Poster session
Roberto Carlos Balcazar Araiza (Universidad Autonoma de Yucatan)

Local fundamental theorems of curves and surfaces in the Euclidean space with fractional
derivatives

Abstract The fundamental theorems of curves and hypersurfaces in Euclidean spaces are key results for the classical Differential Geoemtry, with multiple implications. The aim of this research is to extend such theorems to the case where the fundamental quantities involved, such as the curvatures and the fundamental forms, are expressed in terms of certain (ordinary and partial, respectively) fractional derivatives.

Hyeran Cho (The Ohio State University)
joint work with Jean-Francois Lafont (The Ohio State University) and Rachel Skipper
(The University of Utah)

Hyperbolicity of random branched coverings

Abstract: For a finitely presented group $\Gamma$  with a finite presentation, let  $X_\Gamma$ be the presentation 2-complex. We introduce n-fold random branched coverings of $X_\Gamma$ branched over its 2-cells. With probabilistic notions, we prove that fundamental groups of random branched coverings are asymptotically almost surely Gromov hyperbolic. In other words, for a random branched covering $\overline{X}_\Gamma \rightarrow X_\Gamma$, the probability that $\pi_1(X_\Gamma)$  is Gromov hyperbolic goes to 1 in the limit $n \rightarrow \infty$.

$n\to \mathrm{\infty }$
Nicklas Day (Texas A&M University)

Local Geometry of Rank 2 Distributions: Symplectification and Cartan Connections

Abstract: In 1970, N. Tanaka gave a method for obtaining a canonical frame for distribution with aconstant Tanaka symbol. In 2009, B. Doubrov and I. Zelenko utilized a symplectification procedure toobtain a canonical frame for rank 2 distributions independent of their Tanaka symbol under anadditional assumption called maximality of class. For a rank 2 distribution on an n-dimensionalmanifold, the symplectification can be interpreted as the result of n-4 iterative Cartanprolongations and the canonical frame in the method of Doubrov-Zelenko can be arranged to be aCartan connection. The poster gives a stronger result: one can assign Cartan connection to adistribution of maximal class after applying n-5 Cartan prolongations, and no fewer. The talk isbased on joint work with Igor Zelenko.

Yueqing Feng (University of California, Berkele)

Examples of toric scalar-flat Kahler surfaces with mixed-type ends

Abstract: Given a strictly unbounded toric symplectic 4-manifold, we explicitly construct complete toricscalar-flat Kaehler metrics on the complement of a toric divisor. These symplectic 4-manifoldscorrespond to a specific class of non-compact Kaehler surfaces. Furthermore, we reconstruct toricscalar-flat Kaehler metrics with conical singularity along the toric divisor, following the Abreu and Sena-Dias approach.


Pratit Goswami (University of Oklahoma)

On Distortion in Cubical Virtually Special Groups

Abstract: Our work addresses the question: What distortion can subgroups of virtually specialgroups exhibit? For all integers m>0 and k>0, we construct virtually special group G containing a free group H such that the distortion function of H in G grows like exp^k(x^m). This is jointwork with Maya Verma.


Erhan Guler (Texas Tech University and Bartin University)

On a Class of Helical Hypersurfaces in Seven-Dimensional Euclidean Space

In this research, the differential geometry of helical hypersurfaces defined by six parameters within seven-dimensional Euclidean space is thoroughly examined. This study encompasses the determination of the curvatures of these surfaces, an exploration of their minimality, the provision of illustrative examples, and the computation of the Laplace-Beltrami operator to deepen the understanding of their geometric properties.

Taylor Klotz (University of Colorado at Boulder)

Bi-contact Structures and MHS Integrability

Abstract: A Bi-contact structure on a manifold is two co-orientable contact distributi ons whoseassociated contact 1-forms are everywhere linearly independent. In contrast to classical contactgeometry, maps that simultaneously preserve both contact structures possess a local invariant. Theintersecti on of the two contact distributi ons defi nes a line bundle and when the local bi-contactinvariant is constant, there exists a secti on of this line bundle that provides a natural class of examples ofMagneto-Hydro-Stati c (MHS) integrable vector fi elds. Such vector fi elds are geometrically described aspre-symplecti c structures admitti ng at least one volume-preserving Hamiltonian pair. MHS integrablevector fi elds are important types of magneti c fi elds in plasma physics, especially in relati on to plasmaconfi nement and fusion technology.

This work is parti ally supported by the Simons Collaborati on on Hidden Symmetries and Fusion Energy.

Sri Rama Chandra Kushtagi (University of  Texas at Dallas)

 Volume Renormalizati on for Singular Yamabe Metrics in HigherCodimension

Abstract For a closed embedded submanifold $\Sigma^n$ of a closedRiemannian manifold $(M^{n+k},g)$, with $k < n + 2$, we defi ne extrinsic globalconformal invariants of $\Sigma$ by renormalizing the volume associated to theunique singular Yamabe metric with singular set $\Sigma$. For odd $n$, therenormalized volume is an absolute conformal invariant, while for even $n$,there is a conformally invariant energy term given by the integral of a localRiemannian invariant. We also compute the derivati ves of these quanti ti es withrespect to variati ons of the submanifold. We compare these results with theircounterparts in the CCE and the classical singular Yamabe contexts.

Shiyu Liang (University of  Texas at Austin)

Spherical simple knots in lens spaces

Abstract: A knot in a lens space is said to be spherical if it admits Dehn surgery yielding $S^1\ti mesS^2$. We classify spherical simple knots and thereby confi rm the completeness of a list by Baker,Buck, and Lecuona using rati onal Seifert surfaces and Morse functi ons. Additi onally, we show that thehomology classes of spherical knots are determined by simple knots, analogous to Greene's work inthe context of the Berge Conjecture (i.e., surgeries yielding $S^3$).

Rob McConkey (Colorado State University Pueblo)

Linear Bounds on the Crosscap Number of Links

Abstract: Kalfagianni and Lee found two-sided bounds for the crosscap number of an alternating link in terms of certaincoefficients of the Jones polynomial. We show here that we can find similar two-sided bounds for thecrosscap number of Conway sums of strongly alternating tangles. Then we find families of links for whichthese coefficients of the Jones polynomial and the crosscap number grow independently. These families willenable us to show that neither linear bound generalizes for all links.

Lawrence Mouille(Trinity University, San Antonio)

Positive intermediate Ricci curvature with symmetries

Abstract: The goal of classifying positively curved manifolds with large isometry groups has led to great advances in the setting of positive sectional curvature, but has been less effective in the setting of positive Ricci curvature. By considering intermediate curvature conditions, one can gain more traction in this area. In my poster, I will highlight work focused on studying manifolds with positive intermediate Ricci curvature and large isometry groups. In particular, I will include examples of homogeneous spaces, topological consequences of torus symmetries, and obstructions to isometric cohomogeneity-one actions. (Based on joint work with M. Dominguez-Vazquez, D. Gonzalez-Alvaro, and L. Kennard, and work-in-progress with L. Kennard, J. Nienhaus, and E. Khalili Samani.)
Evangelos Nasta (University of Houston)  Geodesic Behavior on Perturbed Hypersurfaces in Euclidean Spaces: A Hamiltonian Approach
Abstract: Studying geodesics on curved surfaces has been a central theme in differential geometry since at least Gauss's fundamental work,  Gauss (1827}. For nearly spherical hypersurfaces, perturbation methods offer powerful tools for analysis, Arnold (1978) . Here, classical mechanical approaches are fused with modern integral geometry to achieve another classification of geodesic behavior.

Miraj Samarakkody (Texas Tech University)
Closed p−Elastic Curves in 2−Space Forms.

Abstract: Variational problems that involve curves with curvature-dependent energy densities are fundamental in differential geometry, geometric analysis, and mathematical physics. Despite the historical roots of the p-elastic curve problem, tracing back to the Bernoulli family and L. Euler, it remains poorly understood. In this study, we explore p-elastic curves across all 2-dimensional space forms for every p ∈ R. We examine the differential equations that describe these curves and establish the necessary conditions for p that allow the existence of closed p-elastic curves. Additionally, we provide a detailed analysis through examples and numerical simulations to enhance the understanding of p-elastic curves’ properties and applications in these geometric contexts.

Ping Wan (University of illinois Chicago)

Quasi-convex Subgroups of Acylindrically Hyperbolic Groups

Abstract: We introduce a definition of quasi-convex subgroups of acylindrically hyperbolic groups.Such subgroups inherit an acylindrically hyperbolic structure from the ambient group. We also showthat these subgroups have uniformly quasi-convex constants in the coned-off cusped Cayley graph,which facilitates the generalization of results about relatively hyperbolic groups.