Telephone: 263-4753 or 265-3817.

E-mail: *boston@math.wisc.edu*

Office Hours: Tuesdays, 10-11 (in 303 VV), Wednesdays 11-12 (in 3619 EH), Thursdays 2:20-3:20 (in 303 VV).

- Lecture: TR 11:00-12:15, B129 Van Vleck.

Practical applications of IT, including an explanation of the 56K modem

Survey of space-time coding

2. Discussing of codes would not be complete without describing LDPC (and more generally graph-based) codes, which finally answer Shannon's challenge from 1948 of obtaining practical codes with rates close to channel capacity. See my survey lectures, with figures here. We also mention the currently hot topic of network coding, which employs some algebraic geometry.

3. Digital watermarking - in particular Quantization Index Modulation, which uses lattices, and the Atallah-Wagstaff scheme, which uses quadratic reciprocity.

4. Cryptography. Public-key cryptography depends for its security on the difficulty of solving some hard problem. For RSA it's the problem of factoring large integers, but using the number field sieve it's now feasible to factor 576-bit integers. In constrained environments companies are turning to elliptic curve cryptography (ECC). Private-key cryptography recently came up with a new AES (Advanced Encryption Standard), Rijndael, but its structured nature has drawn attacks using commutative algebra.

5. Mathematical Finance. Deterministic numerical estimates for integrals over high-dimensional cubes (as are used in pricing tranches of mortgages) come from finding low-discrepancy sequences. The lowest star discrepancies so far attained come from curves over finite fields with lots of points (the same design criterion as for good codes from algebraic geometry, i.e. Goppa codes). Financial software companies now use these.

6. Signal Processing. Filter design and bounds (see e.g. the Makhoul conjecture challenge). Manipulation of the transfer function of the filter leads to algebraic geometry. Transfer functions also arise in robust control theory.

7. Tomography, generalized principal component analysis. Both use some algebraic geometry.

8. Computer vision. Study of invariants of Lie groups acting on varieties.

Note that mathematically the same topics keep coming up again and again in various contexts, e.g. groups and in particular finite groups of rotations in Euclidean 3-space; lattices and in particular density questions; finite fields and curves defined over them; algebraic number fields particularly cyclotomic ones.

1. Information Theory in Modem Practice.

2. Applications of Geometric Algebra

3P. On the Use of Quasi-Monte Carlo Methods in Computational Finance

4P. Real algebraic geometry and convex optimization

5A. Multiple antenna applications

6P. The Makhoul conjecture challenge

7. Code Division Multiple Access

8. Graphical codes

9. Network coding

10. AES attacks

11. Generalized principal component analysis

12. Cyclotomic fields and space-time codes

13. Implementing elliptic curve cryptography

14. Cryptography and braid groups

15. Algebraic curves and computational vision

16. Steganography and data-hiding

17. Tomography and computer algebra

18. Quantum information processing and geometric algebra

19. Quantum error-correcting codes

20. Algebraic number theory and signal constellations

James P.Cossey: Applications of finite group theory to multiple-antenna design

Kei Hao: Perfect space-time block codes and ultra-wideband

Jarvis Haupt: New sequences from old: a search for low discrepancy sequences

Anders Hendrickson: Space-time block codes from cyclic division algebras: an introduction

Martin Hock: Braid compression

Christopher Holden: Perfect space-time block codes

Eunmo Kang: Review and performance comparison of finite-rate feedback strategies: vector quantization, systematic unitary construction and Grassmannian packing

Sarah Knoop: Supersingular curves and the Weil pairing in elliptic curve cryptography

Wei-Yang Lin: Implementation of elliptic curve cryptography

Wei-Yang Lin: Invariants of isometric transformations

Karl Mahlburg: An overview of braid group cryptography

Harris Nover: Algebraic cryptanalaysis of AES: an overview

Michael Rabbat: Discrete-time signal processing and Makhoul's conjecture

Vasanthan Raghavan: When is limited feedback for transmit beamforming useful?

Changfang Zhu: Algebraic-geometric and probabilistic approaches for clustering and dimension reduction of mixtures of principal component subspaces