14:15
Categorical and K-theoretic Donaldson-Thomas theory of $\mathbb{C}^3$
Abstract
Donaldson-Thomas theory associates integers (which are virtual counts of sheaves) to a Calabi-Yau threefold X. The simplest example is that of $\mathbb{C}^3$, when the Donaldson-Thomas (DT) invariant of sheaves of zero dimensional support and length d is $p(d)$, the number of plane partitions of $d$. The DT invariants have several refinements, for example a cohomological one, where instead of a DT invariant, one studies a graded vector space with Euler characteristic equal to the DT invariant. I will talk about two other refinements (categorical and K-theoretic) of DT invariants, focusing on the explicit case of $\mathbb{C}^3$. In particular, we show that the K-theoretic DT invariant for $d$ points on $\mathbb{C}^3$ also equals $p(d)$. This is joint work with Yukinobu Toda.
The second series of our short films, ‘Me and My Maths’, is now running on our social media with even higher viewing figures than the first series. You can watch a compilation of the first four films via the video below.
Starring: Kylie and Chloe, Andrea, Doyne, and Kate Wenqi.
Me and My Maths. Short films about people who also do maths.
To celebrate LGBTQ+ History Month, we will be joined by MPLS LGBTQ+ Role Model Evan Nedyalkov to discuss LGBTQ+ issues within the department as well as in Maths more generally. We will be in the Quillen Room N3.12 from 1 - 2pm on Wednesday 15th February. Note that this event is open to all and that there will be a free lunch provided. Let us know that you're coming so that we can order enough food.
Our quarterly e-newsletter will go out later today to nearly 13,000 alumni. If you want to know what we are saying about ourselves click here (the composite image below is not a link).
Mapper--type algorithms for complex data and relations
Abstract
Mapper and Ball Mapper are Topological Data Analysis tools used for exploring high dimensional point clouds and visualizing scalar–valued functions on those point clouds. Inspired by open questions in knot theory, new features are added to Ball Mapper that enable encoding of the structure, internal relations and symmetries of the point cloud. Moreover, the strengths of Mapper and Ball Mapper constructions are combined to create a tool for comparing high dimensional data descriptors of a single dataset. This new hybrid algorithm, Mapper on Ball Mapper, is applicable to high dimensional lens functions. As a proof of concept we include applications to knot and game theory, as well as material science and cancer research.
Topological Optimization with Big Steps
Abstract
Using persistent homology to guide optimization has emerged as a novel application of topological data analysis. Existing methods treat persistence calculation as a black box and backpropagate gradients only onto the simplices involved in particular pairs. We show how the cycles and chains used in the persistence calculation can be used to prescribe gradients to larger subsets of the domain. In particular, we show that in a special case, which serves as a building block for general losses, the problem can be solved exactly in linear time. We present empirical experiments that show the practical benefits of our algorithm: the number of steps required for the optimization is reduced by an order of magnitude. (Joint work with Arnur Nigmetov.)