# RTG Seminar

### South Hall 6635, 3:30-4:30pm

## 3/16/17, Guoliang Yu, Powell Professor, Texas A & M "The Novikov conjecture and its applications"Abstract: I will explain what is the Novikov conjecture, why it is interesting,and how it is related to topology of manifolds and geometric group theory. This talk will be accessible to graduate students. ## 10/17/13, Greg Galloway, University of Miami, MSRI "On the topology of black holes and beyond"Abstract: In recent years there has been an explosion of interest in black holes in higher dimensional gravity. This, in particular, has led to questions about the topology of black holes in higher dimensions. In this talk we review Hawking's classical theorem on the topology of black holes in 3+1 dimensions (and its connection to black hole uniqueness) and present a generalization of it to higher dimensions. The latter is a geometric result which places restrictions on the topology of black holes in higher dimensions. We shall also discuss recent work on the topology of space exterior to a black hole. This is closely connected to the Principle of Topological Censorship, which roughly asserts that the topology of the region outside of all black holes (and white holes) should be simple. All of the results to be discussed rely on the recently developed theory of marginally outer trapped surfaces, which are natural spacetime analogues of minimal surfaces in Riemannian geometry. This talk is based primarily on joint work with Rick Schoen and with Michael Eichmair and Dan Pollack.## 10/18/13, Greg Galloway, University of Miami, MSRI "Topological censorship from the initial data point of view"Abstract: Topological censorship is a basic principle of spacetime physics. It is a set of results that establishes the topological simplicity at the fundamental group level of the domain of outer communications (the region outside all black holes and white holes) under a variety of physically natural circumstances. An important precursor to the principle of topological censorship, which serves to motivate it, is the Gannon-Lee singularity theorem. All of these results are spacetime results, i.e., they involve conditions that are essentially global in time. From the evolutionary point of view, there is the difficult question of determining whether a given initial data set will give rise to a spacetime satisfying these conditions. In order to separate out the principle of topological censorship from these difficult questions of global evolution, it would be useful to have a pure initial data version of topological censorship. In this talk we give a brief review of topological censorship, and we formulate and present such an initial data version for 3-dimensional initial data sets. The approach taken here relies on recent developments in the existence theory for marginally outer trapped surfaces, and leads to a nontime-symmetric version of the purely Riemannian results of Meeks-Simon-Yau. Geometrization plays an essential role in the proofs. Results in higher dimensions will also be discussed. This talk is based on joint work with Michael Eichmair and Dan Pollack.## 3/7/13, Karsten Grove, University of Notre Dame "A knot characterization and non-negatively curved 4-manifolds with S^1 symmetry"Abstract: About 25 years ago Hsiang and Kleiner showed that a simply connected positively curved 4-manifold with circle symmetry has euler characteristic at most 3 (at most 4 in non-negative curvature [Kleiner and Searle-Yang]). Utilizing Freedman's topological classification of simply connected 4-manifolds, one concludes that such a manifold is homeomorphic to S^4 or CP^2 (adding S^2xS^2 and CP^2 + -CP^2 when the euler characteristic is 4). Combining work of Fintushel (going back about 35 years) with Perelman's resolution of the Poincare conjecture actually yields the classification up to diffeomorphism. Further geometry including a knot characterization provides a more direct classification up to equivariant diffeomorphism (joint work with Wilking).
## 3/8/13, (3-4pm) Karsten Grove, University of Notre Dame "Positive curvature, symmetry, and Tits geometry"Abstract: During the past two decades investigations of positively curved manifolds in the presence of (large) isometry groups have flourished. In particular this has lead to a number of classification type results as well as to the discovery that the unit sphere bundle of the 4-sphere with an exotic structure has a metric with positive curvature (Dearricott and joint work with Verdiani and Ziller). The new example was discovered via classification work on positively curved manifolds of cohomogeneity one, i.e., with one dimensional orbit space, or equivalently with orbits of codimension one (joint work with Wilking and Ziller). Cohomogeneity one actions are examples of so-called polar actions. It turns out that polar actions of cohomogeneity at least two on positively curved manifolds are intimately related to the theory of buildings by Tits. This in part has lead to a complete classification (joint work with Fang and Thorbergsson). In this talk, I will provide a general background for the investigations alluded to above, and emphasize the link to Tits geometry. All necessary concepts will be discussed.
## 10/5/12, 10/12/12, Lee Kennard, UCSB, see Differential Geometry Seminar ## 10/20/11 Zhenghan Wang, Microsoft Station Q "What is a topological phase of matter?"Abstract: In this informal discussion, we will discuss topological quantum systems using the Hamiltonian formalism and real examples if time permits.
## 10/21/11 Zhenghan Wang, Microsoft Station Q "Topological phases of matter: modeling and classification"Abstract: Motivated by application to quantum computing, we will discuss theoretical modeling and mathematical classification of topological phases of matter. Topological quantum systems are closely related to unitary stable topological quantum field theories. While a general classification is difficult, progress has been made for (2+1)-dimension and short-ranged entangled states. In particular, non-interacting fermion topological quantum systems can be classified completely via K-theory by A. Kitaev.## 12/1/11 Tobias Colding, MIT, visiting MSRI "Monotonicity formulas revisited"Abstract: In this talk I will discuss several new monotonicity formulas for manifolds with a lower Ricci curvature bound. The monotonicity formulas are related to the classical Bishop-Gromov volume comparison theorem and Perelman's celebrated monotonicity formula for the Ricci flow. I will explain the connection between all of these. Moreover, I will explain how these new monotonicity formulas are linked to a new sharp gradient estimate for the Green's function. This is parallel to that Perelman's monotonicity is closely related to the sharp gradient estimate for the heat kernel of Li-Yau. In addition, there are obvious parallels between our monotonicity and the positive mass theorem of Schoen-Yau and Witten. |