Some implications of the 1907 Hurwitz formula

When/Where:

September 9, 2014, at 1:55pm, in LIT 368

Abstract:

I start with a brief review of Siegel’s formula for the number of representations of a positive integer by a genera of a quadratic form. I show how this formula implies the 1907 observation by Hurwitz, who proposed that \(\left|\{ (x,y,z) \in \mathbb{Z}^3 \mid n^2 = x^2+y^2+z^2 \}\right|\) may be expressed as a simple finite function of the divisors of \(n \in \mathbb{N}\). I use this formula together with the Jacobi triple product identity to prove (among other things) that \(9x^2+ 16y^2 +36z^2 + 16yz+ 4xz + 8xy\) represents exclusively all positive integers not of the form

• \(4^a(8m+7)\);
• \(4^a(8m+3)\), \(a=0,1,2\);
• \(4^a(4m+2)\), \(a=0,1,2\);
• \(4^a(8m+5)\), \(a=0,1\);
• \(M^2\);
• or \(4M^2\);

where \(a,m,M\) are non-negative integers and \(M\) is generated by 1 and primes congruent to \(1\pmod{4}\).

Identities missed by Schur and MacMahon

When/Where:

April 22, 2014, at 1:55pm, in LIT 368

Abstract:

The first portion of the talk will describe a new formulation for the generating function for the partitions in Schur’s 1926 partition theorem (joint work with Kathrin Bringmann and Karl Mahlburg). If there is time, I will describe how MacMahon “just missed” proving the Rogers-Ramanujan identities.

Chebotarev Density Theorem – its history and applications

When/Where:

April 15, 2014, at 1:55pm, in LIT 368

Abstract:

The Chebotarev Density Theorem states that the frequency of the occurrence of a given splitting pattern of prime ideals, for all primes \(p\) less than a large integer \(N\), tends to a certain limit as \(N\) goes to infinity. It generalizes Dirichlet’s theorem on arithmetic progressions and the Frobenius Density Theorem. The work of Chebotarev led the way to the Artin Reciprocity Theorem. The theorem yields many important applications including the reduction of the problem of classifying Galois extensions of a number field to that of describing the splitting of primes in extensions.

We will discover the history of the theorem from Gauss Quadratic Reciprocity to Artin Reciprocity Law. We will learn of the struggles and brilliance of Nikolai Chebotarev. Finally, we will catch a glimpse of Chebotarev’s counterintuitive proof to his density theorem.

“I belong to the old generation of Soviet scientists, who were shaped by the circumstance of a civil war. I devised my best result while carrying water from the lower part of town to the higher part, or buckets of cabbages to the market, which my mother sold to feed the entire family.” – Nikolai Chebotarev

Generalization and refinement of the Berkovich-Garvan partition inequalities

When/Where:

April 1, 2014, at 1:55pm, in LIT 368

Abstract:

In section 5 of their paper “Dissecting the Stanley Partition Function”, Profs. Berkovich and Garvan showed how to prove an infinite collection of new partition inequalities by constructing a (non-trivial) injection. I will review these inequalities and the injection they used. Then, I will look at generalizing and refining both the inequalities and the injection, in a manner akin to what Prof. Berkovich and I did in our recent paper “A partition inequality involving two \(q\)-Pochhammer symbols”. Finally, I will discuss how this method of generalization should also work in some other settings.

The truncated Jacobi triple product theorem

When/Where:

March 17, 2014, at 1:55pm, in LIT 339 (the Atrium)

Abstract:

Recently, G. E. Andrews and M. Merca considered the truncated version of Euler’s pentagonal number theorem and obtained a non-negativity result. They asked the same question on the truncated Jacobi triple product identity. In this talk we will discuss this question.

Higher Order Overpartition Spt Functions

When/Where:

April 8, 2014, at 1:55pm, in LIT 368

Abstract:

In 2003 Atkin and Garvan introduced the \(k\)-th rank and crank moments \(M_k(n) = \displaystyle\sum_{m}m^k M(m,n)\) and \(N_k(n) = \displaystyle\sum_{m}m^k N(m,n)\), where \(N(m,n)\) is the number of partitions of \(n\) with rank \(m\) and \(M(m,n)\) is the number of partitions of \(n\) with crank \(m\). A symmetrized version of the rank moments was studied by Andrews in 2006. In 2011 Garvan used symmetrized rank and crank moments to define a higher order \(\mathrm{spt}\) function, noting Andrews’ \(\mathrm{spt}\) function is half the difference of the second crank and rank moment. This also established an inequality between crank and rank moments.

We show this idea extends to the \(\mathrm{spt}\) functions for overpartitions, overpartitions with smallest part even, and partitions with smallest part even and without repeated odd parts. In this talk we give the necessary definitions, in particular what are the symmetrized moments, and review the steps for the ordinary \(\mathrm{spt}\) function. We then continue on with the three additional \(\mathrm{spt}\) functions. The crucial identities follow from a theorem in Garvan’s work that is an application of Bailey pairs and Bailey’s Lemma.

Ramanujan’s Congruences Modulo Powers of 5, 7, and 11 Revisited

When/Where:

March 18, 2014, at 1:55pm, in LIT 368

Abstract:

The number of partitions of \(4\) is \(p(4)=5\), namely: \(4\), \(3+1\), \(2+2\), \(2+1+1\), and \(1+1+1+1\). Ramanujan observed that \(p(5n+4)\) is divisible by \(5\) for all non-negative integers \(n\). More generally, Ramanujan discovered similar congruences modulo \(7\) and \(11\), and also for all powers of these primes. The cases \(5\) and \(7\) were proved by G.N. Watson (1938); in 1984, Frank Garvan was able to simply this proof significantly. In 1967 the case \(11\) was proved by A.O.L. Atkin; in 1983, B. Gordon presented another approach being closer to Watson’s. The talk originates from joint work with Silviu Radu; it describes a new algorithmic setting in the theory of modular functions that gives rise to a new unified frame to prove Ramanujan’s celebrated families of partition congruences.

Bressoud’s “Easy Proof of the Rogers-Ramanujan Identities” and some unexpected consequences

When/Where:

March 11, 2014, at 1:55pm, in LIT 368

Abstract:

A multiple series generalization of the Rogers-Ramanujan identities (analytic form) was given in 1974. We begin this talk with a brief discussion of that result and its proof. Then in 1983, David Bressoud published: “An easy proof of the Rogers-Ramanujan identities,” and the first half of the paper is exactly that. The paper concludes with a multiple series generalization that has unexpected consequences. We shall examine the Bressoud paper in light of the Bailey chain machinery and examine the unexpected consequences.

Congruences for the Fishburn Numbers

When/Where:

February 25, 2014, at 1:55pm, in LIT 368

Abstract:

This talk will present joint work with James Sellers. The Fishburn numbers, \(\xi(n)\), have many interpretations and combinatorial applications. For example, \(\xi(n)\) equals the number of upper triangular matrices with nonnegative integer entries and without zero rows or columns such that the sum of all entries equals \(n\). For example, when \(n = 3\) we have the following matrices:

$$\begin{pmatrix}
1 & 0 & 0\\
0 & 1 & 0\\
0 & 0 & 1
\end{pmatrix},
\begin{pmatrix}
2 & 0\\
0 & 1
\end{pmatrix},
\begin{pmatrix}
1 & 0\\
0 & 2
\end{pmatrix},
\begin{pmatrix}
1 & 1\\
0 & 1
\end{pmatrix},
\text{ and }
\begin{pmatrix}
3
\end{pmatrix}.
$$
Thus \(\xi(3) = 5\).

In this talk I shall describe many of the interpretations of \(\xi(n)\) and will prove an infinite family of congruences for \(\xi(n)\).  In particular, \(5 \mid \xi(5n+3)\).

Essentially Unique Representations by Ternary Quadratic Forms

 

When/Where:

February 4, 2014, at 1:55pm, in LIT 368

Abstract:

We plan to discuss integers which are represented in an essentially unique way by certain ternary quadratic forms. There are 794 positive definite ternary quadratic forms which are alone in their genus, and we discuss a method for determining all integers which are uniquely represented (up to automorphism) by such a form. We then illustrate this method with examples, and close with prospects for future work.