Lorenzo Traldi

Marshall R. Metzgar Professor of Mathematics

Thanks to Ivan Dimitrov for this picture of commencement 2007

Office: 212 Pardee Hall

Phone: (610) 330-5276

email: traldil at lafayette.edu

Spring 2010 teaching schedule

Math 186-01: MWF 10-10:50 in Pardee 102B

Math 186-02: MWF 11-11:50 in Pardee 217

Office hours: MWF 12:00-1:00 and by appointment

Note: Classes will meet on snow days.

A little bit about Lorenzo

I grew up in New York City, a block from the American Museum of Natural History (really!). I went to Bronx Science for high school, and then Queens College. At Queens in 1973 I fell in love twice, once with mathematics -- 88 of the 120 credits I took as an undergraduate were in math. I also met Sharon Richter, and we've been married since 1977. We have four wonderful children -- Arthur, Matthew, Oliver and Rebecca -- and when they're home our house is full of music, chess and conversation. Click here to see some pictures of us.

I got my Ph.D. from Yale in 1980, and I came to Lafayette right after graduation. My Ph.D. dissertation was concerned with the elementary ideals of knots and links in three dimensions. Some dramatic knot-theoretic advances in the 1980s used combinatorics rather than algebraic topology to study knots and links, and since learning about them I've studied several different kinds of combinatorial structures. I'm particularly interested in clutters, which are used to describe network reliability problems; the most familiar examples are the clutters consisting of the sets of edges of a graph which form circuits or spanning trees. The idea which is most relevant to network reliability is to study partitions of the family which contains every set which contains at least one element of a clutter -- for example, the family which contains every subset of the edge-set of a connected graph which contains the edge-set of at least one spanning tree -- and I've studied these partitions using iterative techniques and symbolic logic. Recently I've also been learning about dice games as examples of the fascinating field of voting theory, which is full of counterintuitive results.

I've taught lots of courses here at Lafayette: precalculus, finite math, calculus, statistics, discrete math, a first-year seminar on paradoxes, linear algebra, geometry, graph theory, foundations (logic and set theory), complex calculus, abstract algebra, real analysis, and topology. I've also worked with many independent study students. Five students have written honors theses with me: Daniel L. King '87 wrote “A structure theorem for semi-primitive Artinian rings,” Ashish Bhan '91 wrote “A structure theorem for modules over pids,” Jennifer J. Mandelson '97 wrote “Evariste Galois, the man and his mathematics,” Andrew Hahn '98 wrote “An introduction to cardinality theory and infinity” and Alexandru O. Balan '03 wrote “An enhanced approach to network reliability using Boolean algebra.”

I've been glad to participate in the Research Experiences for Undergraduates program sponsored by the NSF and Lafayette College. Ryan R. Martin, Justin T. Moore, Jessica K. Benashski '96 and I worked on a project in graph theory during the summer of 1994. Jonah Blasiak, Jennifer M. Rowe and Oded Yacobi studied clutters with me during the summer of 2001, and so did Michael H. Dinitz, Jonah M. Gold and Thomas C. Sharkey during the summer of 2003. In the summer of 2005 Jeff Johannes, Karen McCready, Ken Ober and Matthias Youngs and I studied finite-type link invariants. Chelsey Cooley, Billy Ella, Michael Follett '11, Eric Gilson and I learned about dice games as examples of the theory of voting in the summer of 2008.

I've also been glad to work with students supported by Lafayette's EXCEL Scholars program. William R. Graveman '99 worked on clutters with me in the summer of 1997. Alexandru O. Balan '03 and I studied Boolean algorithms for network reliability during the summers of the years 2000 - 2002, and Rebecca Andersen '06 and Jonathan Rowe '06 continued that work during the summers of 2004 and 2005; you can download an algorithm we developed here. Brian Kronenthal '07 and Michael Follett '11 helped me study dice games in 2006, 2008 and 2009. Joe Dudek '09 and Mike Follett '11 worked with Louis Zulli and me in 2007, 2008 and 2009 to understand how certain invariants of knots and graphs are related to linear algebra over the two-element field.

Mathematical publications

Note: Please email me if you would like a copy of a paper which is not “clickable” below, or if clicking does not provide you with the final version of a published paper. (Many publishers prefer that authors not post published articles online.)

(with R. E. Andersen '06, A. O. Balan '03 and J. P. Rowe '06) A note on independent-disjoint analysis of network reliability

[Click here for a program discussed in this paper.]

A bracket polynomial for graphs. III. Vertex weights, Journal of Knot Theory and its Ramifications

Binary nullity, Euler circuits and interlace polynomials, European Journal of Combinatorics

A bracket polynomial for graphs. II. Links, Euler circuits and marked graphs, Journal of Knot Theory and its Ramifications

(with M. H. Dinitz, J. M. Gold and T. C. Sharkey) Graphical representations of clutters, Ars Combinatoria

Weighted interlace polynomials, Combinatorics, Probability and Computing 19 (2010), 133-157

(with C. Anderson, J. Cutler and A. J. Radcliffe) On the interlace polynomials of forests, Discrete Mathematics 310 (2010), 31-36

(with L. Zulli) A bracket polynomial for graphs, I, Journal of Knot Theory and its Ramifications 18 (2009), 1681-1709

(with L. Zulli) Topological realizations of ortho-projection graphs, Topology and its Applications 156 (2009), 2515-2526

(with B. G. Kronenthal '07) Tied dice, Journal of Combinatorial Mathematics and Combinatorial Computing 68 (2009), 85-96

[Click here for the case n = 4.]

(with B. Shtylla '05 and L. Zulli) On the realization of double occurrence words, Discrete Mathematics 309 (2009), 1769-1773

Generalized dice: Many questions and a few answers, Graph Theory Notes of New York LIII (2007), 39-42

A conjecture about sums of disjoint products, Journal of Combinatorial Mathematics and Combinatorial Computing 60 (2007), 3-6

[Click here for symmetric SDP forms of the disjunction of p < 19 variables, or here for p = 19, or here for p = 23.]

(with J. A. Ellis-Monaghan) Parametrized Tutte polynomials of graphs and matroids, Combinatorics, Probability and Computing 15 (2006), 835-854

[Click here for details of the proof of Proposition 7.2.]

Dice games and Arrow's theorem, Bulletin of the Institute of Combinatorics and its Applications 47 (2006), 19-22

On the colored Tutte polynomial of a graph of bounded treewidth, Discrete Applied Mathematics 154 (2006), 1032-1036

Non-minimal sums of disjoint products, Reliability Engineering and System Safety 91 (2006), 533-538

Dice graphs, Congressus Numerantium 172 (2005), 177-191

(with J. Blasiak, J. Rowe and O. Yacobi) Several definitions of matroids, Ars Combinatoria 77 (2005), 33-44

The prevalence of "paradoxical" dice, Bulletin of the Institute of Combinatorics and its Applications 45 (2005), 70-76

Generalized activities and reliability, Journal of Combinatorial Mathematics and Combinatorial Computing 52 (2005), 17-32

Parallel connections and coloured Tutte polynomials, Discrete Mathematics 290 (2005), 291-299

A subset expansion of the coloured Tutte polynomial, Combinatorics, Probability and Computing 13 (2004), 269-275

Rearrangements of generalized shellings, Congressus Numerantium 163 (2003), 27-32

(with A. O. Balan '03) Preprocessing minpaths for sum of disjoint products, Institute of Electrical and Electronics Engineers Transactions on Reliability 52 (2003), 289-295

Clutters and circuits, III, Algebra Universalis 49 (2003), 201-209

A note on delta-wye-delta reductions of plane graphs, Congressus Numerantium 158 (2002), 213-220

Visualizing clutters, Congressus Numerantium 154 (2002), 13-20

Chain polynomials and Tutte polynomials, Discrete Mathematics 248 (2002), 279-282

Commentary on: reliability polynomials and link importance in networks, Institute of Electrical and Electronics Engineers Transactions on Reliability 49 (2000), 322

A note on a theorem of Wu, Kobe Journal of Mathematics 17 (2000), 27-28

Series and parallel reductions for the Tutte polynomial, Discrete Mathematics 220 (2000), 291-297

A note on reliability and expected value, Congressus Numerantium 133 (1998), 95-99

Two universal reliability problems, Congressus Numerantium 132 (1998), 199-204

Clutters and circuits. II, Advances in Applied Mathematics 21 (1998), 437-456

Clutters and circuits, Advances in Applied Mathematics 18 (1997), 220-236

(with J. Rodriguez) (K,j)-domination and (K,j)-reliability, Networks 30 (1997), 293-306

On the Arf invariant of a purely proper link, Journal of Knot Theory and its Ramifications 5 (1996), 417-420; erratum, 741

Polygon matroids for hypergraphs and K-terminal networks, Congressus Numerantium 110 (1995), 153-159

(with J. Benashski '96, R. Martin and J. Moore) On the beta-invariant for graphs, Congressus Numerantium 109 (1995), 211-221

Crapo's beta-invariant and K-terminal networks, in Graph Theory, Combinatorics, and Applications: Proceedings of the Seventh Quadrennial International Conference on the Theory and Applications of Graphs, Y. Alavi and K. Schwenk, eds., J. Wiley and Sons, N. Y., 1995, pp. 1169-1177

Generalized activities and K-terminal reliability. II, Discrete Mathematics 135 (1994), 381-385

(with G. Gordon) Polynomials for directed graphs, Congressus Numerantium 94 (1993), 187-201; addendum, ibid. 100 (1994), 5-6

On the star-delta transformation in network reliability, Networks 23 (1993), 151-157

Generalized activities and K-terminal reliability, Discrete Mathematics 96 (1991), 131-149

Higher Fitting invariants, Houston Journal of Mathematics 17 (1991), 55-61

(with G. Gordon) Generalized activities and the Tutte polynomial, Discrete Mathematics 85 (1990), 167-176

A dichromatic polynomial for weighted graphs and link polynomials, Proceedings of the American Mathematical Society 106 (1989), 279-286

Linking numbers and Chen groups, Topology and its Applications 31 (1989), 55-71

Conway's potential function and its Taylor series, Kobe Journal of Mathematics 5 (1988), 233-264

(with W. S. Massey) On a conjecture of K. Murasugi, Pacific Journal of Mathematics 124 (1986), 193-213

On the Goeritz matrix of a link, Mathematische Zeitschrift 188 (1985), 203-213

Milnor's invariants and the completions of link modules, Transactions of the American Mathematical Society 284 (1984), 401-424

Some properties of the determinantal ideals of link modules, Mathematics Seminar Notes, Kobe University, 11 (1983), 363-380

(with M. Sakuma) Linking numbers and the groups of links, Mathematics Seminar Notes, Kobe University, 11 (1983), 119-132

The determinantal ideals of link modules. II, Pacific Journal of Mathematics 109 (1983), 237-245

Linking numbers and the elementary ideals of links, Transactions of the American Mathematical Society 275 (1983), 309-318

The determinantal ideals of link modules. I, Pacific Journal of Mathematics 101 (1982), 215-222

A generalization of Torres' second relation, Transactions of the American Mathematical Society 269 (1982), 593-610

(with W. S. Massey) Links with free groups are trivial, Proceedings of the American Mathematical Society 82 (1981), 155-156

Lafayette College home page Department of Mathematics home page