Differential Geometry, PDE and Mathematical Physics
Department of Mathematics and Statistics
Texas Tech University
We introduce the Chern–Ricci flow, a parabolic flow of Hermitian metrics on compact complex manifolds. We show that finite time non-collapsing singularities of the Chern-Ricci flow on compact Hermitian manifolds always form along analytic subvarieties, thus partially answering a question of Feldman–Ilmanen–Knopf and Tosatti--Weinkove.
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In his Habilitationsschrift Riemann mentioned the geometric tangent 2D-indicatrix as a relevant extension of the Euclidean unit circle, thus conceiving the simplest Riemann-Finsler metrics. However, in 1818, Gabriel Lamé (1795-1860) had already provided a systematic treatment of supercircles and superellipses, with the aim of describing crystals geometrically.
The generalization of Lamé curves to the Superformula, a generic geometric transformation generalizing conic sections, turns out to be an excellent scientific tool for the study of natural shapes and phenomena, from the very small to the very large. These developments have found their way in numerous applications in technology and science. It also connects many fields of science in a simple way, e.g. shape and form (constant anisotropic mean curvature surface), and inequality in economics and biology.
In a dynamic and kinematic setting, recent developments include a complete differential geometry apparatus, adapted to these shapes. This aligns with Lamé’s Unique Rational Science: find the best fitting coordinate systems and then solve the relevant boundary value problems.
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The Links–Gould (LG) polynomials are two-variable link invariants arising from the quantum supergroup $U_q(\mathfrak{sl}(2|1))$, exhibiting hybrid behavior between the Jones and Alexander polynomials. I will survey some recent results illustrating this theme.
The basic LG invariant provides a lower bound on the Seifert genus and this bound extends to its colored versions. It has been verified by Garoufalidis and Li that the 2-colored Links–Gould polynomial detects the genus for all 352.2 million prime knots with up to 19 crossings. We also prove a conjecture of Geer and Patureau-Mirand that the Links–Gould invariant admits a specialization to the Akutsu–Deguchi–Ohtsuki (ADO) invariant at a sixth root of unity. Finally, we establish analogs of Jones–Wenzl idempotents for LG and formulate an analog of the Fox conjecture, supported by computational evidence.
This is joint work, in various combinations, with Stavros Garoufalidis, Rinat Kashaev, Ben-Michael Kohli, Jiebo Song, Guillaume Tahar, and Emmanuel Wagner.
In weighted Kähler geometry, we consider Kähler manifolds equipped with a torus action and a fixed positive function on the moment polytope. I will introduce this setting, as well as the notions of canonical metrics in weighted Kähler geometry: weighted solitons and weighted cscK metrics. I will then review some results on existence of such metrics, and applications to the more classical Kähler-Einstein metrics, Kähler-Ricci solitons and Calabi's extremal Kähler metrics.
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