Geometric deep learning extends deep learning to incorporate information about the geometry and topology of data, especially in complex domains like graphs. Many existing methods in this field rely on message passing. However in this talk we will take a new approach by combining the theory of differential k-forms in Euclidean space with the geometric information of graphs and complexes given by embeddings of node coordinates. This method offers interpretability and geometric consistency without the use of message passing.
The first lecture is the introduction to cluster algebras (an object introduced by Fomin and Zelevinsky in around year 2000) and related Poisson and quantum algebras of geodesic functions (hyperbolic cosines of half-lengths of geodesics in Poincare uniformization of Riemann surfaces with holes) A.K.A. Goldman brackets. The cluster variables therefore provide the parameterization of Teichmuller spaces T_{g,s} of Riemann surfaces of genus g with $s>0$ holes. This is based on papers by V.V.Fock, M.Shapiro, M.Mazzocco and myself in years 1999-2010.
A representation of a surface group into a Lie group of Hermitian type G is called maximal if it maximizes the Toledo invariant. In a 2017 joint work with N. Treib, we proved that in the case of surfaces with non-empty boundary, all such representations could be obtained by a Ping-pong construction in the Shilov boundary of G. For the specific case of Sp(2n,R), maximal representations admit open cocompact domains of discontinuity in the projective space, and we used the Ping-pong construction to obtain explicit fundamental domains for this action bounded by quadric hypersurfaces. In joint work with Pier-Olivier Rodrigue, we construct similar domains for the action of maximal representations in SO(2,n) on the grassmannian of isotropic 2-planes (also called photon space, as it parametrizes photons in the Einstein Universe). We also prove that the same hypersurfaces can be used to bound fundamental domains for representations obtained by the inclusion of a classical Schottky group in SO(1,n) into SO(2,n) and their small deformations.
In this talk we will discuss some recent work on groups which connect self-similar and Higman-Thompson groups to big mapping class groups via "braiding". We will explain some results on the topological finiteness properties of the resulting groups, which are topological generalizations of the algebraic properties of being finitely generated and finitely presented. The talk will involve recent joint works with Xiaolei Wu (Fudan) and Matthew Zaremsky (Albany).
I will discuss the following recent result that was conjectured by Brunella:
Let $X$ be a compact complex manifold of dimension $\geq 3$. Let $\mathcal{F}$ be a codimension one holomorphic foliation on $X$ with ample normal bundle. Then every leaf of $\mathcal{F}$ accumulates to the singular set of $\mathcal{F}$. This result was obtained in collaboration with M. Adachi.
Classical mapping class groups, i.e. for surfaces of finite type, are well-studied objects: they are discrete groups expressing the symmetries of the surface. When we turn our attention to surfaces of infinite type, the situation changes drastically: In particular, the mapping class groups are now "big" in the sense that they are uncountable, not finitely generated, and not even compactly generated.
This means that we loose some of the tools that are used in the classical case but also that we can ask many new questions, for example such coming from the theory of Polish groups: Are big mapping class groups automatically continuous? Amenable? When considering the conjugacy action of a big mapping class group on itself, can there be comeager, dense, or at least somewhere dense orbits?
In this talk, I will give a short introduction to surfaces of infinite type and big mapping class groups and then answer some of the new questions, based on joint work with Jesús Hernández Hernández, Michael Hrušák, Israel Morales, Manuel Sedano, and Ferrán Valdez.
I will introduce a new algorithm to parallelise the computation of persistent homology of 2D alpha complexes. The algorithm distributes the input point cloud among the cores which then compute a cover based on a rectilinear grid. I will show how to compute the persistence Mayer–Vietoris spectral sequence from these covers and how to obtain persistent homology from it. For this, I introduce second-page collapse conditions and explain how to solve the extension problem. I will give a short overview of an implementation in C++ using Open MPI and discuss some experimental results. Finally, I will give an outlook on the three-dimensional case and some applications.
In this talk, we discuss the classification of hyperbolic Coxeter polyhedral, that is, hyperbolic polyhedra whose dihedral angles are integer submultiples of \pi. We give an overview of the main classification results and discuss the few known examples in higher dimensions. We present a method to construct hyperbolic Coxeter polyhedra with mutually intersecting facets and non-zero dihedral angles. We provide a new Coxeter polyhedron in dimension 9 and complete the classification of hyperbolic Coxeter polyhedra with mutually intersecting facets and dihedral angles \pi/2, \pi/3 and \pi/6.
In 2010, Bonahon and Wong constructed the 2d quantum trace map, connecting two different quantizations of the SL_2(C) character variety of an ideally triangulated hyperbolic surface S. The classical limit of this map expresses the trace of the holonomy of a closed immersed curve in S as a Laurent polynomial in the (square roots of the) shear coordinates for the hyperbolic structure defined with respect to the triangulation. In this talk, we will discuss a new construction of a 3d quantum trace map for ideally triangulated 3-manifolds, focusing on the geometric aspects. This is joint work with Sunghyuk Park.
Profinite groups arise in many guises in mathematics, notably as the quotients of compact Hausdorff groups by the connected component of the identity. It is well known that the cardinality of an infinite profinite group cannot be less than 2ℵ0, the cardinality of the continuum. In 2019, Jaikin-Zapirain and Nikolov proved moreover that each infinite profinite group has at least 2ℵ0 conjugacy classes. After a brief introduction to profinite groups we discuss the consequences of restricting the number of conjugacy classes of elements of various types (such as p-elements or elements of infinite order). In particular, every finitely generated profinite group with fewer than 2ℵ0 conjugacy classes of elements of infinite order is finite.
Despite the apparent simplicity of the questions, the answers seem to depend on results such as the classification of the finite simple groups (and their automorphism groups) and work of Zel’manov on Lie algebras associated with profinite p-groups.
In this lecture I will define a family of finite measures on the moduli space of genus g curves. The measures are defined via a construction analogous to that of the Weil-Petersson metric, using the
extra data of a spin structure. In fact, the measures arise naturally out of the super Weil-Petersson metric defined over the moduli space of super curves. The total measure gives the volume of the moduli space of super curves and can be calculated in many examples.
A real affine structure on a surface is an atlas of charts on a topological surface, taking values in R^2 and whose transitions maps are real affine transformations. I will first try to give a variety of examples to gain familiarity with the concept and then discuss classical questions such as existence and geometry of their moduli space, mapping class group actions etc.
My ultimate goal will be to draw a conjectural picture of the geometry of the moduli space based on information coming from the dynamics of the geodesic foliations of affine surfaces.
The Theorem on the existence of three simple closed geodesics on every Riemannian 2-sphere has a “colorful” history and many applications on closed geodesics. I will give an introduction to the problem and outline a proof for the reversible Finsler case. Finally I will comment on some applications for closed geodesics. The talk is based on a collaboration with De Philippis, Marini, and Mazzucchelli.
We address G.D.Mostow, L.Bers and S.L.Krushkal questions on uniqueness of conformal/spherical CR structures on the sphere at infinity of non-compact (Hermitian) symmetric rank one spaces compatible with the action of a discrete isometry group. We construct such non-rigid discrete isometry groups whose quotients have infinite volumes and whose non-trivial deformations are induced by equivariant homeomorphisms of the (Hermitian) symmetric space with bounded horizontal distortion. This non-rigidity is related to non-ergodic dynamics of our discrete isometry group actions on the limit set which could be the whole sphere at infinity.
Integral probability measures (IPMs) are a family of distances between distributions which return the largest deviation under a specified function class. Many of the central problems in data analysis reduce to comparing distributions, and in most cases an IPM is used.
This talk focuses on coresets; these are small and discrete representations of a large data set or a continuous distribution so that the coreset can be used as proxy, and certain measurements are guaranteed to not deviate too far. Interesting coreset questions ask for a given error tolerance, how small can one make the discrete coreset. Such coresets are one of the main tools for scaling machine learning to massive data sets with guaranteed approximation results.
In this talk, we develop coresets for IPMs when the function class is geometrically defined. This has algorithmic applications to many topics including linear classification, kernel density estimates, Kolmogorov-Smirnov distances, and spatial scan statistics. We will conclude with a deeper dive into how to apply these to spatial scan statistics, and how they provide incredibly scalable algorithms without sacrificing statistical power.
In this talk, we explore the expected topology (measured via homology) and local geometry of two different models of random subcomplexes of the regular cubical grid: percolation clusters, and the Eden Cell Growth model. We will also compare the expected topology that these average structures exhibit with the topology of the extremal structures that it is possible to obtain in the entire set of these cubical complexes. You can have a look at some of these random structures here (https://skfb.ly/6VINC) and start making some guesses about their topological behavior.
We discuss some of our current understanding of Lipschitz maps of least Lipschitz constant between two compact negatively curved Riemannian manifolds and maximal stretched measures arising from Thurston’s length ratio metrics. This is joint work in progress with Gerhard Knieper.
We introduce the notion of extended positive representation of a Fuchsian group into a semisimple Lie group with a $\Theta$-positive (in the sense of Guichard--Wienhard) structure. Examples of such representation include Fock-Goncharov's positive framed representations, maximal representations and classical $\Theta$-positive representations. We show that extended positive representations are divergent and extended geometrically finite (in the sense of Weisman).
Given a second countable locally compact group, G, the set of all subgroups S(G) may be endowed with the Chabauty topology, under which it is a compact space. In general, it is difficult to understand the full topology of the space S(G), and the complete homeomorphism type is known only in very few cases. In the first part of the talk we will give an introduction to the Chabauty topology. Then we will study limits of different kinds of subgroups in SL(n, Qp) using the geometry of the Bruhat–Tits building, and the action of the groups on the building. This will give us insight into parts of S(SL(n, Qp)). We will focus on limits of groups of involutions in SL(2, Qp) (joint with Corina Ciobotaru), and time permitting, discuss limits of other subgroups, based on joint work with Ciobotaru and Alain Valette.
Lagrangian Floer homology is a Hamiltonian isotopy invariant of Lagrangian pairs in a symplectic manifold that counts pseudo-holomorphic curves with boundary conditions on the Lagrangians, and it may be thought of as an infinite-dimensional version of Morse theory. The classical Lagrangian PSS isomorphism relates the Lagrangian Floer homology of the pair $(L,L)$ to the singular homology of $L$. In this talk we will explain the promotion of this isomorphism to an equivalence of stable homotopy types — the Lagrangian Floer homotopy type of the pair $(L,L)$ is equivalent to the stable homotopy type of $L$. Although this is the expected result, there is no written account of such an equivalence.
Artin groups form a large family of groups that contains many other interesting families, such as free groups, free abelian groups, braid groups, etc. Despite a lot of effort, these groups remain quite mysterious in general. In this talk, I will explain how one can recover a lot of information about some Artin groups by looking at their actions on a fitting non-positively curved simplicial complex. In particular, I will present how one might recover from such an action various results of quasi-isometric rigidity for the group.
In this talk, we will introduce Bowditch representations of the free group of rank two (defined by Bowditch in 1998) along with primitive-stable representations (defined by Minsky in 2010) acting on Gromov hyperbolic spaces. Minsky first introduced primitive-stable representations in PSL(2,C) to construct an open domain of discontinuity in the character variety. We will discuss the equivalence between primitive-stable and Bowditch representations. We will also introduced simple-stable representations of a surface group and give a similar result in the case of the four-punctured sphere.
In 2013, C. Rossi established the existence and the uniqueness of the maximal extension of a globally hyperbolic conformally flat spacetime. Her proof, however, has the unsatisfactory feature that it relies crucially on the axiom of choice, specifically through Zorn's lemma, and does not provide a description of the maximal extension. In this talk, I will propose a constructive proof that avoids the need of Zorn's lemma. The key idea is that any simply-connected globally hyperbolic conformally flat spacetime M can be embedded into a bigger conformally flat spacetime E(M), called enveloping space of M, containing all the Cauchy-extensions of M, and in particular the maximal extension. It turns out that M and its Cauchy-extensions can be well described in M, a description that I will specify. In particular, the description of the maximal extension in E(M) involves the concept of eikonal functions, coming from PDE theory. I will illustrate this construction with examples and conclude with discussing some interesting consequences that arise from it.
The classical Minkowski problem asks for the construction of a strictly convex surface with specified Gaussian curvature and has already been extended to the Minkowski space by Bonsante and Fillastre. Through the use of affine differential geometry, one can generalize the Minkowski space to the setting of the affine space containing an open proper convex cone. That new setting gives birth to an equivariant affine Minkowski problem: can one introduce and fix the curvature of a convex surface invariant under the action of an affine deformation of a subgroup of SL(n,R) dividing our convex cone? I will show how that problem reduces to a Monge-Ampère equation and can be solved, generalising results from the work of Barbot, Béguin and Zeghib, of Bonsante and Fillastre, and of Nie and Seppi.
I will also explain how that work can be anchored in much bigger task of generalising the theory of flat Lorentzian spacetimes and its tools (causality, Cauchy surfaces and development, cosmological time, etc), which are linked to Teichmüller theory through the fundamental work of Mess, to a theory of affine spacetimes then linked to higher Teichmüller theory.
Artin groups are a generalization of braid groups and are closely related to Coxeter groups. Thanks mainly to a result of Van der Lek, a family of subgroups of Artin groups called parabolic subgroups are central to their study. One question regarding parabolic subgroups is whether the intersection of parabolic subgroups is a parabolic subgroup. This question is wide open, but the answer is known in some cases. An article by Cumplido, Martin and Vaskou introduced a geometric strategy for approaching it. In this talk we will show how to use this strategy to study it in the two-dimensional case. To do so, we will introduce systolic-by-function complexes, which are a generalization of systolic complexes, and use their non-positive curvature to get a positive answer to the question.
The space of representations of the fundamental group of a closed surface to PSL(n,R) contains a distinguished connected component consisting entirely of discrete and faithful representations: the Hitchin component. For n = 2, this component is identified with the Teichmüller space of the surface and the study of its compactifications has led to many influential results. The aim of this talk is to present the Hitchin component, its real spectrum compactification and how we can interpret its boundary points geometrically as representations to PSL(n,F), where F is a closed real field, which admit positive limit maps in flag varieties over F.
Anti-de Sitter space is a spacetime of constant negative curvature, playing a similar role for spacetimes as hyperbolic space does for Riemannian manifolds. Cauchy hypersurfaces in a spacetime are a way to describe space at an instant of time, motivated by physics. By taking levelsets of the cosmological time function, Barbot constructed Cauchy hypersurfaces in GH-regular domains in Anti-de Sitter space. In this short introductory talk I will speak about how we can construct new Cauchy hypersurfaces using the cosmological time function and crowns.
Any closed, flat Riemannian manifold is finitely covered by the torus, by Bieberbach’s classical theorem. Similar classifications have been pursued for closed, Riemannian conformally flat manifolds, as well as for closed, flat Lorentzian manifolds. I will discuss recent and ongoing work with Nakyung Lee and Bill Goldman to classifying closed, Lorentzian conformally flat manifolds when they have nilpotent holonomy.
Hyperbolic groups form an important class of finitely generated groups that has attracted much attention in Geometric Group Theory. We call a group of finiteness type $F_n$ if it has a classifying space with finitely many cells of dimension at most $n$, generalising finite generation and finite presentability, which are equivalent to types $F_1$ and $F_2$. Hyperbolic groups are of type $F_n$ for all $n$ and it is natural to ask if their subgroups inherit these strong finiteness properties. In recent work with Py, we used methods from Complex Geometry to prove that for every $n>0$ there is a hyperbolic group with a subgroup of type $F_{n-1}$ and not $F_n$. This answers an old question of Brady and produces many finitely presented non-hyperbolic subgroups of hyperbolic groups. In this talk we will explain this result and present other recent progress on constructing coabelian subgroups of hyperbolic groups with exotic finiteness properties. This talk is based on joint works with Kropholler, Martelli--Py, and Py.
Bowen and Margulis independently proved in the 70s that closed geodesics on compact hyperbolic surfaces equidistribute towards the measure of maximal entropy. From a homogeneous dynamics point of view, this measure is the quotient of the Haar measure on PSL(2, R) modulo some discrete cocompact sugroup.
In a joint work with Jialun Li, we investigate the higher rank setting of this problem by taking a higher rank Lie group (like SL(d, R) for d ≥ 3) and by studying the dynamical properties of geodesic flows in higher rank : the so-called Weyl chamber flows and their induced diagonal action. We obtain an equidistribution formula of periodic tori (instead of closed orbits of the geodesic flow).
A slope p/q is characterising for a knot K in the 3-sphere if the oriented homeomorphism type of the manifold obtained by performing Dehn surgery of slope p/q on K uniquely determines the knot K. Sorya showed that for any knot K, there exists a constant C(K) such that any slope p/q with |q|≥C(K) is characterising for K. However, the proof of the existence of C(K) in the general case is non-constructive, which naturally evokes the question of how to compute explicit values for C(K). In this talk, I will explore methods for finding C(K) in the case where K is a knot of hyperbolic type (meaning that the JSJ decomposition of its complement has a hyperbolic outermost JSJ piece). I will begin with the simplest case, in which K is a hyperbolic knot; time permitting, I will also discuss some ongoing joint work with Patricia Sorya on the more general case.
We construct open domains of discontinuity for Anosov representations acting on some homogeneous spaces, including (pseudo-Riemannian) symmetric spaces. This generalizes work of Kapovich-Leeb-Porti on flag spaces. Our results complement those of Gueritaud-Guichard-Kassel-Wienhard, who constructed proper actions of Anosov representations. For Zariski dense Anosov representations with respect to a minimal parabolic subgroup acting on some symmetric spaces, we show that our construction describes the largest possible open domains of discontinuity. This is joint work with Florian Stecker.
In topological data analysis (TDA), functors from a poset category to vector spaces, called persistence modules, are a central object of investigation. Persistence modules can arise from data via several constructions, and often encode complex geometric information on the data. Finding informative and computable descriptors of persistence modules is crucial to extract this information and use it in data analysis.
A recent trend in TDA is to describe persistence modules over finite posets via relative homological invariants. Relative homological algebra extends constructions of standard homological algebra by redefining the notion of projective module, which depends on the choice of a family of “basic” modules. In this talk, we study relative resolutions, which are a way of approximating a given persistence module by a sequence of persistence modules that are projective relative to the chosen family. Under certain conditions, the multiplicities of the basic persistence modules in a relative resolution are unique, and define invariants called relative Betti diagrams. Using Koszul complexes, relative Betti diagrams can be computed in a simple and local way, avoiding the computation of the entire relative resolution.
The talk is based on a joint work with Wojciech Chachólski, Isaac Ren, Martina Scolamiero, and Francesca Tombari.
Real uniformly distributed sequences have been widely studied due to their connections to number theory and other fields of mathematics as well as their practical applications. In this talk, we review some of the most important concepts within this field. However, the notion of uniform distribution does not only make sense for real numbers but also in a p-adic setting. There are interesting parallels and differences to the real case which will be the main focus of the talk. The presentation will be kept accessible for a broad audience.
A Reeb graph is a graphical representation of a scalar function on a topological space that encodes the topology of the level sets. Reeb graphs and their variants are popular tools in topological data analysis and visualization. In this talk, I will discuss theoretical advances in studying Reeb graphs and their variants, as well as their applications in data mining and machine learning. In particular, I will discuss novel constructions of Reeb graphs that incorporate the use of a measure, which capture robust topology in data.
Sometimes the topology of a closed 3-manifold is dominated by a single closed surface together with some gluing data. A general challenge is to extract from such data information about the topology and the geometry of the manifold keeping in mind questions like: When does the manifold satisfy the hypothesis of the hyperbolization theorem of Thurston and Perelman? Is it possible to give a formula that turns the combinatorial data of the gluing into geometric invariants of the hyperbolic metric? In this talk, I will explain some recent joint work with Peter Feller and Alessandro Sisto addressing these questions for hyperbolic Heegaard splittings (a Heegaard splitting is a decomposition of a closed 3-manifold into two handlebodies intersecting along their common boundary, the so-called Heegaard surface). In particular, we are able to detect and control the length of some very short closed geodesics in the manifold (giving some novel information about the thick-thin decomposition of the manifold).
The classical Shannon-McMillan-Breiman (SMB) theorem states that for an ergodic measure preserving transformation on a probability space, the rate of uncertainty measured by mathematical entropy can be observed in almost every orbit of the system.
We explain how to use an abstract skew product construction in order to prove such results for measure preserving actions of negatively curved groups, with refinements being taken along random almost geodesics. Joint work with Amos Nevo.
In this talk we will investigate the maximal possible intersection of closed curves of a given length on a surface, and we will specifically focus on the case of translation surfaces. One way to approach this question is to consider the so-called (algebraic) interaction strength (here denoted KVol). After having introduced a few notions related to KVol as well as translation surfaces, we will give a method to compute KVol on SL_2(R)-orbits of Bouw-Möller surfaces with a single singularity, which are instances of particularly symmetric translation surfaces (their Veech group are triangle groups).
Let S be a compact, connected, oriented surface of genus g with one boundary curve, and for n>=0 denote by F_n(S) and C_n(S), respectively, the ordered and the unordered configuration spaces of n distinct points in S. The mapping class group Mod(S) of isotopy classes of boundary-fixing homeomorphisms of S acts naturally on the homology of F_n(S) and C_n(S) with coefficients in any ring R. I will discuss what the kernel of this action is, emphasizing how the answer depends on n in the ordered case, but (almost) does not depend on n in the unordered case. This combines joint work with Jeremy Miller and Jennifer Wilson, and join work with Andreas Stavrou.
I will present some criteria (necessary or sufficient) for the action on the affine space of a group Gamma of affine transformations to be proper. This is joint work with Fanny Kassel.
The main of these criteria links properness of action to the divergence of a parameter called the Margulis invariant. This invariant measures roughly the translation part of an affine transformation, but in a way that is invariant by conjugation.
This link was already known in some special cases (and has often been exploited to construct proper actions). We tried to establish it in as general setting as possible. We proved it in particular if Gamma has some suitable Anosov property (with respect to some natural parabolic subgroup, that depends on the affine group we are working in).
I will possibly also evoke some other invariants similar to the Margulis invariant, that could lead to criteria that work in even more general settings.
Let us consider a tiling of the Euclidean plane by polygons. We play the billiard on this tiling in the following way. A trajectory goes in straight line in each tile. When it reaches a boundary between two tiles, it crosses the boundary and is refracted in the new tile. We get a zigzaging trajectory in the plane. Our goal is to understand what behaviour it can have: Is it periodic or not? bounded or not? If not, how does it go to infinity?
I will first present this dynamical system, and explain how to study it with interval exchange transformations with flips, which are piecewise isometries of the circle.
This will allows me to state a result of my PhD thesis about deviations from asymptotic direction of some tiling billiards trajectories and to explain its proof.
The study of surface group representations into a Lie group is described by the character variety. I shall present a new approach to study character varieties, by so-called Fock bundles. Although similar to Higgs bundles, the new theory does not need a complex structure on the underlying surface. A Fock bundle induces a higher complex structure and gives a connection between these geometric structures and the Hitchin component. Joint work with Georgios Kydonakis and Charlie Reid.
Kahn and Markovic proved the Surface Subgroup conjecture for closed hyperbolic 3-manifolds more than ten years ago. The surface subgroup they constructed can be as close as possible to Fuchsian. However, a closed hyperbolic 3-manifold can also have surface subgroups far away from being Fuchsian. Our result states that provided any genus-2 quasi-Fuchsian group Γ and cocompact Kleinian group G, then for any K>1, one can find a surface subgroup H of G that is K-quasiconformally conjugate to a finite index subgroup F
Multifarious phenomena in quantum field theories are driven or accompanied by extended field configurations such as topological defects. Defects are also expected to play a prominent role for universal self-similar dynamics in systems far from equilibrium. Yet, in simulations of the paradigmatic O(N) vector model, they often appear vastly suppressed in correlations. Persistent homology can provide complementary information to correlations and reveals signatures of the different N-dependent topological defects as I will show. The Betti numbers of sublevel sets of local energy densities contain clear indications for the self-similar scaling of the defects, which has been hard to detect so far.
This application-oriented talk is based on joint work with Viktoria Noel.
Entire functions with finitely many singular values share many dynamical features with polynomials. This is even more the case in the setting of postsingularly finite (psf) functions, here there is not only a finite set of singular values, but every singular value also has a finite orbit.
I will give a brief overview of the dynamics of psf entire functions and show how to approximate psf entire functions by psf polynomials. The main tool is a combinatorial convergence of topological models related to Thurston's classification of branched covers.
This is based on joint work with Malavika Mukundan and Nikolai Prochorov.
The fundamental group of an n-dimensional closed hyperbolic manifold admits a natural isometric action on the hyperbolic space H^n. If n is at most 3 or the manifold is arithmetic of simplest type, then the group also admits many geometric actions on CAT(0) cube complexes. I will talk about a joint work with Nic Brody in which we approximate the asymptotic geometry of the action on H^n by the actions on these complexes, solving a conjecture of Futer and Wise. The main tool is a codimension-1 generalization of the space of geodesic currents introduced by Bonahon. In the 3-dimensional case, we also use some results about minimal surfaces in hyperbolic 3-manifolds.
For S a closed surface of genus at least 2, Labourie proved that every Hitchin representation of pi_1(S) into PSL(n,R) gives rise to an equivariant minimal surface in the corresponding symmetric space. He conjectured that uniqueness holds as well (this was known for n=2,3), and explained that if true, then the space of Hitchin representations admits a mapping class group equivariant parametrization as a holomorphic vector bundle over Teichmuller space.
In this talk we will discuss the analysis and geometry of minimal surfaces in symmetric spaces, and explain how certain large area minimal surfaces give counterexamples to Labourie’s conjecture. This is all joint work with Peter Smillie.
We present variations of an argument of Shalen for demonstrating linearity of certain amalgamated products of linear groups. Applications include (low-dimensional) linearity of some 3-manifold group amalgams. This is joint work in progress with Konstantinos Tsouvalas.
Goldman Conjectured that for components of the PSL(2,R)-character variety of surface groups which do not contain holonomies of hyperbolic structures, the action by the mapping class group should be ergodic. Recently, March\'e and Wolff completely described the genus 2 case, wherein the euler number zero representations surprisingly split into two ergodic components. In new work with James Farre and Peter Smillie, we use a geometric interpretation of the tangent data to reducible representations within the euler class 0 component to give positive evidence towards Goldman's conjecture in the case where genus excedes two and euler number is zero.
A uniform lattice in SL_n(R) with n at least 3 does not admit non-trivial deformations in SL_n(R), thanks to Margulis super-rigidity. However, such a lattice has a natural action on the Furstenberg boundary. Then, a natural question is whether the lattice can be deformed in this (huge) homeomorphism group of the Furstenberg boundary. We prove a semi-conjugacy rigidity result for such deformations. Surprisingly, we also show that no such rigidity holds for the visual boundary of the symmetric space SL_n(R)/SO_n. This is joint work with Chris Connell, Thang Nguyen, and Ralf Spatzier.
I will explain work in progress on a new class of non-commutative cluster algebras which generalize the work of Berenstein and Retakh on non-commutative surfaces. Time permitting, I will briefly discuss the connection with A-type cluster coordinates (Fock-Goncharov coordinates) for decorated representations into Spin(p,q) and the relation to Theta-positivity.
We will give a short overview of the Nielsen-Thurston classification problem on finite-type surfaces and then move to the big setting and see what we know. The objective is to try and push some more geometric approaches to the classification that have not been widely touched in the infinite-type setting with some examples.
Learning numerical representations of graphs is a fundamental problem in machine learning, with traditional Euclidean approaches often falling short in capturing important features such as hierarchies. Motivated by this limitation, there has been growing interest in developing geometric representation learning frameworks that better reflect the properties of target graphs, such as embedding hierarchies in hyperbolic spaces. In this talk, we discuss two such frameworks that go beyond hyperbolic geometry. This talk is based on work with Federico Lopez, Max Riestenberg, Michael Strube, Steve Trettel, and Wei Zhao.
The semistable Higgs bundle moduli space comes naturally equipped with a C*-action and a C*-invariant proper fibration projecting onto a vector space called the Hitchin base. A C*-fixed point is called very stable if it does not admit any other nilpotent Higgs bundle in its upward flow apart from itself. The classification of connected components in the nilpotent cone containing very stable C*-fixed points is a topic of current research. In this talk I will speak about Hecke transformations of Higgs bundles and demonstrate how they can be utilized to study very stable C*-fixed points in the Higgs moduli space.
Due to the isomonodromic property, the space of solutions of the Painleve equations can be parameterized by the monodromy data. Namely, each of the equations can be associated with an affine cubic that is usually called the monodromy surface. Various examples of non-abelian analogs of the Painleve equations have arisen in recent years. In this talk we will discuss how to derive a non-commutative analog of the monodromy surface for them. We will consider the second Painleve equation as an example.
There is a long tradition of using probabilistic methods to solve geometric problems. I will present one such result. Namely, I will show that if the bottom of the spectrum of the Laplacian on a hyperbolic n manifold M is equal to that of its universal cover (or equivalently the fundamental group has exponential growth rate at most (n-1)/2) then M has points with arbitrary large injectivity radius.
Selberg introduced a fundamental domain for discrete groups acting on SL(n,R)/SO(n) which is convex in the projective model. He showed that for uniform lattices, the domain is a finite-sided convex polyhedron. We give sufficient criteria for Anosov subgroups to admit finite-sided Dirichlet-Selberg domains. This is joint work with Colin Davalo.
The character variety associated to a surface group and a semisimple Lie group comes naturally equipped with the so called Goldman symplectic form, and hence one can study Hamiltonian flows on it. One particular instance of such flows are twist flows along simple closed curves, which correspond to „length functions“. I will talk about ongoing work, where instead of considering functions on a single simple closed curve, I look at functions on multiple simple closed curves.
Every horocycle in a closed hyperbolic surface is dense, and this has been known since the 1940's. We study the behavior of horocycle orbit closures in Z-covers of closed surfaces, and obtain a fairly complete classification of their topology and geometry. The main tool is a solution of a surprisingly delicate geometric optimization problem: finding an optimal Lipschitz map to the circle and the associated lamination of maximal stretch. This is joint work with Yair Minsky and Or Landesberg.
It is interesting to ask which word hyperbolic groups arise as Anosov subgroups of semisimple Lie groups. One way to restrict the class of groups is to study how the boundary map sits inside the boundary of the Lie group. Cluster algebras provide a useful tool to understand the topology of these boundaries.
A question of Misha Kapovich asks whether SL_3(Z) contains a subgroup isomorphic to the free product Z^2\ast Z. Motivated by this question, I am going to discuss a characterization of divergent Z^2 subgroups of SL_3(R). This characterization, combined with results of Oh, shows that a Zariski-dense discrete subgroup Γ of SL_3(R) contains a regular Z^2 if and only if Γ is commensurable to a conjugate of SL_3(Z). This is joint work with Sami Douba.
I will talk about Riemann-Hilbert-Birkhoff correspondence between general meromorphic G-connections and Stokes representations, and some ideas on the nonabelian Hodge correspondence.
Let M be a closed hyperbolic 3-manifold and let Gr(M) be its 2-plane Grassmann bundle. We will discuss the following result: the weak-* limits of the probability area measures on Gr(M) of pleated or minimal closed connected essential K-quasifuchsian surfaces as K goes to 1 are all convex combinations of the probability area measures of the immersed closed totally geodesic surfaces of M and the probability volume (Haar) measure of Gr(M).
The stable, unstable (and central) distributions of (partially) hyperbolic dynamics are a priori only Hölder continuous, and several works seem to suggest that their lack of regularity is in fact the only obstacle to their rigidity. Concerning contact-Anosov flows, successive works of Ghys (in dimension three) and Benoist-Foulon-Labourie (in higher dimensions) have for instance proved that the smoothness of the stable and unstable distributions forces the system to be algebraic. In this talk, I will present an analog rigidity result for three-dimensional volume-preserving partially hyperbolic diffeomorphisms whose stable, unstable and central distributions are smooth, and whose stable-unstable plane field is a contact distribution.
I'll discuss joint work with Sami Douba, Balthazar Flechelles, and Feng Zhu, in which we prove that every hyperbolic group acting geometrically on a CAT(0) cube complex admits Anosov representations.
Dyer groups are a family of groups which includes both Coxeter groups and right-angled Artin groups. The goal is to introduce this family, give some properties and some complexes associated to them.
We discuss past work at the intersection of quantum topology and higher Teichmüller theory. One of the guiding philosophies is "Fock-Goncharov duality".
Thurston’s metric is defined on Teichmüller space, which contains very rich geometry. I am trying to mimic the definition and generalize to other spaces, for example, flat cone metrics and positively ratioed representations.
A group equivariant operator transforms measurements of data into other measurements while respecting certain group actions on them. If such operators are also non-expansive, then we talk about GENEOs. GENEOs have been introduced to model neural networks and inject geometrical knowledge about the data, encoded by the group action, to be preserved by the machine. However, data rarely satisfy rigid symmetries due to noise, incompleteness or symmetry-breaking features, and hence the group axioms may not be satisfied. Thus, we propose a topological model to encode partial equivariance in neural networks. To this end, we introduce a class of operators, called P-GENEOs, that change data expressed by measurements, respecting the action of certain sets of transformations, in a non-expansive way. We then study the spaces of measurements, whose domains are subject to the action of certain self-maps, and the space of P-GENEOs between these spaces.
How do you email a Torus? As a parallelogram with opposite sides identified, as an elliptic curve, or as a product of circles embedded in 3-space? We will discuss some presentations of Riemann surfaces, difficulties in moving between the presentations and interesting open problems that arise when considering the moduli space of Riemann surfaces.
In this talk I will discuss about possible generalizations of the notion of equivariant pleated surfaces in the context of higher rank Lie groups, giving a brief overview of what we presently know (based on joint works with G. Viaggi for representations in SO(2,n), and with S. Maloni, G. Martone, and T. Zhang in PSL(d,C)) and what we would like to learn more about.
Linear submanifolds of strata of holomorphic abelian differentials are rare objects. Up to today, only few series in low dimension are known. The classification of linear submanifolds is an open problem, and the question about the existence of examples in higher dimension is still unanswered.
One approach to their study and a possible classification is to analyze their closure inside as suitable compactification of the ambient stratum. I will describe the phenomena encountered at the boundary for the example of the Gothic locus, one of the few exceptional linear submanifolds recently discovered.
As an application of the compactification, I will describe the intersection theoretic framework to compute invariants of linear submanifolds (for example the Masur-Veech volume and Siegel-Veech constants) that we implemented as a SageMath package.
In recent decades, discrete complex analysis has seen remarkable advancements, despite its relatively late inception compared to its continuous counterpart. While continuous complex analysis boasts a long-established history, the discrete analog has emerged more recently and remains an active area of research and exploration.
This presentation focuses on a pivotal aspect of Riemann surfaces: their period matrices. After providing a concise introduction to the linear theory of discrete Riemann surfaces, we introduce the discrete period matrix—a direct counterpart to its continuous counterpart. We also delve into the larger complete discrete period matrix, whose calculation is based on a broader class of discrete holomorphic differentials, thereby giving more information on the underlying discrete Riemann surface.
Our discussion extends to the convergence of discrete period matrices when a sequence of increasingly fine discretizations is applied to a given compact polyhedral surface. We outline the proof strategy for the convergence of these discrete period matrices to the period matrix of the original surface, and elucidate the limits of the blocks within the complete discrete period matrix.
In closing, we explore a discretization of real Riemann surfaces and their period matrices, a collaborative effort with Johanna Düntsch. As one expects, we find that the corresponding discrete period matrices exhibit the same symmetrical properties as their continuous counterparts, with the complete discrete period matrices following a similar pattern.
A translation surface is a collection of polygons with edge identifications given by translations. In spite of the simplicity of the definition, the space of translation surfaces has connections to different areas of math such as the moduli space of Riemann surfaces and rational billiards in the plane.
In this talk we consider the unstable foliation, that locally is given by changing horizontal components of period coordinates, which plays an important role in study of translation surfaces, including their deformation theory and in the understanding of horocycle invariant measures. We show that measures of "large dimension" equidistribute and give an effective rate. An analogous result in the setting of homogeneous dynamics is crucially used in the recent effective equidistribution results of Lindenstrauss-Mohammadi and Lindenstrauss--Mohammadi--Wang.
No prior knowledge of the words in the title is assumed.
The theme of this lecture centers around two famous books, one by Hermann Weyl on “The Classical Groups: Their Invariants and Representations” (1939), another the six-volume work by I.M. Gelfand et al. on “Generalized Functions” (1960’s). I will discuss some broad ideas related to the theme of these two books, as well as some recent development in classical groups and their smooth representations.
The compactification of the moduli space of super Riemann surface has been studied by Deligne (1987) and more recently by Donagi and Witten (2015). We aim to extend those results by constructing a moduli space of stable maps.
Let M be a super Riemann surface with holomorphic distribution D and N a symplectic manifolds with compatible almost complex structure J. We call a map Φ: M-> N a super J-holomorphic curve if its differential maps the almost complex structure on D to J. Such a super J-holomorphic curve is a critical point for the superconformal action and satisfies a super differential equation of first order. Using component fields of this super differential equation and a transversality argument we construct the moduli space of super J-holomorphic curves as a smooth subsupermanifold of the space of maps M->N. The reduced space of the moduli space coincides with the moduli space of J-holomorphic curves from the reduction of M to N.
Gaussian heat kernel bounds play a fundamental role in geometric analysis. We present recent results on explicit Gaussian upper bounds for non-compact manifolds depending on locally uniform Ricci curvature integral assumptions. Furthermore, we discuss generalizations of integral curvature bounds in terms of the so-called Kato class. If time allows, topological applications of those heat kernel upper bounds will be given.
We start giving an overview of Bakry-Emery and Ollivier curvature and corresponding gradient estimates which require global curvature bounds. We then introduce the perpetual cutoff method to give gradient estimates requiring the curvature bound not everywhere. As an application, we give a sharp distance bound to the set where no curvature bound is assumed.
We study the asymptotic behavior of solutions to subcritical $\sigma_{k}$ Yamabe equation in a punctured ball. We prove that an admissible solution to this equation with a non-removable isolated singular point is asymptotic to a radial solution. Then we are able to obtain higher order expansion of solutions using analysis of the linearized operators. These results generalize earlier pioneering work of Caffarelli, Gidas and Spruck.
On a class of closed oriented Riemannian manifolds with lower bounds on Ricci curvature and injectivity radius we study the parameter given by a characteristic number over the volume of the manifold. The domain of this parameter is endowed with the Benjamini-Schramm (BS) topology. The BS-topology is a weak notion of convergence, that originated from graph theory, where it is insensively studied, but also has applications in various other areas. Using an integral characterization (actually, a generalization of Gauss-Bonnet formula) one can show that the parameter explained is continuous and has a continuous extension to the completion of its domain in the BS topology. From the known fact that this completion is compact one can derive that the parameter is testable in constant time.
Damek-Ricci spaces are some one-dimensional extensions of Heisenberg groups. We prove that there does not exist non-geodesic biharmonic curves in a four-dimensional Damek-Ricci space although there exist such curves in three-dimensional Heisenberg groups.
In this talk, we give a new estimate for the eigenvalues of the Dirac operator on a compact spin manifold in terms of an appropriate endomorphism $E^\psi$ of the tangent bundle associated with an eigenspinor $\psi$. We then show that, for isometric immersions and Riemannian flows (local Riemannian submersions), the limiting case could be achieved. In this case the tensor $E^\psi$ is identified with the second fundamental form of the immersion while it is identified with the O'Neill tensor of the flow.
