Departmental Colloquium

Increasingly, biomedical researchers need to build functional computer models from images (MRI, CT, EM, etc.). The "pipeline" for building such computer models includes image analysis (segmentation, registration, filtering), geometric modeling (surface and volume mesh generation), large-scale simulation (parallel computing, GPUs), large-scale visualization and evaluation (uncertainty, error). In my presentation, I will present research challenges and software tools for image-based biomedical modeling, simulation and visualization and discuss their application for solving important research and clinical problems in neuroscience, cardiology, and genetics.

String theory relates certain seemingly different manifolds through a relationship called mirror symmetry. Discovered about 25 years ago, this story is still very mysterious from a mathematical point of view. Despite the name, mirror symmetry is not entirely symmetric -- several distinct spaces can be mirrors to a given one. When this happens it is expected that certain invariants of these "double mirrors" match up. For a long time the only known examples of double mirrors arose through a simple construction called a flop, which led to the conjecture that this would be a general phenomenon. In joint work with Lev Borisov we constructed the first counterexample to this, which I shall present. Explicitly, I shall construct two Calabi-Yau threefolds which are not related by flops, but are derived equivalent, and therefore are expected to arise through a double mirror construction. The talk will be accessible to a wide audience, in particular to graduate students. There will even be several pictures!

Self-organized dynamics is driven by "rules of engagement", which describe how each agent interacts with its neighbors. They consist of long-term attraction, mid-range alignment and short-range repulsion. Many self-propelled models are driven by the balance between these three forces, which yield emerging structures of interest. Examples range from consensus of voters and traffic flows to the formation of flocks of birds or school of fish, tumor growth etc. We introduce a new particle-based model, driven by self-alignment, which addresses several drawbacks of existing models for self-organized dynamics. The model is independent of the number of agents: only their geometry in phase space is involved. We will explain the emerging flocking behavior of the proposed model in the presence of non-symmetric interactions which decay sufficiently slow, and discuss the difficulties of tracing graph connectivity otherwise. The methodology is based on the new notion of active sets, which carries over from particle to kinetic and hydrodynamic descriptions, and we discuss the unconditional flocking at the level of hydrodynamic description.

All mathematicians have heard that the Riemann Hypothesis is a significant open problem, but why is it such a big deal? One can name plenty of older unsolved problems in number theory that get far less publicity. We care about the Riemann Hypothesis because it is connected to a large number of significant questions. I will discuss the history, scope, and range of consequences of the Riemann Hypothesis, with the aim of convincing mathematicians outside number theory that it deserves to be counted as one of the most important unsolved problems in mathematics.