No seminar this week.
Constructive and Realization Problems — Ben McReynolds
Abstract: A common theme in mathematics is the search for objects with special properties. For example, one might wish to construct a pair of nonisometric isospectral Riemannian manifolds. Or given a finite group, find a Riemannian manifold with the given finite group as its isometry group. Or given a Riemannian manifold, try to realize it as a primitive totally geodesic submanifold in a fixed class of Riemannian manifolds. One approach is to find a random procedure and argue that there is no bias for failure. In this talk, I will discuss examples of constructive/realization problems, how to find a random procedure for solving them, and relate bias in the procedure with geometric/algebraic symmetry.
Instead of FFSS seminar this week, there is an extra geom/top seminar at 4:00 in E308.
A gentle introduction to spectral sequences, part 2 – Jenny Wilson
The objective of this talk is to address the question: How do spectral sequences arise?
We will briefly review the key points from the previous spectral sequences talk, and then walk through an outline of the construction of the spectral sequence associated to a filtered complex — a fairly general construction that includes, as a special case, the Leray-Serre spectral sequence for a fibration, our main example from last time. If there is time, we can look at some more elementary applications of this spectral sequence.
Quasi-isometries of Graphs of Z’s – Daniel Studenmund
Abstract: The classical Baumslag-Solitar groups are two-generated one-relator groups with properties of interest to combinatorial groups theorists. These continue to be excellent testing grounds for phenomena in geometric group theory. A natural generalization of such groups is the class of fundamental groups of graphs of infinite cyclic groups. We will discuss a number of properties of these groups, with focus on the geometric structure used in the quasi-isometry classification due to Farb and Mosher (1998) and Whyte (2001)
Wildness, complexity and geometry – Josh Grochow
Abstract: I will give an expository talk on the interrelationships between geometry, representation theory, and computational complexity. Specifically, I will focus on the representation-theoretic and geometric phenomenon of wildness: roughly, a classification problem is wild if it is as hard as classifying the representations of an arbitrary quiver. I will discuss how wildness arises in and could shed light on several central problems in complexity theory, including testing isomorphism of various structures (Lie algebras, groups, graphs, etc.), matrix multiplication, and the Mulmuley-Sohoni geometric complexity theory approach to P vs. NP. The talk will assume no background in complexity theory.
This will be a practice talk on Alex’s research on Teichmuller curves.
Abstract: Teichmüller curves are the closed complex geodesics in the moduli space of curves of genus . Their study lies at the intersection of dynamics, Teichmüller theory, and algebraic geometry. We will introduce a family of Teichmüller curves generated by abelian
square-tiled surfaces, generalizing work of Eskin-Kontsevich-Zorich and Forni-Matheus-Zorich on cyclic square-tiled surfaces. We will then indicate how the construction of the Bouw-Möller Teichmüller curves may be rephrased in terms of abelian square-tiled surfaces, leading to the surprising result that these curves, which are non-arithmetic
themselves, are images of arithmetic Teichmüller curves under a tautological forgetful map.
Katie Mann – Global fixed points for groups acting on the plane.
The Brouwer Plane translation theorem, in its simplest form, states that if the group acts on the plane by homeomorphisms and has a bounded orbit, then there is a global fixed point for the action. The theorem fails for groups other than , but we can – surprisingly – prove an analogous result if we require that orbits be sufficiently bounded on a bounded set of sufficient size. The philosophy of the proof is that a good way to understand a group acting on the plane is to turn it into an action on the circle.
Strom Borman – Symplectic reduction of quasi-morphisms and quasi-states
Quasi-morphisms on the group of Hamiltonian diffeomorphisms are a convenient way to package and see various rigidity phenomenon in symplectic topology. Their general construction due to Entov and Polterovich uses Hamiltonian Floer homology and requires a certain computation in quantum homology ring. In this talk I will explain how symplectic reduction provides a sort of geometric functoriality that allows quasi-morphisms to descend along to symplectic reductions without further quantum homology computations. As an application, I will present a family of closed symplectic manifolds ( in each even dimension ) that have infinite dimensional spaces of quasi-morphisms. The proof involves a lemma about pulling quasi-morphisms back along `quasi-homomorphisms’.