Math 510 Seminars in Mathematics

Place: Ross 3000, Thursdays 9:00-9:50;

Math-Phys & CS seminar

Place: Ross 3000, Thursdays 10:00-10:50

Topics from Math 510, discussed in more depth.

** Math 510 **- Syllabus

__Topics covered__

- 01/18, 01/25: Hilbert spaces (Ch.1)
- 02/01 Fourier Transform (Ch.2)
- 02/08 Windowed Fourier Transform and Wavelets
- 02/15 IWT, signal processing and constructive filed theory
- 02/22 Linear operators in Hilbert spaces (Ch.III) (I)
- 03/01 Spectral theory; eigenvalues/eigenvectors; examples (II)
- 03/08 Examples; the number of particles operator (III)
- 03/15 Extensions and completions: generalized functions
- 04/05 Classical mechanics and Hamiltonian systems
- 04/12 Postulates of quantum mechanics
- 04/19 Quantum computing I (automata, logical gates, quantum bits and registers)
- 04/26 Quantum computing II (quantum gates, q. evaluations of functions, copying)
- 05/03 Student presentations:

1) Kevin Stump: Shor's algorithm

2) Tara Hart: An undergraduate approach to Bell's Inequalities.

(Mathematical-Physics & Computer Science)

__Time & Place:__ Thusrdays 9:00-9:50, Ross 3000.

The goal of the course is the contact with modern and interesting applications of mathematical theories.

__1. Wavelets__

They are recently used in data compression technology (image processing,
etc.):

“Applications of wavelet theory continues to grow rapidly. Engineers,
working in everything from mathematics and physics to digital signal processing,
image compression, and speech and pattern recognition, need to understand
this exciting subject”.

__2. Quantum physics__

“Low and high energy phenomena” … (too many to list!).

( It is a different way of thinking
about “reality” worth knowing!)

__3. Quantum computing__

The parallel computing is an old subject. The modern one is “infinitely
many parallel computing paths”. It is a new emerging domain of CS, promising
to give us a clue on how we think and what conscience is!

__Textbook:__

“Hilbert Spaces, Wavelets, Generalised Functions and Modern Quantum
mechanics”, by Willi-Hans Steeb, Mathematics and Its Applications, 1998.

We will learn the mathematical tools needed to be able to address these applications: Hilbert space theory, Fourier transform and wavelets, linear operators, generalized functions and quantum mechanics, quantum bits, Q-gates and Q-copying.

__Prerequisites__

Basic knowledge in linear algebra (matrices, determinants, vector spaces)
and calculus is required. An intuitive understanding of Euclidean R3 will
help with Hilbert spaces, a calculus level understanding of differential
equations would suffice and some real analysis would be helpful.

The approach will be rigorous, without being hindered by very technical
issues of analysis, focusing on the conceptual part of the mathematics,
physics or CS involved.

Optional exercises are included as a way to feel we understand the
formalism, being meant to give a feed-back on the hidden difficulties involved.

__Format__

I will start presenting the first couple of lectures. Meanwhile students
will decide what to present and when, oriented on a subject, or a part
of the textbook, or a solution to a problem.

Lucian Ionescu LMIones@ilstu.edu.