Graduate Lectures 1/12

September 24, 2025

Sebastian Baader

Prof. Sebastian Baader (Universität Bern)

Claudia Stadlmayr (Université de Neuchâtel) and Hugo Parlier (Université de Fribourg)

The purpose of these traditional lectures is to present recent important developments in mathematics to a large audience of graduate students and young researchers. The lecturers are chosen as experts in their field, with demonstrated pedagogical talent.

The two lectures are held by Claudia Stadlmayr (Uni Neuchâtel) and Hugo Parlier (Uni Fribourg).

They will cover the classical subjects of hyperbolic surfaces and singularities of algebraic surfaces. The results and methods will range from topology and geometry to algebra and number theory, from classical to contemporary.

There will be approximately 12 double lectures of 90 minutes on

Wednesdays (13:15-14:45 & 15:00-16:30), at the University of Bern.

Meeting point: Wednesday, September 24, 13:15, Room B78, building ExWi
(Sidlerstrasse 5).

Here is the schedule with course titles and abstracts.

Wednesday, 13h15-14-45: Prof. Dr. Hugo Parlier

Title: Getting a handle on hyperbolic surfaces

Abstract: This course is about how to build hyperbolic surfaces-and what to do with them. Hyperbolic surfaces offer a rich setting where geometry, topology, group theory, and dynamics come together. They are geometrically flexible: any two homeomorphic hyperbolic surfaces can be continuously deformed into one another. This leads naturally to a geometric perspective on their moduli spaces, which, roughly speaking, parametrize hyperbolic surfaces up to isometry.

We will begin by constructing hyperbolic surfaces from polygons and other elementary pieces, and see how all such surfaces arise in this way. Along the way, we'll explore deformation spaces and the role of closed geodesics, which help probe the geometry of surfaces and provide a way to measure how close-or far apart-two surfaces are.

The course will start with minimal prerequisites, relying only on basic ideas from Euclidean and hyperbolic geometry. The pace and depth will adapt to the background and interests of participants.

Wednesday, 15h00-16h30: Dr. Claudia Stadlmayr

Title: Singularities of Algebraic Surfaces

Abstract: An algebraic surface is a 2-dimensional (so, over the complex numbers, topologically 4-dimensional) zero locus of a set of polynomials. Smooth points of algebraic surfaces are, as intuition tells us, those points where the tangent space has the expected dimension 2 and surface singularities, the subject of this lecture, are the points that violate this condition.

This course will be an introduction to the theory of surface singularities with a special focus on rational double points, which, through their connection to symmetry groups of platonic solids and to the theory of canonical singularities, weave a thematical thread between the geometry of ancient Greece and modern birational geometry. The methods we employ will be algebro-geometric, with a special focus on positive characteristic, and we will introduce the necessary concepts in commutative algebra and algebraic geometry, such as:

- The spectrum of a ring and its Zariski topology

- Localization and completion

- Resolution of singularities: Normalization and blow-ups

- Invariants of singularities: Multiplicity, rationality, embedding dimension, and discrepancies

Throughout, we will see rational double points appear in various guises and as an outlook we will also see the surprising interaction between their deformation theory and algebraic groups.

University of Bern, building ExWi, Sidlerstrasse 5, Room B78

Doctoral Students from CUSO Universities will get their travel expenses reimbursed (SBB ticket 2nd class, 1/2 tax)

Please register under: math.cuso.ch/activities/internal-program

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