds is one of the central problems in topology. The "homeomorphism problem" is somewhat easier: given a pair of manifolds, we are asked to decide if they are homeomorphic. These problems are solved for zero-, one-, and two-manifolds. Even better, the solutions are "effective": there are complete topological invariants that we can compute in polynomial time. On the other hand, in dimensions four and higher the homeomorphism problem is logically undecidable. This leaves the provocative third dimension. Work of Haken, Rubenstein, Casson, Manning, Perelman, and others shows that these problems are decidable. Sometimes we can do better: for example, if one of the manifolds is the three-sphere then the homeomorphism problem lies in the complexity class NP. In joint work with Marc Lackenby, we show that recognizing spherical space forms also lies in NP. If time permits, we'll discuss the standing of the other seven Thurston geometries. View talk at https://youtu.be/4nu_dSCZ5V8

Thursday, September 3, 2015

1:00 pm in Altgeld Hall 243,Thursday, September 3, 2015

#### On purely loxodromic actions

###### Ilya Kapovich (UIUC Math)

Abstract: Purely loxodromic isometric actions of finitely generated groups on Gromov-hyperbolic spaces (that is, actions where every element of infinite order in the group acts as a loxodromic isometry) appear naturally in many contexts, such as in the theories of hyperbolic, relatively hyperbolic and acylindrically hyperbolic groups, the study of convex cocompact subgroups of mapping class groups, etc. Often one needs to consider non-proper actions, and in that context the notion of an acylindrical action serves as an important substitute of being proper. We construct an example of an isometric action of the free group $F(a,b)$ on a $\delta$--hyperbolic graph $Y$, such that this action is acylindrical, free, purely loxodromic, has asymptotic translation lengths of nontrivial elements of $F(a,b)$ separated away from $0$, has quasiconvex orbits in $Y$, but such that the orbit map $F(a,b)\to Y$ is not a quasi-isometric embedding. View talk at https://youtu.be/y5eXM5Nst0U

Tuesday, September 8, 2015

12:00 pm in 345 Altgeld Hall,Tuesday, September 8, 2015

#### Arithmetic progressions in the primitive length spectrum

###### Nicholas Miller (Purdue University)

Abstract: The length spectrum, i.e. the collection of all lengths of closed geodesics on a hyperbolic manifold, has drawn much attention over the last few decades. Of particular interest has been the question of whether the length spectrum determines the commensurability class of such a manifold. There have also been a host of prime geodesic theorems displaying a surprising analogy between the behavior of primitive, closed geodesics on hyperbolic manifolds and the behavior of the prime numbers in the integers. For instance, just as the prime number theorem dictates the asymptotic growth of primes less than n, there is an analogous asymptotic for primitive, closed geodesics of length less n. In this talk, I will review some basics on the length spectrum and survey some existing results exhibiting this connection. I will then go on to discuss some recent work on arithmetic progressions in the primitive length spectrum extending this relationship. View talk at https://youtu.be/ymQbUWDyPZo

Thursday, September 10, 2015

1:00 pm in 243 Altgeld Hall,Thursday, September 10, 2015

#### Some statistical properties of digits in continued fractions

###### Florin Boca (UIUC Math)

Abstract: Specifically, we will discuss results on generalized Gauss-Kuzmin statistics and on the distribution of partial sums of digits of a random irrational number.

Tuesday, September 15, 2015

12:00 pm in 345 Altgeld Hall,Tuesday, September 15, 2015

#### Random walks and random group extensions

###### Giulio Tiozzo (Yale University)

Abstract: Let us consider a group G of isometries of a delta-hyperbolic metric space X, which is not necessarily proper (e.g. it could be a locally infinite graph). We can define a random walk by picking random products of elements of G, and projecting this sample path to X. We show that such a random walk converges almost surely to the Gromov boundary of X, and with positive speed. As an application, we prove that a random k-generated subgroup of the mapping class group is convex cocompact, and a similar statement holds for Out(F_n). This is joint work, partially with J. Maher and partially with S. Taylor.

Thursday, September 17, 2015

1:00 pm in 243 Altgeld Hall,Thursday, September 17, 2015

#### Null distance on a spacetime

###### Carlos Vega (Saint Louis University)

Abstract: Given any time function on a spacetime M, we define an induced null distance' function, built from and closely related to the causal structure of M. This null distance' is a conformal pseudometric in general, but is positive-definite under natural conditions. Further, in basic model cases, the causal structure is encoded completely in the resulting metric space. In the cosmological setting, a canonical choice of time function was introduced and studied by Andersson, Galloway, and Howard. We show that under their basic niceness' condition, the induced null distance is definite, and hence provides a uniform way of encoding big bang' and related spacetimes as metric spaces. This is joint work with Christina Sormani.

Tuesday, September 22, 2015

12:00 pm in 345 Altgeld Hall,Tuesday, September 22, 2015

#### 4-manifolds can be surface bundles over surfaces in many ways

###### Nick Salter (University of Chicago)

Abstract: An essential feature of the theory of 3-manifolds fibering over the circle is that they often admit infinitely many distinct structures as a surface bundle. In four dimensions, the situation is much more rigid: a given 4-manifold admits only finitely many fiberings as a surface bundle over a surface. But how many is “finitely many”? Can a 4-manifold possess three or more distinct surface bundle structures? In this talk, we will survey some of the beautiful classical examples of surface bundles over surfaces with multiple fiberings, and discuss some of our own work. This includes a rigidity result showing that a class of surface bundles have no second fiberings whatsoever, as well as the first example of a 4-manifold admitting three distinct surface bundle structures, and our progress on an asymptotic version of the “how many?” question. Time permitting, we will discuss some connections with the homology of the Torelli group, (non)-realization problems a la Nielsen and Morita, and symplectic topology.

Thursday, September 24, 2015

1:00 pm in 243 Altgeld Hall,Thursday, September 24, 2015

#### Poisson manifolds of compact types

###### Rui Loja Fernandes (UIUC Math)

Abstract: In this talk I will describe a class of Poisson manifolds which play the role of "compact objects" in Poisson geometry. These Poisson structures of compact type are very rigid compared to general Poisson structures: the symplectic foliation has a transverse integral affine structure, its leaf space is an orbifold with a Duistermaat-Eckman type measure, and they have an associated symplectic gerbe, that obstructs realizing them as isotropic quotients of a symplectic manifold. In this talk I will carefully define these objects and give a small tour of their rich geometry. Joint work with M. Crainic (Utrecht) and D. Martinez-Torres (PUC-Rio de Janeiro).

Tuesday, September 29, 2015

12:00 pm in 345 Altgeld Hall,Tuesday, September 29, 2015

#### Finding paths in graphs of triangulations

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

Abstract: There are many different ways of triangulating a surface using n arcs. As some triangulations are more similar than others, we get a natural topology on the space of triangulations which can be seen as an infinite graph; where two triangulations are connected if and only if they share (n-1) edges. This graph provides a combinatorial model for the surfaces mapping class group as it acts geometrically. We will look at some techniques for efficiently finding paths through this graph, allowing us to efficiently represent and compute with mapping classes.

Thursday, October 1, 2015

1:00 pm in Altgeld Hall 243,Thursday, October 1, 2015

#### Probability groups

###### Anush Tserunyan (UIUC Math)

Abstract: I will introduce a class of groups equipped with an invariant probability measure that respects the group structure in an appropriate sense; call such groups probability groups. This class contains all compact groups and is closed under taking ultraproducts with the induced Loeb measure. I will discuss the use of probability groups as a potential alternative to Furstenberg's correspondence principle. As an example, I will define a notion of mixing for probability groups and mention a double recurrence result for mixing probability groups that generalizes a theorem of Bergelson-Tao proved for ultra quasirandom groups, nevertheless having a considerably shorter proof.

Tuesday, October 6, 2015

12:00 pm in 345 Altgeld Hall,Tuesday, October 6, 2015

#### Combinatorial rigidity of the arc complex

###### Valentina Disarlo (Indiana Math)

Abstract: We study the arc complex of a surface with marked points in the interior and on the boundary. We prove that the isomorphism type of the arc complex determines the topology of the underlying surface, and that in all but a few cases every automorphism is induced by a homeomorphism of the surface. This generalizes a result of Irmak - McCarthy. As an application we deduce some rigidity results for the Fomin-Shapiro-Thurston cluster algebra associated to a surface. Our proofs do not employ any known simplicial rigidity result.

Thursday, October 8, 2015

1:00 pm in 243 Altgeld Hall,Thursday, October 8, 2015

#### Shadows of Teichmueller discs in the curve graph

###### Robert Tang (Oklahoma Math)

Abstract: A Teichmueller disc parameterises the family of metrics obtained by performing SL(2,R)-deformations on a given flat surface. We consider several natural sets of curves associated with a Teichmueller disc from the point of view of the curve graph. We show that these sets agree up to uniform Hausdorff distance, and are all quasiconvex. Furthermore, we extend the notion of balance time along Teichmueller geodesics to Teichmueller discs, and show that it satisfies analogous projection properties to the curve graph. This talk will focus on the tools used to prove the above results. This is a joint work with Richard Webb.

Tuesday, October 13, 2015

12:00 pm in 345 Altgeld Hall,Tuesday, October 13, 2015

#### The topology of local commensurability graphs

###### Khalid Bou-Rabee (City College of CUNY)

Abstract: The p-local commensurability graph of a group has vertices consisting of all finite-index subgroups, where an edge is drawn between two subgroups if their commensurability index is a power of p. While commensurability is a fundamental notion, these graphs also have ties to subgroup growth, another natural invariant. What group-theoretic information can we draw from the topology of these graphs? To initiate the study of this question, we explore these graphs for a number of examples and share some of our findings. It turns out that any such graph for a group with all nilpotent finite quotients is complete. Further, this topological criteria characterizes such groups. In contrast, for any prime p, any large group (e.g. a nonabelian free group or a surface group of genus two or, more generally, any virtually special group) has geodesics in its p-local commensurability graph of arbitrarily long length. This talk covers joint work with Daniel Studenmund.

Thursday, October 15, 2015

1:00 pm in 243 Altgeld Hall,Thursday, October 15, 2015

#### How to make predictions in topology (resp. arithmetic) using arithmetic (resp. topology)

###### Benson Farb (University of Chicago)

Abstract: Weil, Grothendieck, Deligne and others built an amazing bridge between topology and arithmetic. In this talk I will describe some recent attempts (some successful, some still conjectural) to add planks to this bridge. I hope to convince the audience that the question: "Why is $1/\zeta(n)$ the same as the 2-fold loop space of $CP^{n-1}$ ?" is not completely crazy. This is joint work with Jesse Wolfson.

Tuesday, October 20, 2015

12:00 pm in Altgeld Hall 345,Tuesday, October 20, 2015

#### Gap Distributions in Circle Packings

###### Xin Zhang (UIUC Math)

Abstract: Given a configuration of finitely many tangent circles, one can form a packing of infinitely many circles by Möbius inversions. Fixing one circle from such a packing, we study the distribution of tangencies on this circle via the spectral theory of automorphic forms. Specifically, we will use Anton Good's theorem to show that these tangencies are uniformly distributed when naturally ordered by a growing parameter, and the limiting gap distribution exists, which is conformally invariant. This is a joint work with Zeev Rudnick.

Tuesday, October 27, 2015

#### Dynamics and Cauchy-Riemann Geometry

###### Ilya Kossovskiy (Univ. of Vienna)

Abstract: From the point of view of Complex Analysis, Cauchy-Riemann (CR) Geometry is a tool for studying holomorphic functions of several variables. From the point of view of Differential Geometry, CR Geometry lies in the framework of Cartan's moving frame method. Finally, CR Geometry is a tool for studying properties of solutions of linear Partial Differential Equations, as suggested by the celebrated work of Hans Lewy, Nirenberg, and Treves. We have recently discovered a new face of CR Geometry which regards CR manifolds as certain Dynamical Systems, and vice versa. Geometric properties of CR manifolds are in one-to-one correspondence with that of the associated dynamical systems. This technique has enabled us recently to solve a number of long-standing problems in CR Geometry. It also has promising applications for Dynamical Systems. In this talk, we will outline this technique, and describe its recent applications to Complex Analysis and Dynamics. In particular, we will discuss here an approach for studying Painleve Differential Equations.

Thursday, October 29, 2015

1:00 pm in Altgeld Hall 243,Thursday, October 29, 2015

#### Energy growth in discontinuous Hamiltonian systems

Abstract: We consider a family of discontinuous area-preserving twist maps arising naturally in the study of non-smooth switched Hamiltonian systems. An unbounded solution for the special case of the so-called Pinball transformation is constructed. For the generic values of the parameters, in the large energy limit, the Pinball map is shown to behave similarly to another one introduced earlier by Erd\" os and Sz\" usz. This is a joint work with Maxim Arnold (UT Dallas).

Tuesday, November 3, 2015

12:00 pm in Altgeld Hall 345,Tuesday, November 3, 2015

#### Ergodic Theory and Rigidity of Nilpotent Groups

###### Michael Cantrell (University of Illinois at Chicago)

Abstract: Random aspects of the coarse geometry of finitely generated groups both occur naturally and have applications to the deterministic case. First, we describe the asymptotic behavior of certain random metrics on nilpotent groups, which generalizes a theorem of Pansu and implies an asymptotic shape theorem for first passage percolation. Seen from another perspective, this is a subadditive ergodic theorem for nilpotent groups. Second, we describe a measurable cocycle analog of Pansu's Rademacher-type differentiation theorem for Carnot spaces, answering a question of Austin. From this we deduce Pansu's quasi-isometric rigidity theorem.

Thursday, November 5, 2015

1:00 pm in Altgeld Hall 243,Thursday, November 5, 2015

#### Proper holomorphic mappings throughout mathematics

###### John P. D'Angelo (UIUC Math )

Abstract: After a brief review of the relationship between proper mappings and CR geometry, we consider in detail proper holomorphic mappings between balls. We discuss spherical equivalence, homotopy equivalence, and Hermitian analogues of Hilbert's $17$-th problem. If time permits we will introduce a complex variety associated with a rational proper map and state a sharp variational inequality for the volumes of images of proper polynomial mappings between balls.

Thursday, November 12, 2015

1:00 pm in Altgeld Hall 243,Thursday, November 12, 2015

#### Convergence of harmonic maps

###### Zahra Sinaei (Northwestern University)

Abstract: In this talk I will present a compactness theorem for a sequence of harmonic maps which are defined on a converging sequence of Riemannian manifolds. The sequence of manifolds will be considered in the space of compact n-dimensional Riemannian manifolds with bounded sectional curvature and bounded diameter, equipped with measured Gromov-Hausdorff topology.

Tuesday, November 17, 2015

12:00 pm in 345 Altgeld Hall,Tuesday, November 17, 2015

#### Constructing solutions of Hitchin's equations near the ends of the moduli space

###### Laura Fredrickson (University of Texas - Austin)

Abstract: Hitchin's equations are a system of gauge theoretic equations on a Riemann surface that are of interest in many areas including representation theory, Teichmuller theory, and the geometric Langlands correspondence. In this talk, I'll describe what solutions of SL(n,C)-Hitchin's equations near the ends'' of the moduli space look like. This construction generalizes Mazzeo-Swoboda-Weiss-Witt's (2014) construction of SL(2,C)-solutions of Hitchin's equations where the Higgs field is "simple." This is ongoing work.

Thursday, November 19, 2015

1:00 pm in 243 Altgeld Hall,Thursday, November 19, 2015

#### Fibrations, subsurfaces and triangulations

###### Yair Minsky (Yale University)

Abstract: When a hyperbolic 3-manifold fibers over the circle, the stable/unstable laminations of its monodromy map give us a model for its geometric structure. The features of this model are determined by the projections of these laminations into arc complexes of subsurfaces of the fiber. The quality of this model depends on the topological type of the fiber, and we do not have a good global understanding of how this structure behaves for arbitrary fibrations. As a kind of laboratory for testing such questions we consider the (typically infinitely many) different ways that a given 3-manifold can fiber over the circle, as organized by Thurston's norm. It turns out that, using Agol's veering triangulations, we can obtain some more precise answers. This is joint work with Sam Taylor.

Tuesday, December 1, 2015

12:00 pm in 345 Altgeld Hall,Tuesday, December 1, 2015

#### The abelianization of automorphism groups of right-angled Artin groups

Abstract: Automorphism groups of right-angled Artin groups form an interesting class of groups, as they `interpolate" between the two extremal cases of Aut(Fn) and GL(n,Z). In this talk we will discuss some conditions on a simplicial graph which imply that the automorphism group of the associated right-angled Artin group has (in)finite abelianization. As a direct consequence, we obtain families of such automorphism groups that do not have Kazhdan's property (T). This is joint work with Conchita Martinez-Perez.

Thursday, December 3, 2015

1:00 pm in Altgeld Hall 243,Thursday, December 3, 2015

#### Complex deformations of Anosov representations

###### Andy Sanders (University of Illinois at Chicago)

Abstract: An Anosov representation of a hyperbolic surface group is a homomorphism from the surface group into a semi-simple Lie group which satisfies certain dynamical properties: from these properties one deduces that Anosov representations are discrete, faithful and the set of all Anosov representations is an open subset of the space of all homomorphisms. In recent years, Guichard-Wienhard produced examples of co-compact domains of discontinuity for Anosov representations, which lie in various homogeneous spaces, thus giving an answer to the question of whether or not Anosov representations appear as monodromies of locally homogeneous geometric structures on compact manifolds. In this talk, which comprises joint work with David Dumas, I will discuss some of the complex analytic features of these locally homogeneous geometric manifolds in the case the relevant homogeneous space is a generalized complex flag variety. In particular, we give sufficient conditions to compute the space of all infinitesimal deformations of the complex manifold underlying these locally homogeneous manifolds.

Tuesday, December 8, 2015

12:00 pm in 345 Altgeld Hall,Tuesday, December 8, 2015

#### Inifinite unicorn paths and Gromov boundaries

###### Witsarut Pho-On (UIUC Math)

Abstract: I will provide direct elementary proofs of results of Klarreich and Schleimer identifying the Gromov boundaries of the arc and curve graph and the arc graph, respectively. The proofs use the tool called unicorn paths, developed by Hensel, Przytycki and Webb in their elementary proofs of hyperbolicity of the arc and curve graph and the arc graph. More precisely, I extend the notion of unicorn paths between two arcs to the case where one arc is replaced by a bi--infinite geodesic asymptotic to a lamination. Using these modified unicorn paths, I define homeomorphisms from some spaces of laminations to the Gromov boundaries of the arc and curve graph and the arc graph which are also equivariant under mapping class groups.

Thursday, December 17, 2015

1:00 pm in 345 Altgeld Hall,Thursday, December 17, 2015

#### Knot Complement Commensurability

###### Neil Hoffman (Melbourne)

Abstract: Two 3-manifolds are commensurable if they share a common finite sheeted cover. Commensurability partitions the set of hyperbolic 3-manifolds into equivalence classes, called commensurability classes. Hyperbolic knot complements appear to be rare in commensurability class, in fact Reid and Walsh have conjectured that there are at most three hyperbolic knot complements in a commensurability class. I will discuss recent progress on Reid and Walsh's conjecture with a focus on the open problems in this area.