Calendar
- 01April 1, 2024
**CANCELED** CMSA Colloquium: Errors and Correction in Cumulative Knowledge **CANCELED**
CMSA, 20 Garden St, G1020 Garden Street, Cambridge, MA 02138**CANCELED**
Societal accumulation of knowledge is a complex, and arguably error-prone, process. The correctness of new units of knowledge depends not only on the correctness of the new reasoning, but also on the correctness of old units that the new one builds on. If left unchecked, errors could completely ruin the validity of most of this knowledge so there must some error-correcting going on. What are the error-corrections processes employed in nature and how effective are they? In this talk, we describe our attempts to model such phenomena using probablistic models – we describe models for growth of cumulative knowledge, emergence of errors and methods to check for errors and eliminate them. We then analyze in this compound model, when effects of errors may survive, and when they are totally eliminated.
The central discovery in our work is the following optimistic statement: If we do checking correctly (most of the time) investing just a constant factor (<1) of our effort in checking (and saving the remaining constant factor towards deriving new units of knowledge), then effects of errors can be kept in check. Notably the amount of effort expended on checking does not scale with the volume of total knowledge or the depth of dependencies in the new units of knowledge, either of which would be overwhelming.
Based on the papers:
Is this correct? Let’s check!
Omri Ben-Eliezer, Dan Mikulincer, Elchanan Mossel, Madhu Sudan
arXiv:2211.12301Errors are Robustly Tamed in Cumulative Knowledge Processes
Anna Brandenberger, Cassandra Marcussen, Elchanan Mossel, Madhu Sudan
arXiv:2309.05638 - 02April 2, 2024
- 02April 2, 2024
Probability Seminar: The planar Coulomb gas on a Jordan curve
The eigenvalues of a uniformly distributed unitary matrix have the physical interpretation of a system of particles subject to a logarithmic pair interaction, restricted to lie on the unit circle and at inverse temperature 2. In this talk, I will present a more general model in which the unit circle is replaced by a sufficiently regular Jordan curve, at any positive temperature. I will show how to obtain the asymptotic partition function and Laplace transform of a linear statistic. These can be expressed using either the exterior conformal mapping of the curve or its associated Grunsky operator. Based on joint work with Kurt Johansson.
- 02April 2, 2024
CMSA General Relativity Seminar: Linearised Second Law for Higher Curvature Gravity and Non-Minimally Coupled Vector Fields
CMSA, 20 Garden St, G1020 Garden Street, Cambridge, MA 02138Expanding the work of arXiv:1504.08040, we show that black holes obey a second law for linear perturbations to bifurcate Killing horizons, in any covariant higher curvature gravity coupled to scalar and vector fields. The vector fields do not need to be gauged, and (like the scalars) can have arbitrary non-minimal couplings to the metric. The increasing entropy has a natural expression in covariant phase space language, which makes it manifestly invariant under JKM ambiguities. An explicit entropy formula is given for f(Riemann) gravity coupled to vectors, where at most one derivative acts on each vector. Besides the previously known curvature terms, there are three extra terms involving differentiating the Lagrangian by the symmetric vector derivative (which therefore vanish for gauge fields).
- 02April 2, 2024
Harvard-MIT Algebraic Geometry Seminar: Webs and Schubert calculus for Springer fibers
Classical Schubert calculus analyzes the geometry of the flag variety, namely the space of nested subspaces $V_1 \subseteq V_2 \subseteq \cdots \subseteq \mathbb{C}^n$, asking enumerative questions about intersections of linear spaces that turn out to be equivalent to deep problems in combinatorics and representation theory. In this talk, we’ll describe some recent results in the Schubert calculus of Springer fibers. Given a nilpotent linear operator $X$, the Springer fiber of $X$ is the subvariety of flags that are fixed by $X$ in the sense that $XV_i \subseteq V_i$ for all $i$. The top-dimensional cohomology of Springer fibers admits a representation of the symmetric group first discovered by Tonny Springer as the seminal example of a geometric representation. Where classical Schubert calculus describes geometry governed by permutations, that of Springer fibers incorporates the combinatorics both of permutations and of partitions. We’ll describe new results about this geometry in more detail, including evidence that from a geometric and topological perspective, the best combinatorial model for Springer fibers comes from representation-theoretic objects called webs.
For more information, please see https://researchseminars.org/seminar/harvard-mit-ag-seminar