MASS Colloquia 2015

Thursday, September 3 Diane Henderson, Penn State
1:25pm A tour of Pritchard Lab
ABSTRACT The MASS students will be introduced to the Pritchard Fluids Lab, a physics that is a part of the Mathematics Department.
Thursday, September 10 James Sellers, Penn State
1:25pm Congruences for Fishburn Numbers
ABSTRACT The Fishburn numbers, originally considered by Peter C. Fishburn, have been shown to enumerate a variety of combinatorial objects. These include unlabelled interval orders on n elements, (2+2)--avoiding posets with n elements, upper triangular matrices with nonnegative integer entries and without zero rows or columns such that the sum of all entries equals n, non--neighbor--nesting matches on [2n], a certain set of permutations of [n] which serves as a natural superset of the set of 231--avoiding permutations of [n], and ascent sequences of length n. In December 2013, Rob Rhoades (Stanford) gave a talk in the Penn State Algebra and Number Theory Seminar in which he described, among other things, the relationship between Fishburn numbers, quantum modular forms, and Ramanujan's mock theta functions. Motivated by Rhoades' talk, George Andrews and I were led to study the Fishburn numbers from an arithmetic point of view - something which had not been done prior. In the process, we proved that the Fishburn numbers satisfy infinitely many Ramanujan--like congruences modulo certain primes p (the set of which we will easily describe in the talk). In this talk, we will describe this result in more detail as well as discuss how our work has served as the motivation for a great deal of related work in the last year by Garvan, Straub, and many others.
Thursday, September 17 Oleg Viro, Stony Brook University
1:25pm Real enumeration problems
ABSTRACT We will consider problems of mixed setup, in which the initial object belongs to the elementary differential geometry (say, a smoothly immersed generic planar or spherical curve) and we are counting with certain weights the simplest algebraic curves in special position to the original curve. Say, bitangent lines or tritangent circles. The resulting quantity happens to be a topological invariant of the curve, which can be calculated combinatorially.
Thursday, September 24 Aaron Abrams, Washington and Lee University
1:25pm Rectangling the square
ABSTRACT There is a famous problem called ``squaring the square,'' named in reference to 
the classical problem of squaring the circle (though the problems are unrelated). 
To ``square the square'' means to divide a square into smaller squares, all of 
which have different sizes. Many solutions are known, and the problem has many 

In this talk I will discuss this problem and some of its rectangular relatives. One 
question we will answer is sort of dual to the above: rather than using (rectangular) 
tiles with fixed shapes (i.e. squares) and varying sizes, how can you tile a square 
with (rectangular) tiles that have prescribed sizes but variable shapes?

Thursday, October 1 Larry Rolen, Penn State
1:25pm Recurrences for Eisenstein series
ABSTRACT In this talk, we will learn about recursive formulas for Eisenstein series, some of which are classical, and some of which are surprisingly new. In particular, we will see that these important examples of modular forms can be recursively defined in many ways, which directly yields surprising identities between convolution sums of sums of divisor functions as well as relations among the classical Bernoulli numbers. Along the way, we will learn about important examples of doubly periodic, meromorphic functions, also known as elliptic functions, and their connections to modular forms. This talk will be self-contained, and no prior knowledge of modular forms or the related objects mentioned above will be assumed.
Thursday, October 15 Gábor Domokos, Budapest University of Technology and Ecomonics
1:25pm Mono-monostatic bodies: the story of the Gömböc
ABSTRACT In 1995, V.I. Arnold conjectured that convex, homogeneous solids with just two static balance points (so-called mono-monostatic bodies) may exist. Ten years later, based on a constructive proof, the first such object (dubbed "Gömböc") was built. 

The newly discovered objects show various interesting features. We will point out that mono-monostatic bodies are neither flat, nor thin, they are not similar to typical objects with more equilibria and they are hard to approximate by polyhedra. Despite these "negative" traits, there seems to be strong indication that these forms appear in the living Nature due to their special mechanical properties: some turtle species evolved special shell geometries close to the Gömböc to facilitate self-righting. 

The first numbered Gömböc (Gömböc 001) was given to V.I. Arnold on the occasion of his 70th birthday in Moscow. Here Arnold proposed that the Gömböc may play a role in explaining the geometric evolution of pebbles. I will discuss some mathematical and geophysical aspects of this conjecture in the Department Colloquium.

Thursday, October 22 Augustin Banyaga, Penn State
1:25pm Flavor of Morse Theory
ABSTRACT In 1934, Marston Morse initiated a study (now known as Morse Theory) relating critical points of some special function f on a smooth manifold M to some topological invariants of M, which are independent 
of the particular choice of the function f. 
In this talk, I describe these special functions ( called Morse functions) and give some simple examples. Then I proceed with a simple result that the Euler characteristic of a manifold M can be given in terms of critical points of a Morse function. Finally, I will discuss ( and illustrate by a simlpe example) the theorem that a Morse function on M determines a CW complex structure on M. If time permits, I will 
mention some groundbreaking theorems obtained using Morse Theory, like Smale's proof of Poincare conjecture in dimension bigger or equal to 5,and the h-cobordism theorem. I will also mention the construction of Morse Homology, and Floer Homology, which play a central role in many areas of Mathematics and Mathematical Physics. 
I recommend reading the beautiful book by Milnor " Morse Theory" and chapter 3 in the book "Morse Homology" by A.Banyaga and D.Hurtubise.
Thursday, October 29 Ken Stephenson, University of Tennessee
1:25pm Sphere packing in 2.5 dimensions
ABSTRACT The densest packing of unit-diameter spheres (i.e. discs) in 2D 
is hexagonal --- namely, the "penny-packing" wherein every disc is tangent 
to 6 others. The 3D version of the penny-packing is the "grocer-packing", 
the configuration you see with oranges stacked on a grocery counter. Around 
1600 Kepler conjectured that this grocer-packing is the densest possible in 3D, 
and after a mere 400 years, Tom Hales, his collaborators, and clever computer 
work have proven Kepler correct. 

In this talk we consider packings of unit-diameter spheres in 3D, but now 
with the side condition that they all be tangent to a fixed cylinder. Taking 
a cue from history, we focus on hexagonal patterns and speculate on density. 
However, surprising issues enter the picture and suggest that this new problem 
hovers somewhere between the 2D and 3D cases --- hence the 2.5D of our title. 
I will use plenty of pictures and hope to get you to exercise your 
intuition a little as we see if there's a reasonable conjecture to make.

Thursday, November 5 Leonid Bunimovich, Georgia Institute of Technology
1:25pm Mechanisms of Chaos
ABSTRACT Chaotic motion is caused by internal instability of dynamics (system's evolution). the last means that starting with arbitrarily closed to each other states the system will evolve very differently. We explain why systems with chaotic dynamics are typical among real systems as well as of their mathematical models. The main mechanisms generating chaotic motion will be described.
Thursday, November 12 Gary Mullen, Penn State
1:25pm Latin squares
ABSTRACT Good problems in mathematics are often easy to describe but their solutions may be difficult to obtain, or perhaps are even unknown today. Many such problems arise in number theory. Latin squares form another set of such problems. Latin squares are interesting combinatorial objects with numerous properties. Many of these properties are easy to describe, and yet, many of the them are very difficult to prove. We will discuss the number of latin squares, which despite the use of modern computers, is still not known except for some very small cases. We will also discuss sets of mutually orthogonal latin squares for which we have many interesting open problems.
Thursday, November 19 Gil Bor, Centro de Investigación en Matemáticas, Mexico
1:25pm The mathematics of internet search
ABSTRACT An Internet search engine such as Google typically retrieves millions of search results in a fraction of a second, of which only the top few results are ever used by the searcher; the order in which the results are presented must therefore be chosen rather carefully. How is it done? A key ingredient is a web page’s “rank”, reflecting somehow the web page importance. The rank is determined by the “PageRank algorithm”, which counts each link to a page as a “recommendation”, weighing each recommendation by the rank of the recommending page… We will see in this talk how to avoid the obvious circularity of this procedure, as well as some other less-obvious traps. The algorithm is a remarkable application of the mathematical theory of Markov Chains, developed over a century ago. The same algorithm is useful in many other situations, such as the ranking of football teams and DNA’s genes.