| George Andrews (PSU) | Title: | Partition Analysis and the Search for Modular Forms |
Abstract: | This talk will introduce P.A. MacMahon's method of partition analysis. We will the discuss some of its applications including our most recent discovery of certain partitions related to directed graphs that have modular forms as generating functions and consequently have a variety of interesting related arithmetical problems and prospects. This work is joint with Peter Paule and Axel Riese. |
| Ae Ja Yee (PSU) | Title: | (q,t)-binomial coefficients |
Abstract: | Motivated by the invariant theory of GL_n(q) on the polynomial ring in n variables over a finite field, Dennis Stanton has defined a (q,t)-binomial coefficient. In this talk, we will introduce a combinatorial interpretation for the (q,t)-binomial coefficient as a generating function for partitions, and discuss some properties and applications. This is joint work with Dennis Stanton. |
| Don Zagier (MPI-Bonn and College de France) | Title: | New Aspects of Modular Forms I: Modular forms and their derivatives Part of the Distinguished Visiting Professor Lecture Series |
Abstract: | After reviewing basic definitions, we will discuss the derivatives of modular forms (which are not quite modular but something nearly as good called "quasimodular") and some of their applications, e.g. to a problem concerning periodic functions and to a problem coming from percolation theory. |
| Don Zagier (MPI-Bonn and College de France) | Title: | New Aspects of Modular Forms III: Quadratic functions and modular forms Part of the Distinguished Visiting Professor Lecture Series |
Abstract: | This lecture, perhaps the most elementary of the series (and independent of its predecessors), shows how modular forms arise naturally from - and provide the unexpected answer to - a simple question concerning sums of quadratic polynomials. |
| Don Zagier (MPI-Bonn and College de France) | Title: | New Aspects of Modular Forms V: Mock theta functions and theta series of indefinite quadratic forms Part of the Distinguished Visiting Professor Lecture Series |
Abstract: | Theta series are one of the most important sources of modular forms and have many applications in pure and applied mathematics (sphere packing, coding theory,...). They are associated to positive definite quadratic forms. Trying to extend the theory to include indefinite forms leads to new arithmetic questions and connections with the mock theta functions of Ramanujan treated in the colloquium lecture. DUE TO SCHEDULE CHANGE THIS SEMINAR WILL EXTEND FROM 11AM TO 1PM. It will also include lecture 6 of this series, "Quantum modular forms." |
| James Sellers (PSU) | Title: | On Sloane's Generalization of Non-Squashing Stacks of Boxes |
Abstract: | Recently, N.J.A. Sloane and I solved a certain box stacking problem related to non-squashing partitions. At one time, Sloane also hinted at a generalized box stacking problem which is closely related to generalized non-squashing partitions. George Andrews and I recently solved this generalized box stacking problem by obtaining a generating function for the number of such stacks. In this talk, I will prove our result and discuss partition functions which arise via this generating function. The proof utilizes partition analysis (as described by George in his Partitions Seminar talk given a few weeks ago). After proving our result, I will discuss some easily obtained partition congruences satisfied by our functions. |
| Krishnaswami Alladi (Univ. of Florida) | Title: | Some new observations on partitions into distinct parts not congruent to i(mod 4) |
Abstract: | Let Q_i(n) denote the number of partitions of n into distinct parts not congruent to i (mod 4). The well known (little) theorem of Gollnitz states that for i=1 or 3, Q_i(n) is equal to the number of partitions of n into parts differing by $\ge 2$, where the inequality is strict if a part is odd, and the least part is $\ge i$. A similar result for Q_0(n) and Q_2(n) is not known. We will provide a unified approach to all four functions Q_i(n), i=1,2,3,4 by connecting them with partitions into parts differing by $\ge 4$ but with weights attached. These weights are powers of 2. From this point of view, Q_0(n) and Q_2(n) turn out to be more interesting than their more well known counterparts Q_1(n) and Q_3(n). Applications include a new interpretation for Jacobi's triple product identity, congruences modulo powers of 2 for Q_2(n), and combinatorial derivation of some modular identities. |
| Krishnaswami Alladi (Univ. of Florida) | Title: | Gollnitz-Gordon partitions with weights and parity conditions |
Abstract: | Let Q_i(n) denote the number of partitions of n into distinct parts not congruent to i (mod 4). By a Gollnitz-Gordon partition we mean one whose parts differ by at least 2 and where the inequality is strict if a part is even. Here we show that Q_0(n) and Q_2(n) are equal to weighted counts of Gollnitz-Gordon partitions with certain parity conditions attached. This is joint work with Alexander Berkovich. These complement results of Gollnitz on Q_1(n) and Q_3(n) and of the speaker on partitions into parts differing by $\ge 4$ alluded to in the first lecture. Also these results are obtained as a combinatorial consequence of a certain special case of a new infinite hierarchy of q-hypergeometric identities. Although this talk is a continuation of Lecture 1, it will be self contained. |
| Sol Friedberg (Boston College) | Title: | Multiple Dirichlet Series |
Abstract: | In this talk I will survey some recent research on multiple Dirichlet series. Multiple Dirichlet series arise in the study of families of L-functions, and are useful in establishing non-vanishing and mean value results. They also arise by combining the combinatorics of root systems with the properties of n-th order Gauss sums. |
| Cilanne Boulet (Cornell) | Title: | A new combinatorial Rogers-Ramanujan proof |
Abstract: | I will give a combinatorial proof of a generalization of the first Rogers-Ramanujan identity by using two new bijections on partitions with successive Durfee squares. These bijections are related to a new generalization of Dyson's notion of rank. |
| speaker | Title: | TBA |
Abstract: |
| November 22 | The University follows a Friday schedule on this date and thus alters regular attendees' schedules, so there will not be a Combinatorics/Partitions Seminar this day. |
| Alex Borisov (PSU) | Title: | Lattice points in rational polygons and cones |
Abstract: | I will survey some known results on classification of lattice polygons and cones with few lattice points inside them. While belonging to the convex discrete geometry, these objects also naturally appear in algebraic geometry, in the theory of toric varieties. I will also outline some connections to other areas of mathematics. |
| Mike Rowell (PSU) | Title: | Combinatorial Methods Applied to q-series |
Abstract: | I will give a combinatorial proof of a special case of Watson's transformation formula by extending Andrews' previous work on the Rogers-Fine identity. I will also give a new combinatorial proof of the Lebesgue identity. |