# Geometry, Topology, Dynamical Systems Seminar, AY 15-16

Date | Speaker | Affiliation | Title | Abstract |
---|---|---|---|---|

Apr 11 (MONDAY, 3:00pm in CB3 1.314) | Hitoshi Murakami | Tohoku University, Japan | An introduction to the volume conjecture, I | We define the Jones polynomial and show it is an invariant for knots in the three-dimensional space. Then we define the “colored” Jones polynomial. |

Apr 12 (TUESDAY, 11am in JSOM 11.206) | Hitoshi Murakami | Tohoku University, Japan | An introduction to the volume conjecture, II | We quickly introduce the three-dimensional hyperbolic geometry. We also show that the complement of the figure-eight knot has a hyperbolic structure. |

Apr 13 (WEDNESDAY, 2:30pm in CB3 1.314) | Hitoshi Murakami | Tohoku University, Japan | An introduction to the volume conjecture, III | We introduce the volume conjecture and prove it in the case of the figure-eight knot. Indeed, we show that a certain limit of the colored Jones polynomial defines the volume of the complement of the figure-eight knot. |

Feb 29 (MONDAY, 2:30pm in FN 2.202) | Takayuki Morifuji | Keio University, Japan | Twisted Alexander polynomial and its applications I | In this series of 3 one-hour lectures, we explain basic properties of twisted Alexander polynomials and discuss some applications to topology of 3-dimensional manifolds (in particular of knot complements in the 3-sphere). More precisely, we focus on fibering and genus detecting problems, and further we mention a conjecture of Dunfield, Friedl and Jackson for hyperbolic knots. |

Mar 1 (TUESDAY, 11am in JSOM 11.206) | Takayuki Morifuji | Keio University, Japan | Twisted Alexander polynomial and its applications II | |

Mar 2 (WEDNESDAY, 2:30pm in FN 2.202) | Takayuki Morifuji | Keio University, Japan | Twisted Alexander polynomial and its applications III | |

Nov 2 | Cesare Tronci | University of Surrey, UK | Variational and Poisson-bracket approaches to quantum dynamics | Starting from the Dirac-Frenkel Lagrangian for pure quantum states, symmetry methods are applied to provide new variational principles for the dynamics of pure and mixed states in different pictures (SchrÃ¶dinger, Heisenberg, Dirac, Wigner-Moyal, and Ehrenfest). In addition, a hybrid classical-quantum Poisson bracket is provided for expectation value dynamics, which is then shown to be canonical (Hamiltonian) for any quantum state. |

Oct 12 | Razvan Gelca | Texas Tech University | Chern-Simons theory and Weyl quantization | Chern-Simons theory is a quantum theory based on the Chern-Simons Lagrangian, and was introduced by E. Witten to explain the Jones polynomial of knots. Since its introduction, this theory proved to have a unifying nature, bringing together quantum theory, 3-dimensional topology and geometry, representation theory, and algebraic geometry. This talk is based on a discovery made by the speaker in joint work with Alejandro Uribe, which shows that the quantization model introduced by H. Weyl in 1931 plays a central role in Chern-Simons theory. |

Oct 5 | Cynthia Curtis | College of New Jersey | The SL(2,C) Casson invariant for knots and the A-polynomial | Low-dimensional topologists study both knots and 3-dimensional manifolds, and in fact all 3-dimensional manifolds can be constructed using knots. We explain this relationship and discuss how we can study both knots and 3-manifolds by looking at representations of groups associated to the knots and 3-manifolds. We focus on two invariants of knots and 3-manifolds which are constructed from such representations, the SL(2, C) Casson invariant and the A-polynomial. We show that the SL(2, C) Casson invariant predicts the degrees of a variant of the A-polynomial and discuss the computability and power of each. |

Sep 28 | Maxim Arnold | UT Dallas | On the shock function for planar Burgers equation (cont’d) | It is well-known that zero-viscosity Burgers equation posses a finite-time singularity. Such a singularity is often called a shock. To construct a solution after shock formation one needs to define a velocity vector field in the point of the shock. This can be done using various methods. I will describe a geometric construction for this and use it to describe the set of points falling to the shock. |

Sep 21 | Maxim Arnold | UT Dallas | On the shock function for planar Burgers equation | It is well-known that zero-viscosity Burgers equation posses a finite-time singularity. Such a singularity is often called a shock. To construct a solution after shock formation one needs to define a velocity vector field in the point of the shock. This can be done using various methods. I will describe a geometric construction for this and use it to describe the set of points falling to the shock. |

Sep 14 | Susan Aberanathy | Angelo State University | Genus-1 tangles and Kauffman bracket ideals | A genus-1 tangle is a 1-manifold with two boundary components properly embedded in the solid torus. A genus-1 tangle G embeds in a link L if G can be completed to L by a 1-manifold in the complement of the solid torus containing G. A natural question to ask is: given a tangle G and a link L, how can we tell if G embeds in L? We discuss the Kauffman bracket ideal (along with its even and odd versions) which gives an obstruction to embedding, and outline a method for computing a finite list of generators for these ideals. We also examine some specific examples and use our method to compute their Kauffman bracket ideals. |