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This project will apply topological methods to questions in neuroscience. Specifically, the team will study dynamics in neural networks used to model memory storage and rhythm generation in the nervous system. These dynamics are determined by topological properties of the underlying network architecture. Recent advances in the theory of recurrent neural networks have identified key structural features, including motifs, that play a role in shaping the emergent dynamics. However, these features are difficult to identify without more sophisticated tools for network analysis. The team members will thus adapt tools from topology in order to detect the relevant structures inside recurrent neural networks, and use these tools to make predictions about the set of attractors and other aspects of the network's dynamics.
The Blue Brain neurotopology team discovered that applying the tools of algebraic topology to the directed graph underlying a network of neurons in the neocortex revealed an intricate topology of synaptic connectivity. The network of neurons comprises an abundance of cliques of neurons bound together into elaborate topological structures, which guide the emergence of neural activity. In response to stimuli, correlated activity, encoded as a sequence of subgraphs of the ambient directed graph, binds synaptically connected neurons into functional cliques and cavities that evolve in a stereotypical sequence towards peak complexity. The team is currently applying topological methods to analyzing the results of simulations of network plasticity, carried out on supercomputers at Argonne National Laboratory by members of the Blue Brain simulation group. Their goal is to elucidate a structure-property relation between the evolving topology of the microcircuit and the evolving dynamics of its response to stimuli, where the evolving network is represented mathematically as a discrete time series of weighted, directed graphs. The goal of this project is twofold. First, the team will develop appropriate probabilistic and statistical tools for the study of distributions on the space of subgraphs S(G) of a given directed graph G and on the space W(G) of all possible weightings on G. Second, the team will evaluate the relevance of various topological invariants of subgraphs of G (respectively, of possible weightings of G) as random variables on S(G) (respectively, W(G)). Invariants to be considered include the Betti numbers and Euler characteristic of the directed flag complex and the Betti and Euler curves of the filtered directed flag complex.
This project focuses on connecting discrete Morse theory to multiparameter persistent homology or persistence. An efficient computation of multiparameter persistence is still not possible in general. Thus, one of the main challenges for topological data analysis is to make persistence a viable option for data filtered by multiple parameters. This team's project will be concerned with precisely this challenge. The plan is to study the preprocessing of data to feed the rank invariant and matching distance algorithms. The aim of the preprocessing is to reduce the data on which to perform the subsequent computations to a minimum. The project has three main objectives: to design an efficient algorithm to preprocess the data; to achieve optimality in terms of reduction amount; and to achieve the exact computation of the matching distance in reasonable time.
Persistent homology data from micro-CT images of porous materials provides a useful summary of their geometrical and topological structure, and is correlated with physical properties such as permeability and trapping capacity. Three aspects of the analysis of this data have not yet been fully explored. First, how should persistent homology be normalized to enable one to compare persistence diagrams from related data sets of different sizes? Second, how does persistent homology change when an image filter (such as a local averaging operator) is applied? Third, the cost of acquiring a micro-CT image is proportional to the imaging resolution, so we want to understand how persistent homology changes with resolution, and what minimum resolution is required for extracting useable data. This group will investigate one or more of these questions with sample micro-CT images of porous materials provided by the ANU CTLab for context.
Knots are a prime area of research in computational 3-manifolds, and have applications in modeling molecules. For example, embeddings of bonding graphs correspond to knots, and ligands of a coordination polymer correspond to edges of a spatial graph and the coordination entities correspond to its vertices. Problems such as unknot recognition and 3-sphere recognition---the simplest cases of the knot equivalence and homeomorphism problems---have very recently become fast in practice, and are now the subject of significant questions in complexity theory. In particular, both problems are now known to lie in the complexity class NP and also in co-NP if the generalised Riemann hypothesis holds. Whether these problems admit a polynomial-time algorithm remains a major open problem, and at this workshop we aim to explore significant new avenues of approach.
Apply to attend
By filling out the 'Register your interesrt' webform you will be applying to attend and participate in one of the 5 working groups. If you are accepted you will be notified of your group placement and emailed a link to the final registration and online payment system. Your place at the workshop will be confirmed after you complete this registration and payment. Note that group allocations will be made primarily on the order applications are received.
- General registration $125
- AMSI institution members $100
- Student/ retired fellow $80
Apply for workshop accommodation
For participants who do not have access to funding, you can apply for workshop accommodation. The accommodation will be for a room in a shared two bedroom apartment at Liversidge court apartments from Saturday 30th June to Saturday 6th July, 2019. Please note, spaces are limited.
To apply for the grant, please specify 'yes' in the Register your interest webform. You will be notified via email if you have been success in obtaining a room.
This event is sponsored by the Australian Mathematical Sciences Institute (AMSI). AMSI allocates a travel allowance annually to each of its member universities (for list of members, see www.amsi.org.au/members).
Students or early career researchers from AMSI member universities without access to a suitable research grant or other source of funding may apply (with approval of their Head of Mathematical Sciences) for subsidy of travel and accommodation out of their home departmental travel allowance.
USA funding support
We have applied for extra funds to help support researchers from the United States, and we will post more information about applying for this support at a later stage.
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If you are catching a taxi or Uber to the ANU Mathematical Sciences Institute, ask to be taken to Building #145, Science Road, ANU. We are located close to the Ian Ross Building and the ANU gym. A Taxi from the airport will usually cost around $40 and will take roughly 15 minutes. Pricing and time may vary depending on traffic.
Taxi bookings can be made through Canberra Elite Taxis - 13 22 27.
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