Calendar
- 01March 1, 2020No events
- 02March 2, 2020
Enumerative invariants and exponential networks
20 Garden Street, Cambridge, MA 02138I will define and review the basics of exponential networks associated to CY 3-folds described by conic bundles. I will focus mostly on the mathematical aspects and general ideas behind this construction as well as its conjectural connection with generalized Donaldson–Thomas invariants. This is based on joint work with S. Banerjee and P. Longhi.
- 03March 3, 2020
CMSA Special Quantum Matter/Quantum Math Seminar: Cutting and pasting 4-manifolds
20 Garden Street, Cambridge, MA 02138We will discuss techniques topologists use for understanding 4-manifolds obtained by cut-and-paste constructions. The hope is that these techniques may be useful for understanding 4-dimensional topological field theories.Equivariant Degenerations of Plane Curve Orbits
1 Oxford Street, Cambridge, MA 02138 USAIn a series of papers, Aluffi and Faber computed the degree of the GL3 orbit closure of an arbitrary plane curve. We attempt to generalize this to the equivariant setting by studying how these orbits degenerate, yielding a fairly complete picture in the case of plane quartics. As an enumerative consequence, we will see that a general genus 3 curve appears 510720 times as a 2-plane section of a general quartic threefold. We also hope to survey the relevant literature and will only assume the basics of intersection theory. This is joint work with M. Lee and A. Patel.
On quantum distributional symmetries for *-random variables
17 Oxford Street, Cambridge, MA 02138 USAIn this talk, we briefly review the distributional symmetries for *-random variables, which are defined by coactions of corepresentations of quantum groups. We classify all de Finetti type theorems for classical independence and free independence by studying vanishing conditions on the classical and free cumulants. Examples for our de Finetti type theorems and approximation results in the spirit of Diaconis and Freedman are also provided.
Complete Kahler Ricci flow with unbounded curvature and applications
1 Oxford Street, Cambridge, MA 02138 USAIn this talk, we will discuss the construction of Kahler Ricci flow on complete Kahler manifolds with unbounded curvature. As a corollary, we will discuss the application related to Yau’s
uniformization problem and the regularity of Gromov-Hausdorff’s limit. This is joint work with L.F. Tam.— Organized by Professor Shing-Tung Yau
- 04March 4, 2020
Non-random behaviour in sums of modular symbols
1 Oxford Street, Cambridge, MA 02138 USAWe give explicit expressions for the Fourier coefficients of Eisenstein series twisted by Dirichlet characters and modular symbols on $\Gamma_0(N)$ in the case where $N$ is prime and equal to the conductor
of the Dirichlet character. We obtain these expressions by computing the spectral decomposition of automorphic functions closely related to these Eisenstein series. As an application, we then evaluate certain sums of modular symbols in a way which parallels past work of Goldfeld, O’Sullivan, Petridis, and Risager. In one case we find less cancellation in this sum than would be predicted by the common phenomenon of “square root cancellation”, while in another case we find more cancellation.**CANCELED** Informal Geometry & Dynamics Seminar: Pseudo-Anosov mapping classes with large dilatation
I’ll talk about some subclasses of pseudo-Anosov mapping classes whose dilatations are bounded away from 1.Well-ordering principles in Proof theory and Reverse Mathematics
There are several familiar theories of reverse mathematics that can be characterized by well-ordering principles of the form(*) “if X is well ordered then f(X) is well ordered”,where f is a standard proof theoretic function from ordinals to ordinals (such f’s are always dilators). Some of these equivalences have been obtained by recursion-theoretic and combinatorial methods. They (and many more) can also be shown by proof-theoretic methods, employing search trees and cut elimination theorems in infinitary logic with ordinal bounds. One could perhaps generalize and say that every cut elimination theorem in ordinal-theoretic proof theory encapsulates a theorem of this type.One aim of the talk is to present a general methodology underlying these results that enables one to construct omega-models of particular theories from (*).As (*) is of complexity $\Pi^1_2$ such a principle cannot characterize stronger comprehensions at the level of $\Pi^1_1$-comprehension. This requires a higher order version of (*) that employs ideas from ordinal representation systems with collapsing functions used in impredicative proof theory. The simplest one is the Bachmann construction. Relativizing the latter construction to any dilator f and asserting that this always yields a well-ordering turns out to be equivalent to $\Pi^1_1$-comprehension. This result has been conjectured more than 10 years ago, but its proof has only been worked out by Anton Freund in recent yearsCMSA Colloquium: Derandomizing Algorithms via Spectral Graph Theory
20 Garden Street, Cambridge, MA 02138Randomization is a powerful tool for algorithms; it is often easier to design efficient algorithms if we allow the algorithms to “toss coins” and output a correct answer with high probability. However, a longstanding conjecture in theoretical computer science is that every randomized algorithm can be efficiently “derandomized” — converted into a deterministic algorithm (which always outputs the correct answer) with only a polynomial increase in running time and only a constant-factor increase in space (i.e. memory usage).In this talk, I will describe an approach to proving the space (as opposed to time) version of this conjecture via spectral graph theory. Specifically, I will explain how randomized space-bounded algorithms are described by random walks on directed graphs, and techniques in algorithmic spectral graph theory (e.g. solving Laplacian systems) have yielded deterministic space-efficient algorithms for approximating the behavior of such random walks on undirected graphs and Eulerian directed graphs (where every vertex has the same in-degree as out-degree). If these algorithms can be extended to general directed graphs, then the aforementioned conjecture about derandomizing space-efficient algorithms will be resolved.
Joint works with Jack Murtagh, Omer Reingold, Aaron Sidford, AmirMadhi Ahmadinejad, Jon Kelner, and John Peebles.
- 05March 5, 2020
**CANCELED** From automorphisms to graphs complexes
1 Oxford Street, Cambridge, MA 02138 USANo additional detail for this event.
- 06March 6, 2020
**CANCELED** A scale-critical trapped surface formation criterion for the Einstein-Maxwell system
Few notions within the realm of mathematical physics succeed in capturing the imagination and inspiring awe as well as that of a black hole. First encountered in the Schwarzschild solution, discovered a few months after the presentation of the Field Equations of General Relativity at the Prussian Academy of Sciences, the black hole as a mathematical phenomenon accompanies and prominently features within the history of General Relativity since its inception. In this talk we will lay out a brief history of the question of dynamical black hole formation in General Relativity and discuss a recent result, in collaboration with Xinliang An, on a scale-critical trapped surface formation criterion for the Einstein-Maxwell system.A topological approach to convexity in complex surfaces
1 Oxford Street, Cambridge, MA 02138 USAWe will discuss the classical notion of J-convexity of subsets of complex manifolds, and the closely related notion of Stein manifolds. The theory is particularly subtle in complex dimension 2. Surprisingly, progress can be made using topological 4-manifold theory. Every tame CW 2-complex topologically embedded in a complex surface can be perturbed so that it becomes J-convex in the sense of being a nested intersection of Stein open subsets. These Stein neighborhoods are all topologically equivalent to each other, but can be very different when viewed in the smooth category. As applications, we obtain uncountable families of distinct smoothings of R^4 admitting convex or concave holomorphic structures. We can also generalize the notion of J-convex hypersurfaces to the topological category. The resulting topological embeddings behave like their smooth counterparts, but are much more common.
Future schedule is found here: https://scholar.harvard.edu/gerig/seminar
- 07March 7, 2020No events