الجبر

Abstract: There is a natural bijection between basic tilting modules for a linearly oriented quiver of type A_n and triangulations of an (n+2)-gon. I will recall this construction and then discuss a higher-dimensional generalization of it, in which the path algebra is replaced by the d-th higher Auslander algebra of linearly oriented A_n (in the sense of Iyama), and the polygon is replaced by a cyclic polytope of dimension 2d+2. The triangulations of the cyclic polytope then correspond bijectively to tilting objects contained in the (d+1)-cluster tilting subcategory of the module category. The structure of this collection of tilting modules (or equivalently of the collection of triangulations) provides a higher-dimensional analogue of cluster combinatorics, which I will be discussing further in the combinatorics seminar on September 24. The work I will be presenting is joint with Steffen Oppermann, and is contained in arXiv:1001.5437.

Abstract: In work with Daniel Erman and Dustin Cartwright, we describe the defining ideal of the r^th secant variety of P²xPⁿ embedded by O(1,2), for arbitrary n and r ≤ 5. We also present the Schur module decomposition of the space of generators of each such ideal. Our main results are based a more general construction for producing explicit matrix equations that vanish on secant varieties of products of projective spaces. This extends previous work of Strassen and Ottaviani. Our matrix equations are simple to understand, and I will present them at the level of basic linear algebra. Then I will show the invariant presentation of the equations. I will explain why these equations are always necessary defining equations of secant varieties. I will describe our results which say when these equations are also sufficient to define the ideal. In addition, I will give several examples to illustrate the limits of these equations. Finally, I will mention applications of these results to signal processing.

Abstract: Modularity lifting theorems have played an important role in number theory over the last 15 years.  The first such theorem, proved by Wiles, was deduced from a stronger result, an "R=T theorem." Recently, Kisin proved a more general lifting theorem without establishing R=T.  I will outline how one can prove R=T in some of these cases.  The key technical step is establishing that the relevant local deformation rings are Cohen-Macaulay.  This is acheived by producing an appropriate resolution of singularities. In the first part of the talk, I will explain what a modularity lifting theorem is, what an R=T theorem is and why they are important.  In the second part, I will explain the general ideas of the proof of the new result.

Abstract: In this talk I'll report on some recent developments in the study of Galois module structures of Kummer and Artin-Schreier theories. In particular we'll discuss some new lower bounds on realization multiplicities for certain p-groups, and we'll discuss some current work on the structure of completed p-power classes.

Abstract:Sklyanin algebras play an important role in the study of physical phenomenon. We will first review techniques of Artin-Tate-van den Bergh (ATV) that describe the ring-theoretic and homological behavior of these structures. In particular, we highlight the significance of twisted homogeneous coordinate rings. The focus of the talk is to introduce a generalized twisted homogeneous coordinate ring P associated to a degenerate version of the three-dimensional Sklyanin algebra. The surprising geometry of these algebras yields an analogue to a result of ATV; namely that P is a factor of the corresponding degenerate Sklyanin algebra.

Abstract: We use tilting theory and birational geometry to study moduli spaces of quiver representations. From certain "representable" functor, we construct a birational transformation from the moduli space of representations of one quiver to another new quiver with one vertex less. The dimension vector and the stability for the new moduli are determined functorially. We introduce several relative notion of stability to study such a birational transformation. Some of its structure is understood but a more round picture may lead to an algorithm similar to the minimal model program. Moreover, we deduce a formular comparing the induced ample divisors for the two moduli. We will illustrate this theory by some surface examples. If time permit, we may show some higher dimensional examples.