Russell Was Not Naive

Posted on June 27, 2013 by Avery Carr

images-2As one peers at a radiating metropolis of buildings piercing the night sky, it is easy to imagine the thought and engineering that produced such a modern wonder.    The intricate detail and unique designs appeal to the artistic senses.  Before each building’s architectural beauty is manifested, a strong foundation must be poured.    Such a concept is inherent throughout physical law and abstract mathematics.

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Advice for (Berkeley) Ph. D. students in math -Bjorn Poonen

Posted on June 25, 2013 by Shijie Gu
Dr. Bjorn Poonen

Dr. Bjorn Poonen

Occasionally, I have a chance to find this article posted on Dr. Bjorn Poonen‘s personal site. He gave quite a lot suggestions not only for Berkeley PhD students, in my view, but also for all the math graduate students. The article, beginning from the very first preliminary exam, language exams to applying for the jobs, publishing papers, appears to be a good instruction that covers all the academic life of graduate students.  I have to admit that I felt quite surprised when I read some parts of this article. For instance, Dr. Poonen described how cruel the job market was during his graduation time, honestly, which is far beyond my imagination, “When finishing my Ph. D., I applied to about 50 schools, and I knew some people who applied for over 200;…”. How did he survive? Except for the good academic background, there must be some other secret weapons that can be found in his article.

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How “small” can the gap be? -A huge landmark in number theory.

Posted on June 24, 2013 by Shijie Gu
Yitang Zhang, lecturer in mathematics at the University of New Hampshire Photo courtesy of Lisa Nugent/UNH Photographic Services

Yitang Zhang, lecturer in mathematics at the University of New Hampshire
Photo courtesy of Lisa Nugent/UNH Photographic Services

What’s the gap between consecutive primes? One can easily observe that the gap will keep increasing as the primes become far rarer. However, for the bounded gap, there exists infinitely many pairs of primes. This is a form of the twin prime conjecture. Continue reading →

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The Conjecture of Marie-Sophie Germain

Posted on June 20, 2013 by Avery Carr

images-1Numbers pervade our lives in many different venues.  Prime numbers, in particular, weave their way into the very fabric of our daily existence.   From surfing the internet to pseudo-random number generators, primes are found ever present behind a multitude of abstractions.    A prime number is a positive integer such that it is only divisible by one and itself.  In light of this definition, the number one is not considered a prime.

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Newcomb’s Paradox

Posted on June 13, 2013 by Avery Carr

Unknown-1In the realm of mathematical puzzles and thought experiments one can find a stock pile of paradoxes.   The Mariam-Webster Dictionary defines a paradox as “an argument that apparently derives self-contradictory conclusions by valid deduction from acceptable premises.”  One short example of such oddities is the Liar’s Paradox.  It is the deceptively simple declaration, “This sentence is false.”   After some evaluation, one concludes that, if the sentence is false, it is also true.  Likewise, if it is true it must be false, providing a self-contradiction that does not permit the assignment of a single truth value.

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