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Talk 01 — Roberta Shapiro (University of Michigan, USA) — [computing…]Local time: click here to view in your time zoneTitle: Geometry, topology, and combinatorics of fine curve graph variants
Abstract
The fine curve graph of a surface encodes information about curves on a surface and their interaction. In this talk, we will discuss recent developments in the area of fine curve graphs relating to their geometry, topology, and combinatorics.
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Talk 02 — Nan Wu (The University of Texas at Dallas, Department of Mathematical Sciences, USA) — [computing…]Local time: click here to view in your time zoneTitle: Adaptive Bayesian Regression on Manifold
Abstract
We investigate how the posterior contraction rate under a Gaussian process prior is influenced by the intrinsic dimension of the domain and the smoothness of the regression function. Specifically, we consider the setting where the domain is a d-dimensional manifold and the regression function is intrinsically s-Hölder smooth on the manifold. We establish the optimal posterior contraction rate of O(n^{-s/(2s + d)}), up to a logarithmic factor. To eliminate the need for prior knowledge of the manifold's dimension, we propose an empirical Bayes prior on the kernel bandwidth, leveraging kernel affinity and k-nearest neighbor statistics. This talk is based on joint work with Tao Tang, Xiuyuan Cheng, and David Dunson.
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Talk 03 — Mariusz Urbanski (University of North texas, USA) — [computing…]Local time: click here to view in your time zoneTitle: One-Dimensional and Multi-Dimensional Continued Fractions \Geometry & Topology
Abstract
I will present geometric and topological properties of the Gauss map acting on irrational numbers with restricted continued expansion entries. I will also discuss geometric and topological properties of multidimensional Gauss iterated function systems, showing, in particular, that the closures of the corresponding limit sets are homeomorphic to the Wazewski universal dendrite.
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Talk 04 — Sumeyra Sakalli (University of South Florida, USA) — [computing…]Local time: click here to view in your time zoneTitle: Topological Approach to Algebraic Genus Two Singular Fibers
Abstract
Kodaira's classification of singular fibers in elliptic fibrations and its translation into the language of monodromies by Harer, Kas, Kirby are one of the main tools in constructions of exotic 4-manifolds. After Kodaira, Namikawa and Ueno classified singular fibers of genus two families of algebraic curves. A subset of those fibers has finite order monodromies, which, in this sense, is analogous to Kodaira’s genus one fibers, and has received attention in the literature. In this talk, first we will translate between singular fibers of genus two families of algebraic curves with finite-order monodromies, and the positive Dehn twist factorizations of Lefschetz fibrations. Our result can be thought as analogous to Harer, Kas, and Kirby's result on Kodaira's singular fibers. Then we will construct some of these fibers with a different approach and talk about applications. This is based on joint works with J. Van Horn-Morris, arXiv:2303.01554 and arXiv:2311.00264.
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Talk 05 — Tatiana Abdelnaim (University of Oklahoma, USA) — [computing…]Local time: click here to view in your time zoneTitle: Cohomology of congruence subgroup $\Gamma_{0,n}(p)$
Abstract
Let \(\Gamma_{0,n}(p)\subset \SL_n (\mathbb Z)\) be the congruence subgroup of level p whose first column is congruent to \((*,0,\dots,0)^t \mod p\). The cohomology of this subgroup has connections to problems in algebraic K-theory and number theory. Borel and Serre (1973) showed that the cohomology of \(\Gamma_{0,n}(p)\) vanishes above degree \(\binom{n}{2}\). We prove that the top-dimensional cohomology group $\homology^{\binom{n}{2}}(\Gamma_{0,n}(p);\mathbb Q )$ vanishes for all \(p\in \{2,3,5,7,13\}\) when \(n \ge 3\), as well as for all primes \(p \leq 6n-14\). We also reprove the known non-vanishing result that this group is nonzero for n=2 for every prime p, and we establish a new non-vanishing for n=3 for all primes \(p \notin \{2,3,5,7,13\}\). Additionally, using Hopf module structure, we find new cohomology classes in degrees below $\binom{n}{2}$. In this talk, I will outline the ideas behind these results.
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Talk 06 — Zhe Su (Auburn University, USA) — [computing…]Local time: click here to view in your time zoneTitle: De Rham-Hodge-enabled data analysis on manifolds
Abstract
Modern data arise in diverse formats, including point clouds, images, and fields, many of which are naturally modeled as data on manifolds. In this talk, I will present our recently developed de Rham-Hodge-enabled frameworks for data analysis on manifolds. Built on Hodge Laplacians, Hodge decompositions, and persistent Hodge Laplacians, these methods provide effective and efficient ways to extract both the topological and geometric features, and are well-suited for integration with machine learning models. I will illustrate the utility of these frameworks through applications in mathematical biology, including protein-ligand binding affinity prediction, single-cell RNA velocity analysis, medical image classification, and protein B-factor analysis.
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Talk 07 — Dmitri Burago (PennState, USA) — [computing…]Local time: click here to view in your time zoneTitle: Finite Distance Altrenative
Abstract
We will discuss the following situation. Let a group G act cocompactly on X. We say that (X,G) satisfy the Finite Distance Property (FDP) if, whenever two G-invariant metric diverge sublinearly at infinity, they are within finite GH-distance. We will discuss several examples when FDP holds and propose some open problems.
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Talk 08 — Stephen McKean (Brigham Young University, USA) — [computing…]Local time: click here to view in your time zoneTitle: Hyperkähler bases for six rational bordism theories
Abstract
I will discuss joint work with Buchanan, Debray, and Krulewski, in which we describe manifolds that represent bases for rational spin, spin^c, and spin^h bordism, along with their symplectic counterparts. These manifolds are all hyperkähler except in dimensions 4n+2, in which case the manifolds are given by the product of a hyperkähler manifold and a 2-torus.
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Talk 09 — Meenakshy Jyothis (The University of Oklahoma, USA) — [computing…]Local time: click here to view in your time zoneTitle: Isometric rigidity of projective filling currents
Abstract
Among the three well-known metric structures on the Teichmüller space of a surface is the Thurston metric, introduced by Thurston in the 1980s. In this talk, I will introduce a metric on the space of projective filling currents that extends the Thurston metric on Teichmüller space. The space of geodesic currents can be thought of as a measure-theoretic completion of closed curves on a surface and contains several important objects associated with a surface, such as the homotopy classes of closed curves, Teichmüller space, measured laminations, and singular flat structures. Projective filling currents form a dense subset of its projectivization. We show that, for a closed surface S, the geometry of the space of projective filling currents with respect to this metric is determined by the genus of S. This is joint work with Didac Martinez-Granado.
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Talk 10 — Hasitha Anuradha Ekanayake (University of Oklahoma, USA) — [computing…]Local time: click here to view in your time zoneTitle: Minimal volume cusped hyperbolic 3-manifolds with a compact totally geodesic boundary
Abstract
Determining lowest volume examples in different topological classes is an extensive project in the study of topology and geometry of hyperbolic 3-manifolds. Its roots can be tracked down at least to Thurston's notes written in 1979. Thurston observed that the volume is a good measurement of the topological complexity. The focus of this talk will be on manifolds with compact totally geodesic boundary. Topological complexity of such manifolds was studied extensively by Frigerio, Martelli and Petronio in the early 2000s. As a part of their work, they constructed a census of minimal complexity manifolds with totally geodesic boundary. In this talk, I will outline a method to identify the lowest volume non-compact (1-cusped) manifold with a compact totally geodesic boundary and discuss potential lowest volume candidates from Frigerio, Martelli, Petronio census for cases of multi-cusps.
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Talk 11 — Martin Schmoll (Clemson University, USA) — [computing…]Local time: click here to view in your time zoneTitle: Wasserstein Distances and probabilistic frames
Abstract
We recall the notion of probabilistic frame in terms of Wasserstein distances, give a few implications and applications of this viewpoint and some background on the theory of probabilistic frames. Once this is done we will venture into some geometric problems related to Wasserstein distances. This is work together with Dongwei Chen (Colorado State).
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Talk 12 — Henry Adams (University of Florida, USA) — [computing…]Local time: click here to view in your time zoneTitle: Vietoris-Rips complexes of manifolds
Abstract
I will survey what is known (and mostly unknown) about Vietoris-Rips complexes of manifolds. We might consider a dataset to be nice if it were densely sampled from a manifold. But we do not know what the Vietoris-Rips persistent homology of manifolds looks like, typically! I will comment on Vietoris-Rips complexes of the circle, spheres, ellipsoids, tori, and manifolds, and advertise open questions.
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Talk 13 — Leslie Mavrakis (University of Utah, USA) — [computing…]Local time: click here to view in your time zoneTitle: Combinatorial Characterizations and Branched Manifolds
Abstract
A family F of compact n-manifolds is locally combinatorially defined (LCD) if there is a finite number of triangulated n-balls such that every manifold in F has a triangulation that locally looks like one of these n-balls. In joint work with Daryl Cooper and Priyam Patel, we show that LCD is equivalent to the existence of a compact branched n-manifold W, such that F is precisely those manifolds that immerse into W. In this way, W can be thought of as a universal branched manifold for F. In current and future work, we use this equivalence to show that, for each of the eight Thurston geometries, the family of closed 3-manifolds admitting that geometry is LCD. In this talk, I will present the main ideas of the proof of the equivalence and if time permits, construct branched 3-manifolds for a few of the geometries.
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Talk 14 — Nikola Milicevic (The University of Texas at El Paso, USA) — [computing…]Local time: click here to view in your time zoneTitle: A continuum of Kunneth theorems for persistent homology
Abstract
Given a parametrized filtration of topological spaces, the goal of persistent homology is understanding how homology groups of the spaces vary with the choice of parameters. This information is encoded in an algebraic object called a persistence module. If the parameters are one-dimensional such as R, this is called one-parameter persistence, while parameters with more than one dimension such as R^n yield multi-parameter persistence. Persistence modules have been shown to have a particularly rich algebraic structure. We develop new aspects of the homological algebra theory for persistence modules, in both the one-parameter and multi-parameter settings. For each p>0, we define a novel tensor product of persistence modules. We prove that each such tensor product is left adjoint to an internal hom functor that also depends on p>0. Furthermore, we compute their respective derived functors, Tor and Ext, explicitly for interval modules. We prove that every tensor product yields a Kunneth short exact sequence of chain complexes of persistence modules arising from filtered cell complexes. We also prove that every internal hom yields a Universal Coefficients Theorem. We show the resulting short exact sequence computes the Borel-Moore homology of an open complex induced from the original filtration on a cell complex. Finally, we show that for every p≥0 the associated Kunneth short exact sequence can be used to significantly speed up and approximate persistent homology computations in a product metric space X x Y with the distance d^p((x,y),(x',y'))=||d_X(x,x'),d_Y(y,y')||_p.
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Talk 15 — Azer Akhmedov (North Dakota State University, USA) — [computing…]Local time: click here to view in your time zoneTitle: Discrete Subgroups of Diffeomorphism Groups of 1-Manifolds
Abstract
The study of discrete subgroups of Lie groups has experienced phenomenal growth and development over the last 60 years. In the case of SL(2,R) and SL(2,C), the study of their discrete subgroups started in the 19th century in the works of F.Klein and H.Poincare. Despite being very old, this area is still very dynamic. For general Lie groups, the study of their discrete subgroups has culminated in the works of Borel, Selberg, Mostow, Margulis, Zimmer and many others exploring deeply the group-theoretic properties of these subgroups as well as the geometric and dynamical properties of their actions on symmetric spaces, homogeneous spaces, manifolds, etc. If M^1 is a one-dimensional compact manifold (so essentially either a circle S^1 or a closed bounded interval I = [0,1]), then the groups of homeomorphisms and diffeomorphisms of M^1 are infinite-dimensional. Nevertheless, in recent trends, researchers have proposed to view the group Diff_{+}^r(M^1) of C^r-regularity diffeomorphisms as an analogue of finite-dimensional (semi-simple) Lie groups. Especially in regularities r>1, some striking results have been obtained in the last three decades, which indicate a certain tameness in the behavior of finitely generated subgroups. In most recent years, the study of discrete subgroups has been launched as well, and the results obtained already point us toward a very rich theory. In particular, it has led to the solution of several long-standing problems. In this talk, I will briefly summarize some recent results in this area. Then I will focus on the problem of the classification of diffeomorphism groups where every non-identity diffeomorphism has at most N fixed points for some uniform N.
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Talk 16 — Chenxing (University of Pittsburgh, USA) — [computing…]Local time: click here to view in your time zoneTitle: Geometry and Two-Sided Asymptotics in Probability Quantization under Constraints
Abstract
Classical quantization studies how well a probability measure can be approximated by finitely supported measures when the approximants of finite cardinality (quantizers) are free to move in the ambient space. In constrained quantization, however, the quantizers are required to lie in a prescribed closed set, and the asymptotic behavior of the error becomes sensitive to the geometry of both the support and the constraint. In this talk, I discuss recent results on upper and lower asymptotics for constrained quantization of compactly supported measures. The main analytic tool is the metric projection onto the constraint, together with a measurable selector and its pushforward measure, which allow the constrained problem to be compared with quantization on the projected set. On the upper-bound side, this yields estimates controlled by the geometry of the projection and the regularity of the associated pushforward measure. On the lower-bound side, I introduce a perturbative argument, refined by a good-lambda type mechanism over Voronoi cells, to obtain quantitative lower estimates under mild nondegeneracy assumptions. Under suitable regularity hypotheses, including Ahlfors-type conditions, geometric accessibility, and a mild thickness condition along the constraint, the upper and lower bounds match and lead to sharp dimension formulas. A central theme of the talk is how these analytic conditions can be reformulated in geometric and topological terms, and how the lower-bound mechanism suggests further connections with harmonic-analytic ideas.