Topics covered will include:

- Geometry and topology of the visual cortex
- Geometry of segmentation and invariance
- Neural codes and neural rings
- Deep learning neural networks: mathematical aspects, and applications to vision and language
- Mathematics and neuroscience of language and syntax

Books:

- G. Bard Ermentrout, David H. Terman, "Mathematical Foundations of Neuroscience", Springer, 2010.
- Jean Petitot, "Neurogeometrie de la vision", Les Editions de l'Ecole Polytecnique, 2008. ( pdf )
- Giovanna Citti, Alessandro Sarti, "Neuromathematics of Vision", Springer, 2014
- Robert Ghrist, "Elementary Applied Topology", Createspace Independent, 2014.
- David Mumford, Agnes Desolneux, "Pattern Theory: The Stochastic Analysis of Real-World Signals", CRC Press, 2010.
- Derek Bickerton and Eors Szathmary, "Biological Foundations and Origin of Syntax", MIT Press, 2009
- David Marr, "Vision: A Computational Investigation into the Human Representation and Processing of Visual Information", MIT Press, 2010.
- Peter Dayan, "Theoretical Neuroscience", MIT Press, 2005.
- David Spivak, "Category Theory for the Sciences" MIT Press 2014, html

Some Articles: (more will be added)

General overview:

- Misha Gromov, "Structures Learning and Ergosystems", pdf
- Misha Gromov, "Ergostructures, Ergologic and the Universal Learning Problem" pdf (*)
- Felipe Cucker and Steve Smale, "On the Mathematical Foundations of Learning" pdf
- Yuri Manin, Error-correcting codes and neural networks pdf (*)

Dynamical models of the neuron:

- Ryan Siciliano, "The Hodgkin-Huxley Model" pdf (*)
- Tanya Kostova, Renuka Ravindran, Maria Schonbek, FitzHugh-Nagumo Revisited: Types of Bifurcations, Periodical Forcing and Stability Regions by a Lyapunov Functional, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 14 (2004), no. 3, 913-925 pdf (*)

Receptor fields and Gabor frames:

- Karlheinz Gròˆchenig, "Multivariate Gabor frames and sampling of entire functions of several variables" pdf
- Kristian Seip, "Density theorems for sampling and interpolation in the Bargmann-Fock space I" pdf (*)
- Kristian Seip, "Density theorems for sampling and interpolation in the Bargmann-Fock space" pdf
- Bruce MacLennan, "Gabor Representations" pdf (*)

Conformal Geometry:

- Peter Olver "Complex Analysis and Conformal Mapping" pdf (*)
- F. Helein and J.C. Wood, "Harmonic Maps" pdf (*)
- Yalin Wang, Xianfeng Gu, Tony Chan, Paul Thompson, Shing-Tung Yau, "Intrinsic Brain Surface Conformal Mapping using a Variational Method", pdf
- D.Ta, J.Shi, B.Barton, A.BRewer, Z.L.Lu, Y.Wang, "Characterizing human retinotopic mapping with conformal geometry: A preliminary study" pdf
- P. Koehl, J. Hass, "Automatic Alignment of Genus-Zero Surfaces" pdf (*)
- S.J.Gortler, C.Gotsman, D.Thurston "Discrete one-forms on meshes and applications to 3D mesh parameterization" pdf (*)
- N.Aigerman, Y.Lipman, "Orbifold Tutte Embeddings" pdf (*)
- M.Hurdal, P.Bowers, K.Stephenson, D.Sumners, K.Rehm, K.Schaper, D.Rottenberg, "Quasi-conformal flat mapping the human cerebellum" pdf (*)

Contact Geometry:

- John B. Entyre, "Introductory Lectures on Contact Geometry", pdf
- William C. Hoffman, "The Visual Cortex is a Contact Bundle" pdf
- Alessandro Sarti, Giovanna Citti, Jean Petitot, "Functional geometry of the horizontal connectivity in the primary visual cortex" pdf (*)

Segmentation and tracking:

- David Mumford, Jayant Shah, " Optimal Approximations by Piecewise Smooth Functions and Associated Variational Problems" pdf
- Laurent Younes, Peter W. Michor, Jayant Shah, David Mumford, A metric on shape space with explicit geodesics, Rend. Lincei Mat. Appl. 19 (2008) 25-57 pdf (*)
- Mumford, D.; Kosslyn, S. M.; Hillger, L. A.; Herrnstein, R. J. Discriminating figure from ground: the role of edge detection and region growing, Proc. Nat. Acad. Sci. U.S.A. 84 (1987), no. 20, 7354-7358 pdf (*)
- Leah Bar et al. Mumford and Shah Model and its Applications to Image Segmentation and Image Restoration, in Handbook of Mathematical Methods in Imaging, Springer 2011, 1095-1157 pdf (*)
- Thomas J. Tsao, Doris Y. Tsao, "Mathematical analysis of the general conditions of vision reveals a new solution to segmentation and invariance", pdf
- M. Belkin, P. Niyogi, "Laplacian Eigenmaps and Spectral Techniques for Embedding and Clustering pdf (*)
- Frank Sottile, Thorsten Theobald, "Line problems in nonlinear computational geometry", arXiv:math/0610407
- Marco Pellegrini, "Ray shooting and lines in space", Handbook of discrete and computational geometry, 599-614 pdf
- Thorsten Theobald "An enumerative geometry framework for algorithmic line problems in R3", SIAM J. Comput. 31 (2002), no. 4, 1212-1228 pdf

Bistable images:

- Emily J. Wald, Brian J. Scholl, "Stochastic or systematic? Seemingly random perceptual switching in bistable events triggered by transient unconscious cues" pdf (*)
- N. Rubin, M.C. Pugh, "Global effects in Figure/Ground segregation by a model with only local interactions", pdf (*)

Neural Codes and Neural Rings:

- Yuri Manin, "Neural codes and homotopy types: mathematical models of place field recognition", arXiv:1501.00897 (*)
- Carina Curto, Vladimir Itskov, Alan Veliz-Cuba, Nora Youngs, "The neural ring: an algebraic tool for analyzing the intrinsic structure of neural codes", arXiv:1212.4201
- Nora Youngs, "The neural ring: using algebraic geometry to analyze neural codes", arXiv:1409.2544 (*)
- Carina Curto, "What can topology tell us about the neural code?", arXiv:1605.01905
- Carina Curto, Vladimir Itskov, Katherine Morrison, Zachary Roth, Judy L. Walker, "Combinatorial neural codes from a mathematical coding theory perspective", arXiv:1212.518 (*)8
- Carina Curto, Anda Degeratu, Vladimir Itskov, "Encoding binary neural codes in networks of threshold-linear neurons", arXiv:1212.0031 (*)

Neural Networks, Deep Learning:

- Fabio Anselmi, Joel Z. Leibo, Lorenzo Rosasco, Jim Mutch, Andrea Tacchetti and Tomaso Poggio, "Unsupervised learning of invariant representations with low sample complexity: the magic of sensory cortex or a new framework for machine learning?" pdf
- Thomas Wiatowski, Helmut Boelcskei, "A Mathematical Theory of Deep Convolutional Neural Networks for Feature Extraction", arXiv:1512.06293 (*)
- Arnab Paul and Suresh Venkatasubramanian, "Why does unsupervised learning work? A perspective from group theory" pdf
- Cedric Beny, "Deep learning and the renormalization group", arXiv:1301.3124
- Pankal Metha, David J. Schwab, "An exact mapping between the Variational Renormalization Group and Deep Learning", arXiv:1410.3831
- Yann Ollivier, "Riemannian metrics for neural networks I: feedforward networks", arXiv:1303.0818 (*)
- Yann Ollivier, "Riemannian metrics for neural networks II: recurrent networks and learning symbolic data sequences", arXiv:1306.0514 (*)

- Thursday January 5: Gromov's Ergobrain program (Doris/Matilde)
- Tuesday January 10: An introduction to neurons and the brain (Doris)
- Thursday January 12: Mathematical models of neurons via nonlinear dynamics (Matilde)
- Tuesday January 17: An introduction to the neuroscience of vision (Doris)
- Thursday January 19: Receptor fields and Gabor frames analysis (Matilde)
- Tuesday January 24: Conformal Geometry of the primary visual cortex (Matilde)
- Thursday January 26: Contact Geometry of the primary visual cortex (Matilde)
- Tuesday January 31: Topological approach to segmentation and invariance (Doris)
- Thursday February 2: Variational problems in image segmentation, the Mumford-Shah functional (Matilde)
- Tuesday February 7: Segmentation and tracking with computational and enumerative algebraic geometry (Matilde)
- Thursday February 9: Neural codes (Doris/Matilde)
- Tuesday February 14: Neural codes and neural rings, topology and algebraic geometry (Matilde)
- Thursday February 16: Bistable images and dynamical systems (Doris)
- Tuesday February 21: Brain Networks and Topology (Matilde)
- Thursday February 23: Mathematics and Language: generative grammars (Matilde)
- Tuesday February 28: Language, syntax and neuroscience (Matilde)
- Thursday March 2: Deep learning (Doris)
- Tuesday March 7: Deep learning and vision (Doris)
- Thursday March 9: Learning, Vision, and Deep Networks (Matilde)

- Overview: Structures, Learning, and Ergosystems
- Gromov's Ergobrain Program as a Mathematical Promenade
- Neurons and the Brain
- The Neuron as a Dynamical System
- The Visual System
- Receptor Profiles and Gabor Frames
- Example 1 and Example 2 of frame condition (by Pengchuan Zhang and Mason McGill)
- Conformal Geometry of the Visual Cortex
- Contact Geometry of the Visual Cortex
- Image Segmentation: the Mumford-Shah Functional
- Algebraic Geometry of Segmentation and Tracking
- Neural Codes
- Neural Codes and Neural Rings: Topology and Algebraic Geometry
- Brain Networks and Topology
- The Mathematical Theory of Formal Languages
- Geometric Models for Linguistics
- Deep Learning
- Why is Deep Learning so effective?
- Ventral Visual Stream and Deep Networks

In the bibliographical list of papers linked above, all papers marked with (*) are available for student presentations (more will be added shortly). Choose an article on a topic that is most interesting to you (first come first serve) and prepare a 30min seminar presentation about it.

The final student presentations will take place Tuesday, March 14 and Thursday, March 16 in Room SLN 151 (same room where the class takes place), starting at 9am. The tentative schedule of presentations is given below.

- Student Presentations
- Tuesday March 14, 9:00am: Jane Panangaden, Functional geometry of the horizontal connectivity in the primary visual cortex
- Tuesday March 14, 9:30am: Arjun Bose, the Hodgkin-Huxley model
- Tuesday March 14, 10:00am: Aaron Anderson, Neural codes and homotopy types
- Tuesday March 14, 10:30am: Jeremy Bernstein, Gardner analysis of the statistical mechanics of neural networks
- Tuesday March 14, 11:30am: Matthew Rosenberg, Unsupervised learning of invariant representations with low sample complexity
- Thursday March 16, 9:00am: John Thompson, Ergostructures, Ergologic and the Universal Learning Problem
- Thursday March 16, 9:30am: Preston Rasmussen, Complex Analysis and Conformal Mapping
- Thursday March 16, 10:00am: Jialing Song, Laplacian Eigenmaps and Spectral Techniques for Embedding and Clustering
- Thursday March 16, 10:30am: Evan Davis, Error-correcting codes and neural networks
- Thursday March 16, 11:00am: Alvita Tran, Discriminating figure from ground