# PMASS Colloquia 2014

## Manfred Denker, Penn State

Neural networks and labeled trees

Thursday, January 16, 2:30pm

It has been observed in experiments that the number of active neurons in a neural network obeys a power law: The chances of having L of them is proportional to L^{-3/2}. I will discuss the question how this can be derived in a mathematically rigorous way from simple assumptions. It turns out that this question is related to an old theorem of Caylay (in 1889) about the number of labeled trees with a fixed number of vertices.

The mathematical tools used in the talk are taken from combinatorics (placing balls in boxes), binomial coefficients, and analysis (differentiation).

## Mihran Paikian, Penn State

Coverings by systems of linear equations

Thursday, January 20, 2:30pm

Let V be a finite dimensional vector space over a finite field.
Let S be a subset of V. What is the minimal number of systems of
linear equations such that the union of their solutions exactly coincides with S?
The motivation for this question comes from logic and computer science.
I will explain all basic concepts required for understanding this problem, and
then discuss some algebraic and geometric approaches to its solution in special cases.

Probability puzzles

Thursday, Feburary 27, 2:30pm

We will discuss several problems in elementary probability
where intuition may give a wrong answer while the correct one seems,
at first glance, unlikely or even impossible.

## Anatole Katok, Penn State

Billiard table as mathematician's playground

Thursday, February 13, 2:30pm

Motion of an ideal particle ("a billiard ball") inside a bounded plane domain with the usual elastic reflection law `"the angle of incidence is equal to the angle of reflection'' is one of the simplest mechanical problems and attempts to describe this motion lead both to remarkable open problems and to connections with a variety of deep mathematical theories. For example, it is still not known whether inside any triangle there is a periodic billiard motion. It is also not known whether there is twice differentiable convex curve such that some billiard motion inside it is dense. In this talk I will give a very brief and elementary introduction to the beauty and complexity of this problem that looks so deceptively simple.

## Vadim Kaloshin, University of Maryland, College Park

Can you hear the shape of the drum?

Thursday, March 6, 2:30pm

## Joe Roberts, Penn State

Grouping and rearranging terms in infinite series

Thursday, April 3, 2:30pm

When studying infinite series of real numbers, it is tempting to expect the familiar properties of addition to hold, and in some cases they do. Series that converge absolutely are as well behaved as finite series --- they are associative and commutative. However, series that are not absolutely convergent have very different properties. The Riemann Rearrangement Theorem states that any conditionally convergent series can be reordered to sum to any real number or to diverge (illustrating a failure of commutativity), and there are divergent series for which one can choose a subsequence of partial sums that converge to any arbitrary real number (a failure of associativity). I will give examples of these kinds of unexpected outcomes, eventually building to a construction due to Sierpinski of a single power series whose terms can be grouped to converge uniformly to any continuous function on [0,1] vanishing at 0 --- a "universal Taylor series".

## Jason Rute, Penn State

What is a random number?

Thursday, April 17, 2:30pm

Suppose there was a new lottery, Power Coin. Once a week, the lotto agency flips a fare coin 50 times,
converts the heads to ones and the tails to zeros, and publishes that as the winning lottery ticket. Consider these
three lottery ticket numbers:

01010101010101010101010101010101010101010101010101,
11001001000011111101101010100010001000010110100011,
011000110 01001110100110110001011100110110000000111.

Do any of these seem more random than the others? (What if I told you the second was the first 50 binary digits of pi?)
Does it even make sense to say that a finite sequence of bits is more random than another? What about an infinite
sequence of bits? In this talk, I am going to explore different definitions of random number. I will talk about the view
that random numbers are a myth. I will talk about normal numbers. I will talk about Kolmogorov complexity, a way to
measure randomness via computable compression. Last, I will introduce Martin-Löf randomness.