Printer-friendly versionAbstracts
Week of November 7, 2021
| Group Actions and Dynamics | Equidistribution of intersections with homogeneous subspaces |
10:15am -
Zoom
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I will discuss the following general problem in homogeneous dynamics: Let $G$ be a semisimple Lie group, $K$ a compact subgroup of $G$ and $\Gamma$ a lattice in $G$. Let $O_n$ be an equidistributing sequence of locally homogeneous subspaces of $\Gamma \backslash G/K$. What is the distribution of the intersection of $O_n$ with a fixed analytic subspace $V$?
In a joint work with Salim Tayou, we show that, under the correct hypotheses, this intersection equidistributes towards a certain $G$-invariant form on $G/K$. Determining this form leads to an interesting question about the cohomology of compact homogeneous spaces. This result has interesting applications in arithmetics and in Hodge theory, which I will mention briefly if time permits.
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| Geometry, Symmetry and Physics | Examples of Hecke eigen-functions for moduli spaces of bundles over local non-archimedian field |
4:30pm -
Zoom
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Abstract: Let X be a smooth projective curve over a finite field k, and let G be a reductive group. The unramified part of the theory of automorphic forms for the group G and the field k(X) studies functions on the k-points on the moduli space of G-bundles on X and the eigen-functions of the Hecke operators (to be reviewed in the talk!) acting there. The spectrum of the Hecke operators has continuous and discrete parts and it is described by the global Langlands conjectures (which in the case of functional fields are essentially proved by V.Lafforgue). After recalling the above notions and constructions I will discuss what happens when k is replaced by a local field. The corresponding Hecke operators were essentially defined by myself and Kazhdan about 10 years ago, but the systematic study of eigen-functions has begun only recently. It was initiated several years ago by Langlands when k is archimedian and then Etingof, Frenkel and Kazhdan formulated a very precise conjecture describing the spectrum in terms of the dual group. Contrary to the classical case only discrete spectrum is expected to exist. I will discuss what is is known in the case when k is a local non-archimedian field (joint work in progress with D.Kazhdan). Zoom link: https://yale.zoom.us/j/99305994163, contact the organizers (Gurbir Dhillon and Junliang Shen) for the passcode. |
| Geometry & Topology | Hamiltonian flows on spaces of surfaces |
4:15pm -
LOM 214
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The algebraic intersection form makes the homology of a closed surface into a symplectic vector space. A construction of Goldman endows spaces of representations of π_1 into certain Lie groups with a symplectic form remembering the intersection pairing on curves. A regular enough function on a symplectic manifold defines a Hamiltonian vector field—its symplectic gradient. The flow of such a vector field is called a Hamiltonian flow, and defines a family of symmetries of the symplectic structure. We will consider (geometrically defined) functions on (geometric) loci in character varieties and study their Hamiltonian flows. For most infinite order mapping classes, we find explicit functions on the Teichmüller space whose Hamiltonian flow at time one induces the action of that mapping class. Our main tools are Thurston’s space of measured geodesic laminations and the shearing coordinates of Bonahon and Thurston for hyperbolic metrics. |
| Algebra and Number Theory Seminar | On normalization in the integral models of Shimura varieties of Hodge type |
4:30pm -
Zoom
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Shimura varieties are moduli spaces of abelian varieties with extra structures. Over the decades, various mathematicians (e.g. Rapoport, Kottwitz, etc.) have constructed integral models of Shimura varieties. In this talk, I will discuss some motivic aspects of integral models of Hodge type constructed by Kisin (resp. Kisin-Pappas). I will talk about recent work on removing the normalization step in the construction of such integral models, which gives closed embeddings of Hodge type integral models into Siegel integral models. I will also mention an application to toroidal compactifications of such integral models. |
| Applied Mathematics | Manifold learning with non-Euclidean norms, and orbit recovery |
2:30pm -
https://yale.zoom.us/j/5137467379
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Abstract: In this talk, I will discuss a couple general mathematical frameworks that have arisen in the study of the cryo-electron microscopy problem. I will start by recalling that cryo-EM is about determining the 3D shape of a protein from many noisy tomographic images of randomly oriented particles. Motivated by this application, I will mostly focus on the use of non-Euclidean norms in manifold learning. Synthetic results show, that when applying diffusion maps to volumetric data, it can be advantageous to base affinities on the Earth Mover’s Distance rather the Euclidean norm. I will characterize the limiting differential operator that we obtain when we use non-Euclidean norms, and contrast this to the classical Laplace-Beltrami operator. Time permitting, I may also mention the problem of parameter estimation in statistics in the presence of group actions, and some unexpected connections to invariant theory. Based on joint works with several coauthors who will be named precisely during the talk. |
| Undergraduate Seminar | Putnam Seminar |
4:00pm -
LOM 214
LOM 214
LOM 214
LOM 214
LOM 214
LOM 214
LOM 214
LOM 214
LOM 214
LOM 214
LOM 214
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The Putnam seminar meets every Wednesday from 4 to 5:30 in LOM 214. As always, everyone is warmly welcomed to come to hang out, learn more cool math, and meet folks. The seminar is casual, and folks can come and go as they like. See Pat Devlin’s webpage (and/or contact him) for more information. Folks can sign up for the mailing list here: https://forms.gle/nYPx72KVJxJcgLha8 |
| Colloquium | Thresholds | 4:15pm - |
Abstract: Thresholds for increasing properties are a central concern in probabilistic combinatorics and elsewhere. (An increasing property, say F, is a superset-closed family of subsets of some [here finite] set X, and the “threshold question” for F asks, roughly, about how many random elements of X should one choose to make it likely that the resulting set lies in F? For example: about how many random edges from the complete graph on n vertices are typically required to produce a Hamiltonian cycle?) We will try to give some sense of this area and then focus on a few recent highlights. |
| Algebra and Geometry lecture series | Quantizations in characteristic p, lecture 9 |
4:00pm -
https://yale.zoom.us/j/99019019033 (password was emailed by Ivan)
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| Analysis | Additive energy of regular measures in one and higher dimensions, and the fractal uncertainty principle | 4:15pm - |
Abstract: We obtain new bounds on the additive energy of (Ahlfors-David type) regular measures in both one and higher dimensions, which implies expansion results for sums and products of the associated regular sets, as well as more general nonlinear functions of these sets. As a corollary of the higher-dimensional results we obtains some new cases of the fractal uncertainty principle in odd dimensions.This is joint work with Terence Tao.
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| Geometric Analysis and Application | Mirror symmetry for log del Pezzo surfaces | 2:00pm - |
Abstract: If X is a del Pezzo surface and D is a smooth anti-canonical divisor, we can regard the complement X\D as a non-compact Calabi-Yau surface. I will discuss a proof of a strong form of the Strominger-Yau-Zaslow mirror symmetry conjecture for these non-compact surfaces. It turns out the mirror Calabi-Yau is a rational elliptic surface (in particular, it has an elliptic fibration onto P^1) with a singular fiber which is a chain of nodal spheres. I will discuss how we can construct special Lagrangian fibrations on these manifolds, as well as moduli of complex and symplectic structures and how hyper-Kahler rotation allows us to construct an identification of these moduli spaces. This is joint work with A. Jacob and Y.-S. Lin. |