Mathematical Sciences

School of Natural Sciences & Mathematics

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


DateSpeakerAffiliationTitleAbstract
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 (Schrödinger, 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.