| Date: | January 25, 2024 |
| Time: | 3:00pm |
| Location: | BLOC 628 |
| Speaker: | Runjie Hu, Stony Brook University |
| Title: | Galois symmetry and manifolds |
| Abstract: | How to understand the Galois group of Q-bar over Q? It has a natural action on nonsingular complex varieties defined over finite extensions of Q. The action preserves the homotopy type (in the finite sense) but permutes the underlying manifold structures. In 1970, Sullivan proposed that there should be an abelianized Galois symmetry on higher dimensional simply-connected TOP manifolds from the topological construction and claimed that it is compatible with the Galois symmetry on varieties. We complete his proposals and still work on a geometric interpretation of this purely algebraic topological construction via branched coverings. The ongoing work in higher dimensions also agrees with Grothendieck's discussion of dessin d'enfants on Riemann surfaces in the 1980's. |
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| Date: | February 1, 2024 |
| Time: | 3:00pm |
| Location: | BLOC 628 |
| Speaker: | Hao Zhuang, Washington University in St. Louis |
| Title: | Invariant Morse-Bott-Smale complex, the Witten deformation and Lie groupoids |
| Abstract: | In this talk, we will first introduce an invariant Morse-Bott-Smale chain complex for closed $T^l$-manifolds with a special type of $T^l$-invariant Morse-Bott functions. Then, we will establish a quasi-isomorphism between the invariant Morse-Bott-Smale complex and the Witten instanton complex. Finally, if time permits, we will explain an ongoing project, which is a generalization of Mohsen’s Witten deformation via Lie groupoids to our $T^l$-invariant Morse-Bott functions. |
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| Date: | February 8, 2024 |
| Time: | 3:00pm |
| Location: | ZOOM |
| Speaker: | Zhicheng Han, University of Göttingen |
| Title: | Spectra of Lie groups and application to L^2-invariants |
| Abstract: | In this talk, I will explore the Laplace operator and Dirac operator on semisimple Lie groups. While the parallel problem on symmetric spaces has been well-studied in the last century, the corresponding problem is much less understood in general homogeneous spaces. We will examine the obstacles in extending existing techniques and discuss how some of them can be resolved in the case of group manifolds. Towards the end, we will see how the spectra data shall aid in computing certain topological L^2-invariants.
ZOOM number: 99377691303 |
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| Date: | March 7, 2024 |
| Time: | 3:00pm |
| Location: | BLOC 628 |
| Speaker: | Jingwen Chen, University of Pennsylvania |
| Title: | Mean curvature flow with multiplicity $2$ convergence |
| Abstract: | Mean curvature flow (MCF) has been widely studied in recent decades, and higher multiplicity convergence is an important topic in the study of MCF. In this talk, we present two examples of immortal MCF in $\mathbb{R}^3$ and $S^n \times [-1,1]$, which converge to a plane and a sphere $S^n$ with multiplicity $2$, respectively. Additionally, we will compare our example with some recent developments on the multiplicity one conjecture and the min-max theory. This is joint work with Ao Sun.
The talk is in person and also broadcast at https://tamu.zoom.us/s/94046447051. |
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| Date: | March 28, 2024 |
| Time: | 3:00pm |
| Location: | BLOC 628 |
| Speaker: | Liang Guo, East China Normal University |
| Title: | Hilbert-Hadamard spaces and the equivariant coarse Novikov conjecture |
| Abstract: | The equivariant coarse Novikov conjecture synthesizes all the Novikov-type conjectures, including the strong Novikov conjecture for groups and the coarse Novikov conjecture for metric spaces. In a recent work of Sherry Gong, Jianchao Wu, and Guoliang Yu, a notion of Hilbert-Hadamard space is introduced to study the Novikov conjecture for specific groups. To generalize their idea to the equivariant coarse Novikov conjecture, in this talk, we will study the equivariant coarse Novikov conjecture for a dynamic system which admits an equivariant coarse embedding into an admissible Hilbert-Hadamard space. This is joint work with Qin Wang, Jianchao Wu, and Guoliang Yu. |
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| Date: | March 29, 2024 |
| Time: | 10:00am |
| Location: | BLOC 306 |
| Speaker: | Zhengwei Liu , Tsinghua University |
| Title: | Alterfold Topological Quantum Field Theory |
| Abstract: | We will introduce the 3-alterfold Topological Quantum Field Theory (TQFT). It is a 3D TQFT with space-time boundary, which encodes Jones' theory of planar algebras as a local theory on the 2D boundary. Both Turaev-Viro (TV) TQFT and the Reshetikhin-Turaev (RT) TQFT can be naturally embedded in the alterflold TQFT through blow-up procedures. We provide 3D topologization of various key concepts, such as the Drinfeld center, connections, Frobenius-Schur indicators, etc. Many remarkable results become apparent in this approach, including the equivalence between TV TQFT and RT TQFT. This is recent work joint with Shuang Ming, Yilong Wang and Jinsong Wu, see arXiv:2307.12284 and arXiv:2312.06477.
For lectures in the next week, we will introduce the alterfold theory in all dimensions and construct infinite TQFT. We will discuss its connections with operator algebras, topological orders, higher categories, etc. We propose an approach to Poincare conjecture through renormalizations.
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| Date: | March 29, 2024 |
| Time: | 11:10am |
| Location: | BLOC 306 |
| Speaker: | Fan Lu , Tsinghua University |
| Title: | Classification of exchange relation planar algebras of rank 5 |
| Abstract: | Exchange relation planar algebras are natural generalizations of Kac algebras from skein theoretical point of view. We show that its classification is essentially solving a system of algebraic equations, but too complicated to solve directly. Then we introduce a key concept, the type of the fusion rule, which completely detects exchange relations as forest types. According to types, the system of equations reduces to exponentially many subsystems, which are solvable individually. In addition, we propose new analytic criteria to rule out most types from being subfactor planar algebras. Eventually, we are able to classify exchange relation planar algebras of rank 5. This method recovers the previous classification up to rank 4 of Bisch, Jones, and Liu with quick proofs. This is joint work with Zhengwei Liu.
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| Date: | March 29, 2024 |
| Time: | 2:00pm |
| Location: | BLOC 306 |
| Speaker: | Zishuo Zhao, Tsinghua University |
| Title: | Relative entropy between bimodule quantum channels |
| Abstract: | We propose a notion of relative entropy between bimodule quantum channels on finite von Neumann algebras, generalizing the remarkable Pimsner-Popa entropy for subfactors. We will discuss various inequalities of this relative entropy. In particular, the relative entropy of bimodule quantum channels is bounded by the relative entropy of their Fourier multipliers, which is a higher analogue of relative entropy of states. The equality holds if the inclusion of von Neumann algebras admits a downward Jones basic construction. |
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| Date: | April 4, 2024 |
| Time: | 3:00pm |
| Location: | BLOC 628 |
| Speaker: | Zhengwei Liu , Tsinghua University |
| Title: | Quantum Fourier Analysis and Categorification Criteria |
| Abstract: | We give a quick review of recent developments in quantum Fourier analysis. We derive the primary categorification criteria from complete positivity and apply it to answer three questions proposed by Jones, Wang and Etingof in 2015, 2017 and 2019 respectively. |
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| Date: | April 5, 2024 |
| Time: | 3:00pm |
| Location: | BLOC 628 |
| Speaker: | Ningfeng Wang, Tsinghua University |
| Title: | 3d connections in alterfold TQFT and embedding theorems |
| Abstract: | We give a 3d topological representation of flat connections within 3-alterfold TQFT. We give a quick proof of the embedding theorem saying that a multi-fusion category as a planar algebra can be canonically embedded into its graph planar algebra. Moreover, the image is the flat part of the connection. It generalizes the corresponding results for subfactors to any field. Furthermore, we developed an embedding theorem for ground states in the configuration spaces of the Levin-Wen model on a surface with/without boundary. This is joint work with Zhengwei Liu. |
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| Date: | April 18, 2024 |
| Time: | 3:00pm |
| Location: | BLOC 628 |
| Speaker: | Qiaochu Ma, Washington University in St. Louis |
| Title: | Mixed quantization and quantum ergodicity |
| Abstract: | Quantum Ergodicity (QE) is a classical topic in spectral geometry and quantum chaos, it states that on a compact Riemannian manifold whose geodesic flow is ergodic with respect to the Liouville measure, the Laplacian has a density-one subsequence of eigenfunctions that tends to be equidistributed. In this talk, we present a uniform version of QE for a certain series of unitary flat bundles using a mixture of semiclassical and geometric quantizations. We shall see that even if analytically unitary flat bundles are similar to the trivial bundle, the holonomy provides extra fascinating geometrical phenomena.
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| Date: | April 19, 2024 |
| Time: | 3:00pm |
| Location: | BLOC 306 |
| Speaker: | Henri Moscovici, Ohio State University |
| Title: | Prolate wave operators beyond the archimedean place |
| Abstract: | As we have seen, the negative spectrum of the prolate spheroidal wave operator tries to simulate the nontrivial zeros of Zeta. To improve the degree of approximation one needs to go beyond the archimedean place and involve the primes, i.e. the finite places of the adeles over Q. This talk, based on joint work with A. Connes and C. Consani, discusses a process for obtaining such an extension in the semilocal adelic framework, which involves finitely many primes at a time. |
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