Department of

Mathematics


Seminar Calendar
for events the day of Thursday, February 27, 2014.

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Thursday, February 27, 2014

11:00 am in 241 Altgeld Hall,Thursday, February 27, 2014

Partitions with non-repeating odd parts - q-hypergeometric and combinatorial identities

Krishna Alladi (University of Florida)

Abstract: By studying partitions with non-repeating odd parts using representations in terms of 2-modular graphs, we first derive a Lebesgue tupe q-series identity and use this to give a unified treatment of several fundamental identities in the theory of q-hypergeometric series. Next we study these partitions combinatorially and obtain new weighted partition identities. Consequences include a combinatorial proof of a modular relation for the Gollnitz-Gordon functions, and a new derivation of a shifted partition identity due to Andrews. Finally we discuss some new parity results and hint at a theory of basic partitions.

1:00 pm in Altgeld Hall 347,Thursday, February 27, 2014

Equivalent trace sets for arithmetic Fuchsian groups

Grant Lakeland (UIUC Math)

Abstract: Let $M = \mathbb{H}^2 / \Gamma$ be a finite area hyperbolic Riemann surface. The length spectrum of M is the set of closed geodesic lengths counted with multiplicities, and the length set is the same set without multiplicities. While isospectral surfaces have the same area, results of Schmutz and Leininger-McReynolds-Neumann-Reid show that this is not the case if we only require equal length sets. The length set of M is closely related to the trace set of $\Gamma$, and thus one may ask how different two cofinite Fuchsian groups with a given trace set may be. In this talk, I'll show that if $\Gamma$ is the modular group (or one of certain families of commensurable groups) then $\Gamma$ contains infinitely many finite index subgroups with the same trace set as itself.

2:00 pm in 140 Henry Administration Building,Thursday, February 27, 2014

Number-Theoretic Random Walks

Yiwang Chen and Tong Zhang (UIUC Math)

Abstract: Exponential sums of the form $(1) S(N)=S(N,f,\alpha)=\sum_{n=0}^{N-1} f(n)e^{i\alpha n}$, where $f(n)$ is some number-theoretic function and $\alpha$ a real number, arise in many problems in number theory and form a key tool in understanding the properties and behavior of the function $f(n)$. In this talk, we focus on the geometric behavior of these sums, interpreting the sums (1) as a random walk in the complex plane whose $n$th step is given by $f(n)e^{i\alpha n}$. We call such a random walk a number-theoretic random walk. This geometric point of view was pioneered by D.H. and Emma Lehmer in a series of papers entitled ``Picturesque Exponential Sums I, II". Specifically, they considered the case when $f(n) = e^{2\pi is_b(n)/b}$, where $s_b(n)$ denotes the sum of the digits of $n$ in base $b$, and $(\ast) \alpha= 2\pi j/b$, where $j$ is an integer, and showed that in this case the graphs of the sums (1) are periodic and exhibit interesting symmetry patterns. Motivated by Lehmers' work, we consider the geometric behavior of sums of this type with general values of $\alpha$, i.e., without the restriction $(\ast)$ on $\alpha$. Dropping the condition $(\ast)$ turns out to dramatically change the nature of the problem: The sums (1) are in general unbounded, and their behavior appears, at first glance, much like that of a true random walk. In some cases the resulting ``random walk" is a known fractal curve such as the Koch curve, while in others it has distinctive fractal-like features that can be explained theoretically; in most cases, however, its behavior remains largely mysterious and seems to defy any explanation. In this talk, we report on experimental and theoretical results on such generalized Lehmer type sums. In particular, to quantify the degree of randomness inherent in a random walk of the type (1), we introduce the scaling exponent as $\mu = \limsup_{N\to \infty} \log|S(N)|/\log N $. We prove exact theoretical formulas for $\mu$ in the case when $\alpha$ is a rational multiple of $2\pi$, and we obtain numerical results in many other cases. These results suggest that such number- theoretic ``random walks", while exhibiting many random-like features, grow at a much slower rate than a true random walk.

2:00 pm in 243 Altgeld Hall,Thursday, February 27, 2014

Distributional Limits, Doubling Metric Spaces, and a Lemma

Jim Gill (St. Louis University)

Abstract: In 2001 I. Benjamini and O. Schramm proved that the distributional limit of a random graph with bounded degree is almost surely recurrent with respect to a random walk. To prove this fact they devised an interesting lemma about finite point sets in the plane. Since then this same lemma has been used several times in different contexts. In particular, in joint work with S. Rohde, we used it to show that the uniform infinite planar triangulation is almost surely a parabolic Riemann surface. In further investigation by the speaker it has recently been found that the conclusion of this lemma holds in any doubling metric space. In fact, the metric doubling condition is equivalent to the property described in the lemma.

4:00 pm in 245 Altgeld Hall,Thursday, February 27, 2014

Göllnitz's (Big) partition theorem and its dual

Krishnaswami Alladi (University of Florida)

Abstract: A Rogers-Ramanujan (R-R) type partition identity is one which equates an infinite series to an infinite product, the series being the generating function of partitions whose parts satisfy difference conditions and the product being the generating function of partitions whose parts satisfy congruence conditions. The history of R-R type identities is rich and they arise in a variety of settings ranging from Lie algebras to statistical mechanics. The 1967 (Big) partition theorem of Göllnitz is one of the deepest R-R type identities. Extending the 1993 "method of weighted words" approach of Alladi-Gordon to generalizations of Schur's fundamental partition theorem, Alladi-Andrews-Gordon in 1995 provided a new approach to Göllnitz' theorem and obtained substantial generalizations and refinements. They viewed the theorem as emerging out an intricate three parameter q-hypergeometric identity called the "key identity". Consequences included new combinatorial proofs of Jacobi's triple product identity and partition congruences modulo powers of 2. In the 1960s Andrews discivered two infinite hierarchies of partition theorems emanating from Schur's theorem -- the two hierarchies being duals. In that spirit, Alladi-Andrews have recently found a dual of Göllnitz's theorem. We will describe the combinatorics behind the construction of the dual and determine its analytic representation.