As is known from the 1980s, CFT should describe scaling limits of critical lattice models in statistical mechanics (i.e., fixed points of the renormalization group flow). One might argue that two-dimensional (2D) CFTs -- at least minimal models and Liouville theory -- are completely understood since the pioneering works in the 1980s. However, a closer inspection reveals that CFTs rele- vant to models containing lattice interfaces, or random curves in the continuum, cannot be described by minimal models, seem to exhibit logarithmic phenomena, and even their spectrum (operator content) seems not to be completely clear. In the random geometry community, about 25 years ago a breaktrough idea came along: Schramm suggested that probabilistically, such interface models could be rigorously described by combining classical complex analysis (namely, Loewner theory) with stochastic analysis (namely, Brownian motion and martingale theory). This provided a wonderful description of scaling limits of lattice interfaces, and turned out to have deeper roots than perhaps originally anticipated. Indeed, Schramm's random SLE curves also share an intrinsic connection with the geometric and algebraic content in CFT. On the one hand, such curves emanating at boundary or bulk points relate to specific Virasoro modules in the theory -- in particular, in the case of boundary phenomena they correspond to degenerate field insertions, which can be studied completely rigorously in terms of probability and PDE theory. On the other hand, the action functional for SLE loops is closely related to the universal Liouville action and Kähler geometry on Teichmüller space, although the associated correlation functions therein remain more mysterious.
As is known from the 1980s, CFT should describe scaling limits of critical lattice models in statistical mechanics (i.e., fixed points of the renormalization group flow). One might argue that two-dimensional (2D) CFTs -- at least minimal models and Liouville theory -- are completely understood since the pioneering works in the 1980s. However, a closer inspection reveals that CFTs rele- vant to models containing lattice interfaces, or random curves in the continuum, cannot be described by minimal models, seem to exhibit logarithmic phenomena, and even their spectrum (operator content) seems not to be completely clear. In the random geometry community, about 25 years ago a breaktrough idea came along: Schramm suggested that probabilistically, such interface models could be rigorously described by combining classical complex analysis (namely, Loewner theory) with stochastic analysis (namely, Brownian motion and martingale theory). This provided a wonderful description of scaling limits of lattice interfaces, and turned out to have deeper roots than perhaps originally anticipated. Indeed, Schramm's random SLE curves also share an intrinsic connection with the geometric and algebraic content in CFT. On the one hand, such curves emanating at boundary or bulk points relate to specific Virasoro modules in the theory -- in particular, in the case of boundary phenomena they correspond to degenerate field insertions, which can be studied completely rigorously in terms of probability and PDE theory. On the other hand, the action functional for SLE loops is closely related to the universal Liouville action and Kähler geometry on Teichmüller space, although the associated correlation functions therein remain more mysterious.
At the turn of the century, Khovanov upgraded the Jones polynomial into a homology theory of knots, which is sensitive to smooth structures in 4D. The Jones polynomial can be recovered from Khovanov homology, so Khovanov homology is at least as hard to compute as the Jones polynomial, and it is an open question how much harder it is. In this talk, we will try to explain why mathematicians should care about fast computations of Khovanov homology. We will also explore polynomial and non-polynomial time algorithms for both Jones polynomial and Khovanov homology of braids. Joint work with Dirk Schütz.
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