Mathematics
http://events.berkeley.edu/index.php/calendar/sn/math.html
Upcoming EventsTopology Seminar (Introductory Talk), Apr 25
http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=117204&date=2018-04-25
Polyfold theory, as developed by Hofer, Wysocki, and Zehnder, is a relatively new approach to resolving transversality issues that arise in the study of J-holomorphic curves in symplectic geometry. In this talk I explain the polyfold theoretic approach to defining the Gromov-Witten invariants for all closed symplectic manifolds.http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=117204&date=2018-04-25The weak Pinsker property, Apr 25
http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=117180&date=2018-04-25
This talk is about the structure theory of measure-preserving systems: transformations of a finite measure space that preserve the measure. Many important examples arise from stationary processes in probability, and simplest among these are the i.i.d. processes. In ergodic theory, i.i.d. processes are called Bernoulli shifts. Some of the main results of ergodic theory concern an invariant of systems called their entropy, which turns out to be intimately related to the existence of `structure preserving' maps from a general system to Bernoulli shifts.<br />
<br />
I will give an overview of this area and its history, ending with a recent advance in this direction. A measure-preserving system has the weak Pinsker property if it can be split, in a natural sense, into a direct product of a Bernoulli shift and a system of arbitrarily low entropy. The recent result is that all ergodic measure-preserving systems have this property. Its proof depends on a new theorem in discrete probability: a probability measure on a finite product space such as A^n can be decomposed as a mixture of a controlled number of other measures, most of them exhibiting a strong `concentration' property. I will sketch the connection between these results and the proof of the latter, to the extent that time allows.<br />
<br />
I will assume a basic graduate-level background in real analysis and measure-theoretic probability, but little beyond that.http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=117180&date=2018-04-25Applied Math Seminar, Apr 25
http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=117189&date=2018-04-25
The numerical simulation of multiphysics problems is significant in many engineering and scientific applications, e.g., aircraft flutter in transonic flows, biomedical flows in heart and blood vessels, mixing and chemically reacting flows, reactor fuel performance, turbomachinery and so on. These problems are generally highly nonlinear, feature multiple scales and strong coupling effects, and require heterogeneous discretizations for the various physics subsystems. Due to dramatic improvement of single-physics solvers during the last two decades, partitioned procedures for multiphysics system become dominant, which exploit single-physics software components and facilitate mathematical modeling. However, these schemes are often low-order accurate (second order accuracy) and suffer from lack of stability (subiteration is needed).<br />
<br />
To relieve these issues, we introduce a general framework for constructing high-order, linearly stable, partitioned solvers for multiphysics problems. The coupled ODE system of the multiphysics problems is taken as a monolithic system and discretized using an implicit-explicit Runge-Kutta (IMEX-RK) discretization based the concept of a predictor for the coupling term. We propose four coupling predictors inspired by basic ideas of weak/strong coupling effects, Jacobi method, and Gauss-Seidel method, which enable the monolithic system to be solved in a partitioned manner, i.e., subsystem-by-subsystem, and preserve the design order of accuracy of the monolithic scheme. We also analyze the stability on a coupled, linear model problem and show that one of the partitioned solvers achieves unconditional linear stability, while the others are unconditionally stable only for certain values of the coupling strength. Furthermore, a fully-discrete adjoint solver derived from our partitioned solvers, is applied for time-dependent PDE constraint optimization. (Joint work with Per-Olof Persson and Matthew J. Zahr)http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=117189&date=2018-04-25Topology Seminar (Main Talk), Apr 25
http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=117205&date=2018-04-25
In 1994 Kontsevich and Manin stated the Gromov-Witten axioms, given as a list of formal relations between the Gromov-Witten invariants. In this talk I prove several of the Gromov-Witten axioms for curves of arbitrary genus for all closed symplectic manifolds.http://events.berkeley.edu/index.php/calendar/sn/math.html?event_ID=117205&date=2018-04-25