Harvard University Math Department

Talk begins at 4:30, reception begins at 5:30 in the Common Room, snacks provided.

**Speaker:**Curt McMullen (Harvard)**Title:**Negatively curved crystals**Abstract:**Imagine the Universe is a periodic crystal. If gravity makes space negatively curved, the thin walls of the crystalline structure might trace out a pattern of circles in the sky, visible at night. In this talk we will describe how to generate pictures of these patterns and how to think like a hyperbolic astronomer. We also touch on the connection to knots and links and arithmetic groups. The lecture is accompanied by an exhibit of prints in the Science Center lobby.

**Speaker:**Peter Kronheimer (Harvard)**Title:**Max Dehn and the Dehn twist**Abstract:**Max Dehn made many remarkable contributions to mathematics, and his name pops up in lots of places, most often in topology, where we have "Dehn surgery", the "Dehn twist", and "Dehn's lemma". Famously, Dehn supplied an incorrect proof of the lemma that bears his name. The mistake wasn't noticed for nearly a decade, and took nearly another four decades to fix. In this talk, I won't mention the lemma, but I will say a few words about Dehn himself, a few more about his early work on "scissors congruences", and then yet more on the Dehn twist, closing with a recent result about Dehn twists in four dimensions.

**Speaker:**Dori Bejleri (Harvard)**Title:**Moduli spaces and their compactifications**Abstract:**Moduli spaces are geometric spaces whose points parametrize geometric shapes of various flavors. The geometry of the moduli space reflects how the shapes of interest can be continuously deformed. In this talk, I want to give an introduction to the notion of moduli spaces via the moduli space of triangles. The classical congruence theorems and laws of trigonometry we learn in grade school describe the geometry of this moduli space. I will then explain how analogous ideas are used to understand moduli spaces of elliptic curves, configurations of points on the Riemann sphere, and other objects in complex and algebraic geometry.

**Speaker:**Sebastien Vasey (Harvard)**Title:**Open games, ordinals, and infinite chess**Abstract:**I will give a brief introduction to the theory of infinite games, including how to use ordinal numbers (an extension of the natural numbers) to precisely count the number of steps it takes to win such games. These concepts will be illustrated with positions from infinite chess.

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**Organizers:** Ana Balibanu (ana@math.harvard.edu) and Sebastien Picard (spicard@math.harvard.edu)