Calendar
- 02April 2, 2024
CMSA General Relativity Seminar: Linearised Second Law for Higher Curvature Gravity and Non-Minimally Coupled Vector Fields
20 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).
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.
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
Harvard-MIT Algebraic Geometry Seminar: Curves with many degree d points (Joint with the MIT number theory seminar, note the special time and location)
Joint with the MIT number theory seminar, note the special time and location
When does a nice curve $X$ over a number field $k$ have infinitely many closed points of degree $d$?
Faltings’ theorem allows us to rephrase this problem in purely algebro-geometric terms, though the resulting geometric question is far from being fully solved. Previous work gave easy to state answers to the problem for degrees $2$ (Harris-Silverman) and $3$ (Abramovich-Harris), but also uncovered exotic constructions of such curves in all degrees $d \geqslant 4$ (Debarre-Fahlaoui). I will describe recent progress on the problem, which answers the question in the large genus case. Along the way we uncover systematic explanations for the Debarre-Fahlaoui counstructions and provide a complete geometric answer for $d \leqslant 5$. The talk is based on joint work with Isabel Vogt.For more information, please see https://researchseminars.org/seminar/harvard-mit-ag-seminar