Differential Geometry
Differential geometry is a branch of mathematics that uses the tools of calculus and linear algebra to study the geometry of smooth shapes, such as curves, surfaces, and higher-dimensional spaces. It focuses on properties like curvature, distances, angles, and how objects bend or change in space, often by representing these objects as differentiable manifolds. Differential geometry has deep applications in physics, especially in general relativity, where it provides the language for describing spacetime curvature, as well as in fields like engineering, computer graphics, and robotics.
Featured Publications
Cohesive modules give a dg-enhancement of the bounded derived category of coherent sheaves on a complex manifold via superconnections. In this paper we discuss the deformation theory of cohesive modules on compact complex manifolds. This generalizes the deformation theory of holomorphic vector bundles and coherent sheaves. We also develop the theory of Kuranishi maps and obstructions of deformations of cohesive modules and give some examples of unobstructed deformations.
We give a complete classification of local and global conformal biharmonic maps between any two space forms by proving that a conformal map between two space forms is proper biharmonic if and only if the dimension is 4, the domain is flat, and it is a restriction of a Möbius transformation. We also show that proper k-polyharmonic conformal maps between Euclidean spaces exist if and only if the dimension is 2k and they are precisely the restrictions of Möbius transformations. This provides infinitely many simple examples of proper k-polyharmonic maps with nice geometric structure.
In this paper we first prove a characterization formula for biharmonic maps in Euclidean spheres and, as an application, we construct a family of biharmonic maps from a flat 2-dimensional torus \(\mathbb {T}\) into the 3-dimensional unit Euclidean sphere \(\mathbb {S}^3\). Then, for the special case of maps between spheres whose components are given by homogeneous polynomials of the same degree, we find a more specific form for their bitension field. Further, we apply this formula to the case when the degree is 2, and we obtain the classification of all proper biharmonic quadratic forms from \(\mathbb {S}^1\) to \(\mathbb {S}^n\), \(n \geq 2\), from \(\mathbb {S}^m\) to \(\mathbb {S}^2\), \(m \geq 2\), and from \(\mathbb {S}^m\) to \(\mathbb {S}^3\), \(m \geq 2\).
BCV spaces are a family of 3-dimensional Riemannian manifolds which include six of Thurston's eight geometries. In this paper, we give a complete classification of proper biharmonic Riemannian submersions from a 3-dimensional BCV space by proving that such biharmonic maps exist only in the cases of \( H^2\times \mathbb {R}\rightarrow \mathbb {R}^2 \) , or \( {\widetilde{SL}}(2,\mathbb {R})\rightarrow \mathbb {R}^2 \)
. In each of these two cases, we are able to construct a family of infinitely many proper biharmonic Riemannian submersions. Our results, on one hand, extend a previous result of the authors which gave a complete classification of proper biharmonic Riemannian submersions from a 3-dimensional space form, and, on the other hand, they can be viewed as the dual study of biharmonic surfaces (i.e., biharmonic isometric immersions) in a BCV space studied in some recent literature.
This monograph addresses two significant related questions in complex geometry: the construction of a Chern character on the Grothendieck group of coherent sheaves of a compact complex manifold with values in its Bott-Chern cohomology, and the proof of a corresponding Riemann-Roch-Grothendieck theorem. One main tool used is the equivalence of categories established by Block between the derived category of bounded complexes with coherent cohomology and the homotopy category of antiholomorphic superconnections. Chern-Weil theoretic techniques are then used to construct forms that represent the Chern character. The main theorem is then established using methods of analysis, by combining local index theory with the hypoelliptic Laplacian
We study the descent problem of cohesive modules on complex manifolds. For a complex manifold X we can consider the Dolbeault dg-algebra
on it and Block in 2006 introduced a dg-category
called cohesive modules, associated with
. The same construction works for any open subset \(U\subset X\) and we obtain a dg-presheaf on X given by
We prove that this dg-presheaf satisfies the descent property for any locally finite open cover of a complex manifold X. This generalizes part of the results of Ben-Bassat and Block in 2012, who studied the case that X is covered by two open subsets.
We consider the BGG category \( O \) of a quantized universal enveloping algebra \( U_{q}(g) \). It is well-known that \( M \otimes N\in O \) if M or N is finite dimensional. When g is simple and of type ADE, we prove in this paper that \( M \otimes N \notin O \) if M and N are both infinite dimensional.
In this paper, we give an explicit second variation formula for a biharmonic hypersurface in a Riemannian manifold similar to that of a minimal hypersurface. We then use the second variation formula to compute the normal stability index of the known biharmonic hypersurfaces in a Euclidean sphere and to prove the nonexistence of unstable proper biharmonic hypersurface in a Euclidean space or a hyperbolic space, which adds another special case to support Chen's conjecture on biharmonic submanifolds.
We consider the BGG category O of a quantized universal enveloping algebra Uq (g). We call a module M ∈ O tensor-closed if M ⊗ N ∈ O for any N ∈ O. In this paper we prove that M ∈ O is tensor-closed if and only if M is finite dimensional. The method used in this paper applies to the unquantized case as well.
We consider the dg-category of twisted complexes over simplicial ringed spaces. It is clear that a simplicial map \( f:({\mathscr {U}},{\mathscr {R}})\rightarrow ({\mathscr {V}}, {\mathscr {S}}) \) between simplicial ringed spaces induces a dg-functor \( f^*:\mathrm{Tw}({\mathscr {V}}, {\mathscr {S}})\rightarrow \mathrm{Tw}({\mathscr {U}}, {\mathscr {R}}) \) where \( \mathrm{Tw}({\mathscr {U}}, {\mathscr {R}}) \) denotes the dg-category of twisted complexes on \( ({\mathscr {U}},{\mathscr {R}}) \). We prove that for simplicial homotopic maps f and g, there exists an \( A_{\infty } \)-natural transformation \( \Phi :f^*\Rightarrow g^* \) between induced dg-functors. Moreover, the 0th component of \( \Phi \) is an objectwise weak equivalence. If we restrict ourselves to the full dg-subcategory of twisted perfect complexes, then we prove that \( \Phi \) admits an \( A_{\infty } \)-quasi-inverse when \( ({\mathscr {U}},{\mathscr {R}}) \) satisfies some additional conditions.
News
ETAMU Students Turn Data into Real-World Solutions at Alteryx Datathon
Students from across East Texas A&M University applied their analytical skills to real-world challenges during this fall's Alteryx Datathon
From ETAMU to Amazon: Computer Science Alum Shares Career Journey
East Texas A&M University alumna Tejaswini Atluri recently spoke with the campus ACM chapter about her new role as a software development engineer at Amazon.
Students Connect with Industry Leaders to Explore “The Future of Work in the Age of AI”
East Texas A&M recently hosted its first Industry Connect event in partnership with the Paris Economic Development Corporation, giving students look at how leading companies are using artificial intelligence to transform the modern workplace.
