# Topological persistence in geometry and analysis

*Part I: Oscillations and bars in persistence barcodes* (see Part II here)

**Reading group - Mathematical Institute - Michaelmas term 2023/2024 **

*Organised with Luis Scoccola*

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**Abstract:**
The classical Bézout theorem states that the number of common zeros of n polynomials in n variables is generically bounded by the product of their degrees. For many reasons (that we could explore), it is interesting to generalize this to the transcendental Bézout problem, concerned with the count of zeros of complex analytic maps. However, this problem turns out to be more than challenging: the natural generalisation of the notion of degree for analytic maps does not satisfy a transcendental Bézout theorem, as shown by Cornalba and Shiffman in 1972. In a recent preprint, Polterovitch et al. [3] proved the striking result that this notion of degree does satisfy a Bézout theorem if the number of zeros is replaced by a coarse count of zeros studying short bars of associated persistence barcodes. The goal of the reading group would be to understand this coarse version of the transcendental Bézout theorem, as well as a similar bound on coarse nodal count in terms of Sobolev norms [2].

### References

[1] Topological persistence in geometry and analysis - Polterovitch, Rosen, Samvelyan, Zhang

[2] Coarse nodal count and topological persistence - Buhovsky, Payette, Polterovich, Polterovich, Shelukhin, Stojisavljević

[3] Persistent transcendental Bézout theorems - Buhovsky, Polterovich, Polterovich, Shelukhin, Stojisavljević

[4] Persistent modules and Hamiltonian diffeomorphisms - Lecture by Polterovich (video)

### Calendar

*Unless specifically stated otherwise, meetings will take place on Fridays at 11:00 (UK time).*

13/10/2023 - Week 1 - **Persistence modules and persistence barcodes** [1, Chaps. 1, 2]

**Room:** C2 **Speaker:** Marc Fersztand

20/10/2023 - Week 2 - **What can we read from a barcode?** [1, Chap. 4]

**Room:** C5 **Speaker:** Ximena Fernández

27/10/2023 - Week 3 - **Topological function theory** [1, Chap. 6]

**Room:** S2.37 **Speaker:** David Beers

03/11/2023 - Week 4 - **From classical to persistent transcendental Bézout theorem** [3, Sec. 1]

**Room:** S2.37 **Speaker:** Luis Scoccola

10/11/2023 - Week 5 - **Proof of the persistent transcendental Bézout theorem** [3, Secs. 3 & 4]

**Room:** S2.37 **Speaker:** V. L.

17/11/2023 - Week 6 - **Subadditivity of number of bars in persistence barcodes** [2, Sec. 3]

**Room:** S2.37 **Speaker:** María Torras Pérez

24/11/2023 - Week 7 - **Multiscale polynomial approximations** [2, Sec. 4]

**Room:** S2.37 **Speaker:** Darrick Lee

01/12/2023 - Week 8 - **Proof of the main coarse nodal count result** [2, Sec. 5]

**Room:** S2.37 **Speaker:** Otto Sumray (online)