Algebra and Cryptography Seminar, Spring 2014

Organizers: Robert Gilman, Delaram Kahrobaei, Alexei Myasnikov, and Vladimir Shpilrain


2:30-3:30 pm
Room 8405, CUNY Graduate Center
365 Fifth Avenue at 34th Street


11:00 am-12:00 pm
Room Peirce 220, Stevens Institute of Technology
Hoboken, NJ


February 21, Graduate Center: Vladimir Shpilrain (City College), Solving Yao's millionaires' problem without one-way functions
Abstract: The ``two millionaires problem" introduced by Yao is: Alice has a private number a and Bob has a private number b, and the goal of the two parties is to solve the inequality a < b? without revealing the actual values of a or b, or more stringently, without revealing any information about a or b other than a < b or a > b. We offer efficient and practical physical as well as mathematical solutions of Yao's millionaires' problem without using any one-way functions.

February 28, Graduate Center: Mohammad Mahdian (Google), Online bipartite matching with random arrivals
Abstract: In the online bipartite matching problem, we have n boys and n girls, with girls arriving one by one. When a girl arrives, we learn the set of boys she likes, and must match her to one such boy who is still unmatched. This decision cannot be changed later, and must be made without the knowledge of what comes later. In a classical result, Karp, Vazirani, and Vazirani showed that a simple ranking algorithm achieves a competitive ratio of 1-1/e for this problem, i.e., it guarantees that no matter in what order the girls arrive, the algorithm produces a matching whose size is at least 1-1/e times the size of the optimal matching. Their result also implies that in the random arrivals model defined by Goel and Mehta, where the girls arrive in a random order, a simple greedy algorithm achieves a competitive ratio of 1-1/e. In this talk, we discuss the ranking algorithm in the random arrivals model, and show that it has a competitive ratio of at least 0.696, beating the 1-1/e ~= 0.632 barrier in the adversarial model. Our analysis has two main steps. First, we exploit properties of the ranking algorithm to derive a family of factor-revealing linear programs (LPs). Second, to obtain a good lower bound on the optimal values of all these LPs and hence on the competitive ratio of the algorithm, we derive a family of modified LPs such that the optimal value of any single one of these LPs is a lower bound on the competitive ratio of the algorithm. This enables us to leverage the power of computer LP solvers to solve for large instances of the new LPs to establish bounds that would otherwise be difficult to attain by human analysis.
This is based on joint work with Qiqi Yan.

April 25, Graduate Center: Giovanni Di Crescenzo (Applied Communication Sciences), TBA

May 2, Graduate Center: Yuri Bahturin (Memorial University of Newfoundland), Growth of subalgebras and subideals in free Lie algebras
Abstract: The aim of the talk is to show how the technique of generating functions can be used for the study of subgroups of free groups and subalgebras of free Lie algebras. Most of the results are from joint work with Alexander Olshanskii.

May 9, Graduate Center: Jamshid Derakhshan (Oxford University), p-adic model theory uniformly in p, and applications to algebra and number theory
Abstract: In 1949, Julia Robinson proved that the field of rational numbers is undecidable. Later, in 1965, motivated by a conjecture of Artin on zeroes of p-adic forms, Ax and Kochen proved that the field of p-adic numbers is decidable. This enabled development of model theory for p-adic and related fields. These theories have turned out to have a surprising feature, namely, that most of the properties that hold do not depend on the prime p, and are in turn controlled by other universal theories. Later, Ax developed a model theory of finite fields for almost all p, and this theory beautifully relates to the theory of p-adics for almost all p. The uniform logical behavior was then showed by Pas-Denef-Loeser to govern many properties of p-adic integrals uniformly in p, which enabled a theory of motivic integration, and it is is believed that this is a feature of many of the naturally occurring concepts and structures in number theory.
Recently, in continuation of this line of developments, new results have been obtained on the following topics:
1. Counting conjugacy classes and representations (and other counting problems) in algebraic groups over local fields (this is joint work with Mark Berman, Uri Onn, and Pirita Paajanen) where one can translate asymptotic questions in group theory formulated in terms of a generating Poincare series to questions on p-adic and motivic integrals approachable by model theory,
2. A model theory has been developed for the adeles of a number field (this is joint work with Angus Macintyre). The ring of adeles of the rational numbers is a locally compact ring made of all the p-adic fields for all p and the real field and enables using results on the local fields to derive results for the global rational field. It is intimately related to questions on various kinds of zeta functions in arithmetic and geometry.
Going from the local (p-adic and real) fields back to the rationals has long been a fundamental local-global transition both for the logic and the algebra. The above results give new tools and results on this. I will give a survey of some of the results and challenging open problems.


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