Vorträge

Donnerstag, 16.10.2025: Curve Shortening Flow on a Surface with Ricci Flow Background

Tejal Bhattarai (Tübingen)

We will study the evolution of embedded closed curves by a curve shortening flow on a smooth closed surface whose background metric evolves by the volume-normalized Ricci flow. We consider the ratio of extrinsic and intrinsic distances along the curve and show that under this coupled flow, this ratio can deteriorate at most exponentially at local minima. For curves that survive for infinite time, using integral estimates, we show that the curvature and all its higher derivatives converge to zero. As an application, we consider the evolution of curves foliating the topological two-sphere and show that there exists exactly one curve that survives the coupled flow for infinite time and smoothly (sub)converges to a great circle on the round sphere.

Mittwoch, 22.10.2025: Tropical Tevelev degrees

Erin Dawson (Tuebingen)

Tropical Hurwitz spaces parameterize genus g, degree d covers of a tropical rational curve with fixed branch profiles. Since tropical curves are metric graphs, this gives us a combinatorial way to study Hurwitz spaces. Tevelev degrees are the degrees of a natural finite map from the Hurwitz space to a product \Mgnbar{g,n} \times \Mgnbar{0,n}. In 2021, Cela, Pandharipande and Schmitt presented this interpretation of Tevelev degrees in terms of moduli spaces of Hurwitz covers. In this talk I will define tropical Tevelev degrees and as well as explore a method to perform this calculation of Tevelev degrees of P^1 using the moduli spaces of tropical Hurwitz covers.

Donnerstag, 23.10.2025: Refined invariants in tropical geometry

Uriel Sinichkin (Tel Aviv)

Refined invariants were introduced into tropical geometry by Block and Gottsche in 2016. Since then various authors have investigated their properties and considered several generalizations of them. In this talk I will explore some of the approaches to investigating refined invariants and present a version of refined invariants in positive genus introduced jointly with Eugenii Shustin.

Donnerstag, 23.10.2025: The Spacetime Penrose inequality with non-optimal constant

Edward Bryden (Antwerp)

Loosely speaking, the Penrose inequality is a relationship between the total mass present and the area of black holes. When working with initial data sets, we need a good stand in for black holes; these will be outermost apparent horizons. Thus, for initial data sets the Penrose inequality can be interpreted as the following statement: mass is bounded below by a constant times the least area required to enclose an outermost apparent horizon. There are physical arguments which suggest a particular constant. In fact, G. Huisken and T. Ilmanen used inverse mean curvature to establish the optimal result in the time symmetric case. Additionally, H. Bray used a conformal flow to obtain another proof of the Penrose inequality in the time symmetric case. We will outline a proof of the Penrose inequality for general initial data sets, but for a constant which is far smaller than what should be optimal.

Donnerstag, 23.10.2025: TBA

Dr. Matthew de Courcy-Ireland (Stockholm University)

Donnerstag, 13.11.2025: TBA

Dr. Barbara Roos (L'Aquila)