Wednesday, Apr 25 1:30 pm - 2:30 pm
Title: The j-multiplicity of edge ideals Abstract: The notion of j-multiplicity extends the classical Hilbert-Samuel multiplicity to arbitrary ideals. In this talk, we will describe connections between the j-multiplicity of certain ideals arising from graphs and the underlying combinatorial structures. This leads to formulas for the j-multiplicity of such ideals in terms of the graph-theoretic data. This is joint work with A. Alilooee and I. Soprunov.
Wednesday, Apr 25 3:25 pm - 4:25 pm
Rigidity and F-isocrystals.
Abstract: We explain that rigid connections yields F-overconvergent isocrystals (in the projective case) and discuss a few points concerning the p-curvature conjecture. (Joint work with Michael Groechenig)
Thursday, Apr 26 1:30 pm - 2:20 pm
Q-systems: Discrete integrability and cluster algebrasI’ll introduce a family of remarkable recursion relations, originally found in the context of the Bethe ansatz of generalized Heisenberg modules and quantum groups. I will reconsider this system as a discrete integrable system in its own right, and show some of the remarkable properties it and its quantization, via a cluster algebra formulation, exhibit. This will be a basic talk with no prior knowledge of any of the above mentioned topics assumed, with the purpose of introducing some of these structures and showing why they might be useful in mathematical physics.
Thursday, Apr 26 1:30 pm - 2:20 pm
A glimpse of the history and some problems in the Langlands program for covering groupsAbstract: The long-lasting significance of the Langlands program for linear reductive groups and its deep connection with number theory, arithmetic of variety etc are well-known. On the other hand, for finite-degree central coverings of a linear reductive group, the analogous framework has not been fully established despite many important works for special families of coverings, e.g,. the classical double cover of the symplectic group. In this expository talk, we will give a brief overview of the analogous Langlands program for covering groups. We highlight several concrete questions which pertain to a Langlands-Shahidi theory (yet to be developed) for covering groups.
Thursday, Apr 26 3:30 pm - 4:30 pm
Some recent work on conformal biharmonic maps
Abstract: Biharmonic maps are maps between Riemannian manifolds that are critical points of the bienergy functional, they are solutions of a system of 4thorder PDEs. Biharmonic maps include harmonic maps, biharmonic functions and biharmonic submenisolds as special examples. The talk will be focused on biharmonic conformal immersions, biharmonic conformal submersions and their relations to the maps between manifolds that preserve solutions of bi-Laplace equations, and biharmonic conformal maps between manifolds of the same dimension and their links to Yamabe-type equations.
Friday, Apr 27 10:30 am - 11:20 am
Title of Talk: On combinatorial invariants of Minkowski spaces
Special Colloquium, Professor Erin Carson, Courant Institute of Mathematical Sciences, New York University, REC 113
Friday, Apr 27 3:30 pm - 4:20 pm
Sparse Linear Algebra in the Exascale Era
Abstract: Sparse linear algebra problems, typically solved using iterative methods, are ubiquitous throughout scientific and data analysis applications and are often the most expensive computations in large-scale codes due to the high cost of data movement. Approaches to improving the performance of iterative methods typically involve modifying or restructuring the algorithm to reduce or hide this cost. Such modifications can, however, result in drastically different behavior in terms of convergence rate and accuracy. A clear, thorough understanding of how inexact computations, due to either finite precision error or intentional approximation, affect numerical behavior is thus imperative in balancing the tradeoffs between accuracy, convergence rate, and performance in practical settings.
In this talk, we focus on two general classes of iterative methods for solving linear systems: Krylov subspace methods and iterative refinement. We present bounds on the attainable accuracy and convergence rate in finite precision s-step and pipelined Krylov subspace methods, two popular variants designed for high performance. For s-step methods, we demonstrate that our bounds on attainable accuracy can lead to adaptive approaches that both achieve efficient parallel performance and ensure that a user-specified accuracy is attained. We present two such adaptive approaches, including a residual replacement scheme and a variable s-step technique in which the parameter s is chosen dynamically throughout the iterations. Motivated by the recent trend of multiprecision capabilities in hardware, we present new forward and backward error bounds for a general iterative refinement scheme using three precisions. The analysis suggests that on architectures where half precision is implemented efficiently, it is possible to solve certain linear systems up to twice as fast and to greater accuracy.
As we push toward exascale level computing and beyond, designing efficient, accurate algorithms for emerging architectures and applications is of utmost importance. We discuss extensions to machine learning and data analysis applications, the development of numerical autotuning tools, and the broader challenge of understanding what increasingly large problem sizes will mean for finite precision computation both in theory and practice.
Research Area: High Performance Computing, Numerical Linear Algebra and Parallel Algorithms
Automorphic Forms and Representation Theory Seminar, Prof. John Bergdall, Michigan State University, UNIV 317
Thursday, May 3 1:30 pm - 2:20 pm