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Register a new userStudent-centric graduate training in mathematics: A commentary
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Career opportunities for PhDs in the mathematical sciences have never been better. Traditional faculty positions in mathematics departments in colleges and universities range from all teaching to combined teaching and research responsibilities. Beyond those, a wide array of careers has now opened up to freshly minted graduates, in academics, industry, business, and government. It is well-understood that these all require somewhat different preparations for PhDs to be competitive. This commentary compares and contrasts mathematics graduate programs with Ph.D. programs in the life and biomedical sciences, which are structured in a way that allows considerable customization around each students career goals. While these programs may not be appropriate templates for the mathematical sciences, they have some features that might be informative. This commentary is intended to add perspective to the ongoing discussion around PhD training in the mathematical sciences. It also provides some concrete proposals for changes.
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Problem-based learning (PBL) is a constructivist learner-centered instructional approach based on the analysis, resolution and discussion of a given problem. It can be applied to any subject, indeed it is especially useful for the teaching of mathematics. When compared to traditional teaching, the PBL approach requires increased responsibility for the teachers (in addition to the presentation of mathematical knowledge, they need to engage students in gathering information and using their knowledge to solve given problems). It thus become crucial that the future teachers become aware of its effectiveness. One of the main obstacle to this awareness lies usually on the fact that future teachers did not find this methodology in their own pre-service training. In this paper we will describe the attempt to introduce PBL in University courses so to have future maths teacher experience mathematics themselves.
I am an industrial mathematician. When asked to identify my profession or academic field of study, this is the most concise answer I can provide. However, this seemingly straightforward statement is commonly greeted by a blank stare or an uncomfortable silence, regardless of whether I am speaking to a fellow mathematician or a non-mathematician. I usually follow up with the clarification: I am an applied mathematician who derives much of my inspiration from the study of industrial problems that I encounter through collaborations with companies. This dispels some confusion, but unfortunately still leaves a great deal open to interpretation owing to the vagueness of the words mathematics, industry and company, each of which covers an extremely broad range of scientific or socio-economic activity. To those academics who actually work in the field of industrial mathematics (and whose perspective referred to in the title is the focus of this article) this ambiguity is familiar and untroubling. However, for anyone less acquainted with the work of industrial mathematicians, some clarification is desirable especially for anyone who might be considering entering the field. This essay therefore aims to shed light upon the nature of research being done at the interface between mathematics and industry, paying particular attention to the following questions: What is industrial mathematics? Where is industrial mathematics? How does one do industrial mathematics? Why (or more precisely, what value is there in doing) industrial mathematics? I will attempt to answer these questions by means of several case studies drawn from my own experience in tackling mathematical problems from industry.
An introduction to the On-Line Encyclopedia of Integer Sequences (or OEIS, https://oeis.org) for graduate students in mathematics
This is an expanded version of my review of Nina Engelhardts book Modernism, Fiction and Mathematics, Edinburgh University Press 2018. A considerably shortened version will appear in the Notices of the AMS.
We survey recent results on the mathematical stability of Bitcoin protocol. Profitability and probability of a double spend are estimated in closed form with classical special functions. The stability of Bitcoin mining rules is analyzed and several theorems are proved using martingale and combinatorics techniques. In particular, the empirical observation of the stability of the Bitcoin protocol is proved. This survey article on the mathematics of Bitcoin is published by the Newsletter of the European Mathematical Society, vol.115, 2020, p.31-37. Continuation of arXiv:1601.05254 (EMS Newsletter, 100, 2016 p.32).
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