Columbia University

SEMINARIO DE GEOMETRÍA

Conferenciante: Raphael Tsiamis (Columbia University)

Abstract: Translators in \(\mathbb{R}^3\) are solitons of the mean curvature flow for embedded 2-surfaces. In the semigraphical case, where the translators are allowed graphical as well as vertical components, Hoffman-Martín-White classified the surfaces into six types. They conjectured the uniqueness of the objects within two families contained in slabs, the "helicoids" and the "pitchforks," for any given width. We present the proof of the conjecture by combining an arc-counting argument motivated by Morse-Radó theory for translators with a rotational application of the maximum principle. We then discuss applications of this result to the classification of semigraphical translators in \(\mathbb{R}^3\) and their limits, related to the work of Hoffman-Martín-White and Gama-Martín-Møller. This is joint work with F. Martín and M. Sáez.

Geometry seminar

Speaker: Mu Tao Wang (Columbia University)

Abstract: A map is said to be homotopic equivalent to another map if there is a continuous path of maps connecting the two. Surprisingly a geometric flow such as the mean curvature flow provides natural paths to deform a map to a canonical representative in its homotopy class. I shall discuss this approach and its applications, and in particular some recent results regarding the homotopy class of maps between complex projective spaces of different dimensions. This is based on joint work with Chun-Jun Tsai and Mao-Pei Tsui.