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matematiske perler (og kuriositeter)
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In 1919 – a century ago – Viggo Brun succeeded in constructing a sieve that opened up new avenues to make progress on some famous conjectures about prime numbers. These conjectures, e.g., the Goldbach Conjecture and the Twin Prime Conjecture, were considered by the experts at the time to be unattainable. Remarkably, Brun started with the Eratosthenes' sieve from antiquity, and by elementary methods, using the so-called inclusion–exclusion principle, he was able to modify that sieve so that he could obtain highly non-trivial results. We will try to explain Brun`s reasoning that led to his sieve.
(Incidentally, Viggo Brun was professor at NTH from 1923 to 1946.)
Adapted from an image by Dr Shwarz.
Mathematicians like to bring order to chaos, for example by classifying the structures we invent/discover. In abstract algebra, the fundamental kind of structure is the group: a set with one operation satisfying a few natural conditions. Groups show up everywhere in mathematics, often as sets encoding symmetries. For example, the groups that we now call Lie groups were introduced by Sophus Lie in connection with his work on differential equations.
Can we classify all groups in a meaningful way? This is hopeless unless we restrict ourselves to finite groups, that is, groups containing only finitely many elements. But even here the task is generally considered to be impossible. However, finite groups are built out of building blocks, namely finite simple groups, like integers are built out of prime numbers. The finite simple groups have been completely classified, in an effort that has involved lots of mathematicians and many thousand pages over fifty years. In this talk, we take a look at the history of this classification theorem.
At several points in mathematical literature, a constant called the ubiquitous constant with approximate value 0.847213 appears. In this talk, we are going to investigate some siblings of the ubiquitous constant, arising from theories going back to Gauss and, later on, Ramanujan.
Most probably, the constants appear at far more places in mathematics (and maybe even other sciences) than covered by this talk; nonetheless, we will try to discuss some of the following topics.
We will start with an engineering problem about optimal sampling strategies for Gaussian Gabor systems, the Strohmer and Beaver conjecture. This is, to the speaker's knowledge, the most recent example of the appearance of the siblings.
In addition, we will study an old unsolved problem from geometric function theory, concerning the true value of Landau's “Weltkonstante”. Further topics may include the temperature distribution on the 2-dimensional torus with varying metric, a coloring problem for lattices, a generalized arithmetic-geometric-mean method, Gauss' hypergeometric function 2F1 and complete elliptic integrals.
Despite the heavy nature of the topics, this is going to be a fun talk where we will just scratch the surface of each topic a bit.
Communication in medical research is storytelling by statistics. As a biostatistician at Oslo University Hospital, my job is to cooperate with clinicians to translate stories about clinically interesting blood, sweat and tears into numbers. When they tell their stories, clinicians must use methods and a language originating in a foreign culture: the statistical culture. It takes knowledge, practice and courage to succeed at a top level internationally, in disciplines which communicate differently from your mother tongue.
An important task of the statistical community is to approach researchers with other professional background and training, to achieve the best communication and consequently, the best results. “The best results” in interdisciplinary communication and advising comprise more than choosing the best statistical methods for a problem at hand. Statisticians should meet non-statisticians half-way and enhance their understanding and confidence in the foreign language of statistics. Advising, teaching, articles, books, popular science, TV-shows, TED-talks, apps, blogging, knitting – no means should remain unused. And to find the best channels, we must increase our awareness of our own superpowers.
Never stop having fun!
There are different descriptions of mathematical knowledge. Two quite opposite characterizations are given by two English philosophers and scientists, Roger Bacon (c. 1214–1294) and Bertrand Russell (1872–1970):
“The knowledge of mathematical things is almost innate in us … this is the easiest of sciences, a fact which is obvious in that no one's brain rejects it; for layman and people who are utterly illiterate know how to count and reckon.”
R. Bacon (cited in Dehaene, 2011, p. 260)
“Mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true.”
B. Russell (1918, p. 75)
How is it possible simultaneously to view mathematics as a system of transparent truths and a tangle of relations among opaque entities? Why is mathematics both so easy and so hard? The lecture will draw on recent research on neuroscience – it will give a brief overview of possible sources of learning disability in mathematics, and an explanation of how the brain works when we study and learn mathematics. This takes us to a semiotic perspective on mathematical activity – the impact of representations of mathematical object and transformations between them. Finally, we will see how this might be implemented in mathematics instruction.