Curriculum Vitae for Boaz Tsaban
Basic
information.
Born in Israel 6 February 1973; citizen of Israel (I.D. 02509157-0); Married +
3.
Current
position. A
post-doctoral student at Einstein Institute of Mathematics, the Hebrew
University of Jerusalem, Israel; supervisor: Professor Saharon Shelah.
Contact
details.
E-mail: tsaban@math.huji.ac.il ; regular mail: 4 Bialik St., Givat-Shmuel
54011, Israel.
Academic
studies.
Completed B.Sc. 1994 (Mathematics extended), M.Sc. 1997 (Mathematics,
supervised by Martin Goldstern and Hillel Furstenberg), and Ph.D. 2003
(Mathematics, supervised by Hillel Furstenberg), all at Bar-Ilan University,
with distinction. The doctoral dissertation won the 2003 Nessyahu
Prize for the best Ph.D. in mathematics in Israel.
Teaching. Taught various courses at
all academic levels, in universities and colleges.
Solved open
problems and conjectures of: Menger (1925), Hurewicz (1927), Preneel (1994),
Steprans (1996), Miller-Just-Scheepers-Szepticky (1996), Scheepers (1999),
Kocinac-Scheepers (2000),
Tkacenko-Hernandez (2000), Bukovski-Reclaw-Repicky (2001),
Kocinac-Scheepers (2002), Fremlin (2002), Naor-Reingold (2002).
Collaborators. Austria: Heike
Mildenberger. Poland: Tomasz Weiss. Israel: David Garber, Hillel
Furstenberg, Adi Shamir, Saharon Shelah, Mina Teicher, Uzi Vishne. USA:
Tomek Bartoszynski, Charles Holland, Marion Scheepers.
Seminar and
conferences talks. Lectured in more than 20 seminars, workshops and conferences,
including: BGU Symposium in mathematics (Israel, 2001), Coverings, selections, and games in topology (Italy, 2002), Israel Mathematical Union (Israel,
2003), The Barcelona
Conference on Set Theory (Spain, 2003).
Research interests
Infinite-combinatorial
topology: This is a new branch with old roots in the field of
set-theoretic topology. We use tools from infinite combinatorics to study
classical and new problems in topology, usually when the problems involve a
diagonalization principle in an explicit or implicit manner. We use this
approach in a systematic manner and obtain an aesthetically pleasing theory,
which is then used to solve problems in the field in an elegant manner. The
approach is often to develop “dictionaries” translating topological properties
into combinatorial ones, and then applying the known methods of infinite
combinatorics. The major tools are from set theory (in particular, we use the
method of forcing), measure and category theory, general topology, and
infinite game theory.
In
addition to solving several open problems in the field and settling several
conjectures (see above), we have also carried several rather complete
classification projects of some of the prototypes of properties in the field,
and introduced several new notions to the field.
Topological
algebra: We
applied the above-mentioned framework to solve a problem of Tkacenko and
Hernandez on a generalization of sigma-compact groups defined in terms of a
diagonalization procedure.
We
also studied parametric equations in lattice-ordered groups from a
computational point of view.
Pseudo-randomness: We made several
contributions to this field, dealing with huge (but finite) randomly-looking
objects, including: New, provably secure modes of operations, word-oriented
linear generators with maximal period, fast-forward pseudorandom permutations.
Algebraic
geometry and combinatorial group theory: We found a method which yields a probabilistic
solution to a given system of equations in “random” subgroups of the braid
group. This undecideable problem implies many of the difficult problems
in the field, like the conjugacy problem, and the group membership problem.
Books. I wrote the books for some
of the large courses at Bar-Ilan University: Linear Algebra, Infinitesimal
Calculus, Set Theory, and Mathematical Logic.
Refereeing. I have refereed papers for
the following journals: Proceedings of
the American Mathematical Society, Jounal of Number Theory, Note di Matematica,
Information Processing Letters, IEEE Transactions on Information Security,
Proceedings of the LATIN 2004 Conference on Theoretical Computer Sciences.
Reviewing. I review papers for Mathematical Reviews and for
Zentralblatt Math.
Editing. I edit (voluntarily) the SPM Bulletin, a
(monthly to quarterly) bulletin devoted to selection principles and
diagonalization arguments in mathematics.
Publications. Following are my published works and works in press.
1
Special classes
of strongly null sets: A journy into the continuum, M.Sc. Thesis, Bar-Ilan University, 1997.
2
On the Rabbinical approximation of л
(with David Garber), Historia Mathematica 25 (1998), 75-84.
3
A topological
interpretation of t, Real
Analysis Exchange 25 (1999),
391-404.
4
A
mechanical derivation of the area of a sphere (with David Garber), The American Mathematical Monthly 108
(2001), 10-15.
5
Guaranteeing the
diversity of number generators (with
Adi Shamir), Information and Computation 170 (2001), 1-14.
6
A diagonalization property between Hurewicz and Menger, Real Analysis Exchange
207 (2001/02), 757-763.
7
The combinatorics
of Borel covers (with Marion
Scheepers), Topology and its Applications 121 (2002), 357--382.
8
Efficient linear feedback shift registers with maximal period (with Uzi Vishne), Finite
Fields and their Applications 8 (2002), 256-267.[1]
9
Infinite Combinatorial Topology, Doctoral Dissertation, Bar-Ilan University,
2003.
10
Bernoulli numbers and the probability of a birthday surprise, Discrete Applied
Mathematics 127 (2003), 657-663.[2]
11
Permutation graphs, fast forward permutations, and sampling the cycle
structure of a permutation, Journal of Algorithms 47 (2003), 104-121.[3]
12
Critical cardinalities and additivity properties of combinatorial
notions of smallness (with Saharon Shelah), Journal of Applied Analysis 9 (2003),
149-162.
13
Additivity properties of topological diagonalizations (with Tomek Bartoszynski
and Saharon Shelah), Journal of Symbolic Logic 68 (2003), 1254-1260.
14
The minimal cardinality where the Reznichenko property fails, Israel Journal of
Mathematics 140 (2004), 367-374.
15
The combinatorics of splittability, Annals of Pure and Applied
Logic, to
appear.
16
Hereditary topological diagonalizations and the Menger-Hurewicz
Conjectures
(with Tomek Bartoszynski), Proceedings of the American Mathematical Society,
to appear.
17
Selection principles and the minimal tower problem, Note di Matematica,
to appear.
18
Topological diagonalizations and Hausdorff dimension (with Tomasz Weiss), Note
di Matematica, to appear.
19
Selection principles in Mathematics: A milestone of open problems, Note di Matematica, to appear.
20
o-bounded groups and other topological groups with strong combinatorial
properties,
Proceedings of the American Mathematical Society, to appear.
21
The Hurewicz covering property and slaloms in the Baire space, Fundamenta Mathematicae,
to appear.
22
The conjugacy problem and related problems in lattice-ordered groups (with W. Charles Holland), International
Journal of Algebra and Computation, to appear.
Citing papers. Most of my works are cited in published papers and works in progress
by other authors. Following are some examples.
|
[An] |
Vladimir Anashin,
Pseudorandom number generation by p-adic ergodic transformations,
preprint. |
|
[An2] |
Pseudorandom Number
Generation by p-adic Ergodic Transformations: An Addendum, preprint. |
|
[Ba] |
Taras Banakh, Cardinal
characteristics of the ideal of Haar null sets, Commentationes
Mathematicae Universotatis Carolinae, to appear. |
|
[BNS] |
Taras Banakh, Peter
Nickolas, and Manuel Sanchis, Filter games and pathological subgroups of a
countable product of lines, Australian Journal of Mathematics, to
appear. |
|
[BZ] |
Taras Banakh and Lubomyr
Zdomsky, Games on multicovered spaces and their applications in
topological algebra, preprint. |
|
[BVPCS] |
Martin Boesgaard, Mette
Vesterager, Thomas Pedersen, Jesper Christiansen, Ove Scavenius, Rabbit: A
New High-Performance Stream Cipher, FSE 2003, 307-329. |
|
[Bu] |
Lev Bukovski, Hurewicz properties, not distinguishing
convergence properties, and sequence selection properties, Topology
and its Applications, to appear. |
|
[DP] |
Michael Dewar and Daniel
Penario, Linear transformation shift registers, IEEE Transactions
on Information Theory, to appear. |
|
[EJ] |
Todd Eisworth and Winfried
Just, NCF and the combinatorics of open covers, Fundamenta
Mathematicae, to appear. |
|
[GGN] |
Oded Goldreich, Shafi
Goldwasser and Asaf Nussboim, On the implementation of huge random objects,
preprint. |
|
[Je] |
Jesse Jenkins, CipherStream Protocol – how CoolRunner-II
CPLDs protect FPGA IP, Xilinx White Pages 197 (2003), 1-14. |
|
[Mi] |
Arnold W. Miller, A nonhereditary Borel-cover g-set, Real Analysis Exchange, to appear. |
|
[NR] |
Moni Naor and Omer Reingold, Constructing Pseudo-Random
Permutations with a Prescribed Structure, Journal of Cryptology 15
(2002), 97-102. |
|
[Sa] |
Masami Sakai, The weak Fréchet-Urysohn property in function
spaces, preprint. |
|
[SBPCR] |
Ove Scavenius, Martin Boesgaard, Thomas Pedersen, Jesper Christiansen,
and Vincent Rijmen, Periodic Properties of Counter Assisted Stream Ciphers, to
appear in the Proceedings of
the RSA Conference 2004. |
|
[Sc] |
Marion Scheepers, Selection
principles and covering properties in topology, Note di Matematica,
to appear. |
|
[Tha] |
R. Thangadurai, Adams
theorem on Bernoulli numbers revisited, Journal of Number Theory 106
(2004), 169-177. |
Our paper: |
Cited in: |
|
3 |
[Sc], [Mi] |
|
5 |
[NR], [Je], [BVPCS], [SBPCR], [An],
[An2] |
|
6 |
[Sc], [EJ] |
|
7 |
[Bu], [Sc], [Mi] |
|
8 |
[DP], [SBPCR] |
|
10 |
[Tha] |
|
11 |
[GGN] |
|
12 |
[Mi], [Sc] |
|
14 |
[Sa] |
|
16 |
[Sc], [EJ], [Sa] |
|
13, 15, 16, 17 |
[Sc] |
|
19 |
[Sa] |
|
20 |
[Ba], [BNS], [BZ] |