Index of elliptic operators for a diffeomorphism

We develop elliptic theory of operators associated with a diffeomorphism of a closed smooth manifold. The aim of the present paper is to obtain an index formula for such operators in terms of topological invariants of the manifold and of the symbol of the operator. The symbol in this situation is an element of a certain crossed product. We express the index as the pairing of the class in K-theory defined by the symbol and the Todd class in periodic cyclic cohomology of the crossed product.Comment: 37 page

Classifying music perception and imagination using EEG

This study explored whether we could accurately classify perceived and imagined musical stimuli from EEG data. Successful EEG-based classification of what an individual is imagining could pave the way for novel communication techniques, such as brain-computer interfaces. We recorded EEG with a 64-channel BioSemi system while participants heard or imagined different musical stimuli. Using principal components analysis, we identified components common to both the perception and imagination conditions however, the time courses of the components did not allow for stimuli classification. We then applied deep learning techniques using a convolutional neural network. This technique enabled us to classify perception of music with a statistically significant accuracy of 28.7%, but we were unable to classify imagination of music (accuracy = 7.41%). Future studies should aim to determine which characteristics of music are driving perception classification rates, and to capitalize on these characteristics to raise imagination classification rates

On the homotopy classification of elliptic operators on stratified manifolds

We find the stable homotopy classification of elliptic operators on stratified manifolds. Namely, we establish an isomorphism of the set of elliptic operators modulo stable homotopy and the $K$-homology group of the singular manifold. As a corollary, we obtain an explicit formula for the obstruction of Atiyah--Bott type to making interior elliptic operators Fredholm.Comment: 28 pages; submitted to Izvestiya Ross. Akad. Nau