Resúmenes de minicursos de la primera semana (22 de julio  26 de julio)
A) Introducción a la geometría hiperbólica, Sebastián HurtadoSalazar (Yale), Plinio Pino Murillo (UFF)
Resumen: Discutiremos aspectos básicos (y no tan básicos), de geometría hiperbólica, algunos de los topicos que posiblemente discutiremos son:

Conceptos básicos como lineas (geodésicas), triángulos, áreas, polígonos. Frontera al infinito. Isometrías.

Grupos discretos y sus cocientes. Conjuntos limites en la frontera.

Reticulados en dimensión dos y tres.

Clasificación de superficies hiperbólicas (Espacio de Teichmuller), y Teorema de Rigidez de Mostow en dimensión tres.
Estos tópicos son discutidos en el libro Fuchsian groups (S. Katok) y con mas profundidad en el libro Outer Circles (Marden). Tambien muy recomendado el libro de Thurston (Three dimensional geometry and topology).
B) Geometría simpléctica, Poisson y más allá, Iván Contreras (Amherst), Nicolas Martínez (UNAL sede Bogotá)
Resumen: Geometría simpléctica y de Poisson ha sido usada como herramienta para estudiar y entender sistemas físicos. En este minicurso introduciremos los objetos y técnicas fundamentales en geometría simpléctica (e.g. coordenadas de Darboux, subvariedades Lagrangianas, haz cotangente) y geometría de Poisson (foliaciones simplécticas, diferentes ejemplos de estructuras de Poisson). Motivaremos estos objetos geométricos con ejemplos provenientes de física clásica, y al final discutiremos una aplicación de geometría de Poisson a la teoría de Lie, usando algebroides y grupoides.
C) Nudos y superficies, Juanita Pinzón (University of Notre Dame)
Resumen: La teoría de nudos es la subárea de la topología que estudia los embebimientos del círculo S^1 en un espacio tridimensional. Demostrar que dos nudos son distintos (o equivalentes) es el principal problema de la teoría de nudos. Usando la existencia de superficies de Seifert y la clasificación de superficies es posible a) definir invariantes de nudos que detecten diferencias, y b) definir una relación de equivalencia sobre el conjunto de nudos que lo convierte en un grupo. En este mini curso discutiremos los métodos utilizados para distinguir nudos, principalmente desde la perspectiva de las superficies en con frontera un nudo fijo.
D) Teoría de homotopía usando conjuntos simpliciales, Angélica Osorno (Reed College), Manuel Rivera (Purdue University)
Resumen: El tema central de la topología algebraica es el estudio de espacios topológicos salvo alguna noción de equivalencia (eg homeomorfismo, equivalencia homotópica, etc…) usando invariantes algebraicos. Una manera de asociar estructuras algebraicas a espacios topológicos es primero describiéndolos combinatorialmente, por ejemplo, mediante una serie de instrucciones de como pegar discos, símplices, o poliedros. En este minicurso ofreceremos una introducción a la teoría de espacios salvo equivalencia homotópica usando conjuntos simpliciales. Un conjunto simplicial es una abstracción del concepto de un complejo simplicial: un espacio topológico construido pegando símplices de diferentes dimensiones siguiendo una serie de reglas. Sucede que, por sus particularidades combinatoriales, los conjuntos simpliciales no solo son útiles para modelar espacios topologícos sino también para organizar y estudiar estructuras algebraicas y categóricas. Por esta razón, las técnicas simpliciales aparecen en diferentes áreas de la matemática moderna como la teoría de categorías, álgebra homológica, geometría algebraica y topología geométrica.
Comenzaremos con una discusión general y clásica de la teoría de homotopía de espacios topológicos, continuaremos abstrayendo los elementos básicos de ésta e interpretándolos en el contexto combinatorial de conjuntos simpliciales y finalizaremos explicando el sentido en que ambas teorías son equivalentes. En el camino, desarrollaremos técnicas y herramientas básicas de la teoría de categorías que son útiles en diferentes contextos. Asumiremos conocimientos básicos de topología y álgebra abstracta.
Abstracts for the second week (July 29th August 2nd)
Camilo Arias Abad (Universidad Nacional de Colombia, sede Medellín)
Title: The multiple facets of the Associahedron
Abstract: The title is borrowed from Loday, who wrote about the polytopes that Stasheff encountered in 1963 while studying the associativity properties of the composition of loops. Already in 1951 Tamari had constructed these polytopes in terms of the partial order in the space of bracketings on n letters. Stasfheff observed that associativity is an optional property that can be relaxed. This observation led to homotopical algebraic structures that found many applications to topology, algebra, mathematical physics and representation theory. Associahedra are also fascinating from the point of view of combinatorics and discrete geometry, and are related in a deep way to formal power series via the Lagrange inversion formula. The goal of the talk is to tell you about some of these stories.
Mikhail Belolipetsky (IMPA)
Title: Counting lattices
Abstract: Let H be a simple Lie group. It is well known that H has lattices, i.e. discrete subgroups of finite covolume, which can be cocompact or noncocompact. Apparently, for higher rank Lie groups most lattices are cocompact. This result, which follows from our joint work with Alex Lubotzky, is less well known and appears to be somewhat counterintuitive. In the talk I will discuss the ideas behind its proof and some related open problems.
Henrique Bursztyn (IMPA)
Title: Dirac structures and integration of Poisson homogeneous spaces
Abstract: The integration of a Poisson manifold to a symplectic groupoid can be regarded as a "non linear" generalization of the problem of integrating a Lie algebra to a Lie group. A key difference is that, in contrast with (finitedimensional) Lie algebras, not every Poisson manifold is integrable in this sense. In this talk, I will show that every Poisson homogeneous space of any Poisson Lie group is integrable (joint work with Iglesias and Lu). This result is obtained through a concrete construction making use of Dirac structures. Time permitting, I will also indicate how elements of "shifted symplectic geometry" are present in this construction.
Mauricio Bustamante (Pontificia Universidad Católica de Chile)
Title: Symmetries of exotic tori
Abstract: An important feature of the torus $T^d=\mathbb{R}^d/\mathbb{Z}^d$ is that it has an effective smooth action of $SL_d(\mathbb{Z})$ induced by the linear action on $\mathbb{R}^d$. A natural question is to what extent this action depends on the smooth structure of the torus. Specifically, if $\mathfrak{T}$ is an exotic torus, i.e. a smooth manifold homeomorphic but not diffeomorphic to $T^d$, does $SL_d(\mathbb{Z})$ act effectively on $\mathfrak{T}$? In this talk, we will discuss this question and we will see that many exotic tori do not admit any nontrivial smooth action.
Julie Bergner (University of Virginia)
Title: Combinatorial examples of 2Segal sets and their Hall algebras
Abstract: The notion of a 2Segal set encodes an algebraic structure that is similar to that of a category, but for which composition need not exist or be unique, yet is still associative. The fact that such structures give rise to Hall algebras, generalizing constructions in representation theory and algebraic geometry, is one of the primary motivations for studying them. In this talk, we look at 2Segal sets that arise from trees and graphs and their associated Hall algebras. These examples are discrete versions of the analogous 2Segal spaces of trees and graphs developed by GálvezCarrillo, Kock, and Tonks, and provide a way to explore various general constructions quite explicitly. Much of this work is joint with Borghi, Dey, GálvezCarrillo, and Hoekstra Mendoza.
Michelle Chu (University of Minnesota)
Title: Surfaces in twobridge knots and components in the character variety
Abstract: The SL(2,C)character variety of the fundamental group of a hyperbolic 3manifold encodes a lot of topological and geometric information about the manifold. As an algebraic set, this character variety can have multiple algebraic components. In this talk I will mention both old and new results on when characters at the intersection of distinct components in the character variety of twobridge knot groups produce splittings of the fundamental group over surface subgroups.
Paul Kirk (Indiana University)
Title: An endomorphism in the symplectic category induced by a marking via traceless SU(2)character varieties.
Abstract: From the perspective of the (1990s) AtiyahFloer conjecture, traceless SU(2) character varieties provide a halfway point in the construction of a TQFT, more precisely they determine a (partially well defined) functor from the (2,0)+1 cobordism category to the (partially well defined) Weinstein symplectic category, whose objects are symplectic manifolds, with Hom(M,N)=Lagrangian immersions L > M x N. Following this with the LagrangianFloer homology functor is conjectured to produce a Floer gauge theory/TQFT equivalent to FloerDonaldsonKronheimerMrowka instanton homology, defined via the Morse theory of the ChernSimons function. In these theories, as with most gauge theories, much of the difficulty lies in transversality issues and the related deformation problems.
I will begin by reviewing the definitions and ideas of the preceding paragraph, then focus on character varieties of knots and links (and webs) in 3manifolds decomposed along a Conway sphere (a 4punctured 2sphere), give a few examples, and finally focus on one “utility” tangle with boundary a disjoint union of two Conway spheres. I will then outline the proof that image of this tangle in the symplectic category via the traceless character function is a Lagrangian bifold immersion of a genus 3surface into S^2 x S^2 and give a description of the induced endomorphism of the Fukaya category of the traceless character variety of the Conway sphere. I will outline how one can extract a Floertheoretic construction of Khovanov homology, and discuss conjectures concerning an immersed curve calculus for instanton homology. (joint work with Cazassus, Herald, and Kotelskiy)
Rui Loja Fernandes (University of Illinois, UrbanaChampaign)
Title: Extremal Kähler metrics for toric fibrations
Abstract: The question of finding 'best possible' metrics on a manifold is a classical problem in Riemannian geometry. For Kähler manifolds, Calabi proposed the notion of an extremal Kähler metric. I will discuss the problem of finding extremal Kähler metrics on toric manifolds, covering the pioneering work of Abreu and Guillemin in the late 1990s, the stability theory of Donaldson, and recent progress that places this theory in the realm of general Lagrangian fibrations admitting only elliptic singularities. This presentation is based on ongoing joint work with Miguel Abreu (ISTLisbon) and Maarten Mol (Max PlanckBonn).
Rajan Mehta (Smith College)
Title: Frobenius objects in the category of spans
Abstract: Frobenius algebras can be given a categorytheoretic definition in terms of the category of vector spaces, leading to a more general definition of Frobenius object in any monoidal category. In this talk, I will describe Frobenius objects in a category where the objects are sets and the morphisms are isomorphism classes of spans. The main result is that it is possible to construct a simplicial set (with some additional maps) that encodes the data of the Frobenius structure. The simplicial sets that arise in this way can be characterized by a few relatively nice conditions.
Commutative Frobenius objects give rise to surface invariants that can be natural number valued. I will give some explicit examples where the invariants can be computed. Time permitting, I will also describe the coherent highercategorical analogue in the bicategory of spans, where the corresponding structures on simplicial sets are even nicer.
This work is part of a bigger program aimed at better understanding the relationship between Poisson/symplectic geometry and topological field theory. Part of the talk will be devoted to giving an overview of this program, and in particular explaining how the category of spans is related to the symplectic category. This is based on work with Ivan Contreras and Molly Keller (arXiv:2106.14743), Contreras and Walker Stern (arXiv:2311.15342), and work in preparation with Sophia Marx.
Heber Mesa (Universidad del Valle, Cali)
Title: Stereographic model of the de Sitter space
Abstract: This talk introduces a stereographic model for the de Sitter space. In this model, the geodesics are characterized by Euclidean lines and circles, also we can identify the asymptotic boundary (boundary at infinity) of the de Sitter space as two disjoint spheres. Besides, spacelike hypersurfaces of the de Sitter space are studied; in particular, a correspondence between a certain family of totally umbilical spacelike hypersurfaces and geodesic balls on one of the components of the asymptotic boundary of the de Sitter space is shown.
Pavel Mnev (University of Notre Dame)
Title: Combinatorial 2d topological conformal field theory from a local cyclic Ainfinity algebra
Abstract: I will explain a construction of a combinatorial 2d TCFT, assigning partition functions to triangulated cobordisms (as chain maps between spaces of states), in such a way that a Pachner flip induces a Qexact change. More generally, the partition function becomes a nonhomogeneous closed cochain on the “flip complex.” One has a combinatorial counterpart of the BV operator G_{0,} arising from evaluating the theory on a special 1cycle on the flip complex of the cylinder. The local input for the model is a cyclic Ainfinity algebra, with the operation m_3 playing the role of the BRSTprimitive G of the stressenergy tensor T=Q(G).
I will also describe a way to incorporate invarianceuptohomotopy with respect to the second 2d Pachner move (stellar subdivision/aggregation). This version of the model is based on secondary polytopes of GelfandKapranovZelevinsky and uses a certain enhancement (by extra homotopies) version of a cyclic Ainfinity algebra as input. The talk is based on a joint work with Andrey Losev and Justin Beck, https://arxiv.org/pdf/2402.04468.pdf.
Eric Samperton (Purdue University)
Title: Some limits on efficient topological quantum state preparation
Abstract: Quantum state preparation is one of the overarching theoretical issues in the study of quantum computation, especially as it arises in quantum error correction protocols and in quantum simulation. Some of the more interesting states one might try to prepare come from (fully extended and unitary) (d+1)dimensional TQFTs. This is especially so when d=2, since such states are expected to be some of the best candidates for performing faulttolerant quantum computing (via the topological quantum computing paradigm). I'll give as thorough an overview of all of this that time permits, and then argue that as soon as d=3, there can be no efficient way to prepare "most" TQFT states, even for relatively simple TQFTs arising from finite gauge theory (aka DijkgraafWitten theory) based on solvable groups. No background in quantum will be necessary, as I plan to review the pertinent points at the beginning of the talk.
Maria Amelia Salazar (Universidade Federal Fluminense, Brasil)
Title: Going relative in cohomology
Abstract: Motivated by the attempt to understand characteristic classes of Lie groupoids and geometric structures, we are brought back to the fundamentals of the cohomology theories of Lie groupoids and algebroids. One element that was missing in the literature was the notion of relative cohomology in this setting. In this talk I will explain this notion, the relation between the relative cohomology of groupoids and that of algebroids via van Est maps, and to indicate how it can be used to provide an intrinsic definition of characteristic classes.
Anthony Sanchez (UCSD)
Title: Quantitative finiteness of hyperplanes in hybrid manifolds
Abstract: The geometry of nonarithmetic hyperbolic manifolds is mysterious in spite of how plentiful they are. McMullen and Reid independently conjectured that such manifolds have only finitely many totally geodesic hyperplanes and their conjecture was recently settled by BaderFisherMillerStover in dimensions larger than 3. Their works rely on superrigidity theorems and are not constructive. In this talk, we strengthen their result by proving a quantitative finiteness theorem for nonarithmetic hyperbolic manifolds that arise from a gluing construction of Gromov and PiatetskiShapiro. Perhaps surprisingly, the proof relies on an effective density theorem for certain periodic orbits. The effective density theorem uses a number of ideas including Margulis functions, a restricted projection theorem, and an effective equidistribution result for measures that are nearly full dimensional. This is joint work with K. W. Ohm.
Bernardo Uribe (Universidad del Norte, Barranquilla)
Title: Topological invariants of crystals
Abstract: The modeling of electromagnetic properties of crystals through band theory and density functional theory has proven to be an amazing tool in the quest of distinguishing interesting crystals. In this talk I will explain the role of ktheory invariants on this quest.
Franco Vargas Pallete (Yale University)
Title: On the topology and index of minimal/Bryant framed surfaces
Abstract: In this talk we'll discuss a 1to1 correspondence between Euclidean minimal and Bryant surfaces, known in the literature as Lawson's correspondence. Along with this correspondence we will describe framed surfaces, which is a class of Euclidean minimal and hyperbolic CMC1 surfaces that generalize immersed Euclidean minimal surfaces and Bryant surfaces. For this class we prove a lower bound on the (unrestricted) Morse index by a linear function of the genus, number of ends and number of branch points (counting multiplicity), generalizing previous work in the literature. This is based on joint work with Davi Maximo.
Contributed talks
Emilia Alves (Universidade Federal Fluminense)
Title: Intersection of real Bruhat cells
Abstract: We examine arbitrary intersections of real Bruhat cells. Arising in various contexts
across several disciplines – such as in the KazhdanLusztig theory and the study of the spaces of locally convex curves – these objects have attracted the attention of many authors. We present a stratification of an arbitrary pairwise intersection of two real Bruhat cells. Also, we show the dual CWcomplex of such stratification is homotopically equivalent to the intersection under analysis. Finally, both classical and new topological results about such intersections stem from our methods. We include a wide latitude of examples and perform explicit computations to illustrate our methods. Includes work in progress with N. Saldanha (PUCRio), G. Leal (PUCRio) and B. Shapiro (Stockholm University).
References:
[1] E. Alves and N. Saldanha, On the Homotopy Type of Intersections of Two Real Bruhat Cells. Int. Math. Res. Not. IMRN, v. 00, p. 157, 2022.
[2] B. Shapiro, M. Shapiro, A. Vainshtein, Connected components in the intersection of two open opposite Schubert cells in SLn(R)/B, Int. Math. Res. Not. IMRN, no. 10, (1997) 469493.
[3] B. Shapiro, M. Shapiro, A. Vainshtein, Skewsymmetric vanishing lattices and intersection of Schubert cells, Int. Math. Res. Not. IMRN no. 11, (1998) 563588.
Fabricio Valencia (Universidade de Sao Paulo)
Title: On the Novikov type inequalities
Abstract: In the early 80s Novikov started a generalization of Morse theory in which instead of critical points of smooth functions on a compact manifold M he dealt with closed 1forms and their zeros. He introduced numbers b_j(ξ) (Novikov Betti numbers) and q_j(ξ) (Novikov torsion numbers) depending on a real cohomology class ξ ∈ H_1(M, R), which, in the special case ξ = 0, respectively equal the Betti numbers of M and the minimal number of generators of the torsion subgroup of H_1(M, Z). The Novikov inequalities state that any closed 1form ω of Morse type has at least b_j(ξ) + q_j(ξ) + q_{j−1}(ξ) zeros of Morse index j, where ξ = [ω] stands for the real cohomology class associated to ω. These generalize the classical Morse inequalities, the ones obtained when considering ξ = 0. The main aim of this talk is to present the ingredients needed to better stating the Novikov inequalities, thus exhibiting a proposal which aims at extending those inequalities for certain kind of singular spaces whose geometry can be encoded by the structure of a Lie groupoid.
Brian Grajales (Universidade Estadual de Campinas)
Title: Einstein metrics on cohomogeneity one manifolds
Abstract: Let M be a differentiable nmanifold equipped with a smooth action of a compact connected Lie group G. The action is said to be of cohomogeneity one if the orbit space M/G is onedimensional. A manifold with such a property is referred to as a cohomogeneity one manifold. When M is compact, M/G is diffeomorphic to either the circle S^1 or a closed interval
I. In this talk, we provide an overview of existing findings on the existence and properties of Einstein Riemannian metrics on compact cohomogeneity one manifolds where the corresponding orbit space is a closed interval. In particular, we provide a detailed exposition for the case where the isotropy representation of the principal orbits, corresponding to interior points of I, decomposes into three irreducible and inequivalent subrepresentations.
References:
[1] Bohm, C. Inhomogeneous Einstein metrics on lowdimensional spheres and other lowdimensional spaces. Invent. math. 134, 145–176 (1998).
[2] Eschenburg, J.H., and Wang, M.Y. The initial value problem for cohomogeneity one Einstein metrics. J. Geom. Anal. 10, 109–137 (2000).
[3] Grove, K., and Ziller, W. Cohomogeneity one manifolds with positive Ricci curvature. Invent. math. 149, 619–646 (2002).
Sofía Martínez Alberga (Purdue University)
Title: Coalgebraic Structures Describing Gspaces
Abstract: Given a commutative ring R, a π_1Requivalence is defined to be a continuous map of spaces inducing an isomorphism on fundamental groups and an Rhomology equivalence between universal covers. If R is the ring of integers, then this notion coincides with that of a weak homotopy equivalence. When R is an algebraically closed field, Rivera and Raptis described a full and faithful (co)algebraic model for the homotopy theory of spaces up to π_1Requivalence by means of simplicial coalgebras considered up to a notion of weak equivalence created by the cobar functor. Their work extends previous algebraic models for spaces considered up to Rhomology (Kriz, Goerss, Mandell) by including the information of the fundamental group in complete generality. In this talk, I will describe Gequivariant analogs of this statement obtained through generalizations of a celebrated theorem of Elmendorf.
Elizaveta Vishnyakova (Universidade Federal de Minas Gerais)
Title: Applications of graded coverings of supermanifolds
Abstract: Our talk is devoted to several applications of a graded covering of a supermanifold.
Camilo Rengifo (Universidad de La Sabana)
Title: Courant algebroids and differential graded geometry
Abstract: In this talk I will present the transgression functor for a Courant algebroid and the inverse image functor for Courant algebroids. The given functors write down the Courant algebroid structure and morphisms between two of them in terms of marked Lie algebroids and Courant to Lie morphisms in suitable categories. I will give some examples for both constructions and show its relations with the Poisson geometry and the Lie theory. If time permits, I will talk about the relationship between Courant and Vertex algebroids following definitions from Kac, Bressler, and BeilinsonDrinfeld.
Javier Gargiulo (Universidade Federal Fluminense)
Title: Stability of pullbacks of foliations on weighted projective spaces
Abstract: In this talk, we will discuss the problem of describing irreducible components and studying the geometry of moduli spaces of codimension one foliations of a given degree on a classical projective space. The classification of the irreducible components of these quasiprojective algebraic varieties is a very challenging open problem, only solved for very low degrees. It has gained popularity since the 90s, especially after the renowned article [1] by Prof. Cerveau and Prof. Lins Neto, where they obtained the latest complete classification result. In particular, we will present some results given in [2]. This work began as an effort to obtain new irreducible components whose generic points are pullbacks of foliations on a weighted projective plane and other pullbacks of foliations induced by Lie group actions. By combining tools from complex and algebraic geometry with commutative algebra, we produced a general theorem to describe a larger family of irreducible components whose generic elements are pullbacks under rational maps of foliations with split tangent sheaf on a general weighted projective space. As an application, we were able to show an infinite number of previously unknown components, establish a recipe for producing new ones, and construct a unified proof for some other wellknown families of foliations.
References:
[1] D. Cerveau and A. Lins Neto. Irreducible components of the space of holomorphic foliations of degree two in CP(n), n ≥ 3. Annals of Mathematics (2) 143.3 (1996), pp. 577–612.
[2] J. Gargiulo Acea, A. Molinuevo, F. Quallbrunn and S. L. Velazquez. Stability of pullbacks of foliations on weighted projective spaces. Available at https://doi.org/10.48550/arXiv.2212.12974. Submitted for publication (2023).
Sebastián MuñozThon (Purdue University)
Title: Scattering rigidity for standard stationary Lorentzian manifolds via timelike geodesics
Abstract: We consider Standard Stationary Lorentzian Manifolds (SSM), that is, smooth manifolds of the form M = \mathbb{R} \times N, where N is a compact smooth manifold with smooth boundary, and endowed with Lorentzian metrics of the form
g_{ij}= −λ(x)(dt – ω_i dx^i)^2 + h_{ij} (x)dx^i dx^j,
where ω is a smooth 1form on N and h is a smooth Riemannian metric on N. These manifolds appear in relativity and acoustics. In this talk, we consider the scattering rigidity problem for SSM, i.e., we study to what extent the Scattering map determines the metric. The definition of the scattering is the following: given a point x on the boundary and a timelike vector v tangent at x, we follow the geodesic flow so that we obtain (assuming some geometric and topological conditions) another point on the boundary and another vector tangent at it. From this information, we recover the metric up to some group of transformations. We obtain results in the analytic case, when dim N = 2, and working for metrics on the same conformal case as in h. We also show a generic result. The proof relies on the theory of Hamiltonian reduction of Hamiltonian Gspaces with symmetry to reduce the problem to the case of MPsystems (which consist of a compact Riemannian manifold with boundary, endowed with a magnetic field and a potential). Then, we relate the scattering from the SSM to the one of some MPsystem obtained in the reduction. The results now follows from previous works of the author.
References:
[1] Assylbekov, Yernat M.; Zhou, Hanming, Boundary and scattering rigidity problems in the presence of a magnetic field and a potential, Inverse Probl. Imaging 9, (2015). 935950.
[2] MuñozThon, Sebastián, The boundary and scattering rigidity problems for simple MP systems (2023), preprint, available at arXiv:2312.02506
[3] MuñozThon, Sebastián, The linearization of the boundary rigidity problem for MPsystems and generic local boundary rigidity, (2024), preprint, available at arXiv:2401.11570
[4] MuñozThon, Sebastián, Scattering rigidity for stationary manifolds via timelike geodesics, (2024), preprint, available at arXiv:2404.09449
Juan Camilo Arosemena (Rice University)
Title: Rigidity of Codimension One Higher Rank Group Actions
Abstract: We prove that any closed manifold M admitting a locally free action of a split higher rank semisimple Lie group G, with codimension one orbits, there exists Γ ≤ G a uniform lattice and a finite cover \tilde{M} of M such that M ̃ is Gequivariantly diffeomorphic to G/Γ × S^1. This result is in the spirit of the Zimmer program. The proof uses ergodic and probabilistic methods, and is based on ideas from the recent proof of non leftorderability of higher rank lattices by Deroin and Hurtado, and in a paper by Deroin and Kleptsyn where they study the dynamics of conformal foliations.
Valentina Zapata Castro (University of Virginia)
Title: Compatible transfer systems for C_{p^r q^s}
Abstract: Transfer systems are mathematical objects that encode transfers within algebras over specific types of structures known as equivariant operads. These systems allow us to employ combina torial tools to investigate equivariant homotopy theory. One significant aspect is the study of compatible pairs of transfer systems, which correspond to multiplicative structures that align with an underlying additive structure.
In this talk, I will introduce Gtransfer systems, which are transfer systems defined for a given group G. I will discuss the fundamental concepts of saturation and pairs of compatible transfer systems. Additionally, I will present joint work with Kristen Mazur, Angelica Osorno, Constanze Roitzheim, Rekha Santhanam, and Danika Van Niel on the compatibility of C_{p^r q^s}transfer systems, where C_{p^r q^s} denotes the cyclic group of order p^r q^s. Specifically, we will outline a criterion for determining when transfer systems only form trivially compatible pairs.
By delving into these topics, we aim to shed light on the intricate relationships between transfer systems and their compatibility, providing valuable insights for further exploration in the field of equivariant homotopy theory.