GRK-Seminar Wintersem. 2024/25
RTG Seminar Winter term 2024
The Thursday morning seminar (10:15-11:45 in WSC-N-U-3.05) will be the “Research Training Group Seminar” where members of the RTG (PhD students, post-docs,…) present their results. Sometimes, we also have speakers from other places. Depending on the number of speakers and on the proposed topic, a speaker could use one or two sessions.
| 31.10.2024 | Carolina Tamborini | Non-tautological double cover cycles |
| 7.11.2024 | Andreas Pieper | Newton meets Torelli |
| 14.11.2024 | Sebastian Bartling | Some remarks on Brauer groups of proper and smooth Deligne-Mumford stacks over the integers. |
| 21.11.2024 | |
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| 28.11.2024 | reserved | general assembly |
| 5.12.2024 | Robert Franz | Barcodes and the Witten Deformation on a Singular Space |
| 19.12.2024 | Pietro Gigli | Generators of the algebraic symplectic cobordism ring |
| 9.1.2025 | Thiago Solovera y Nery | A stratification on the special fiber of ramified local models |
| 16.1.2025 | Maximilian Hauck | Syntomic cohomology and stacks in p-adic geometry |
| 23.1.2025 | Louisa Bröring | $\mathbb{A}^1$-Euler Characteristic of Low Symmetric Powers |
| 30.1.2025 | Giorgio Navone | Transcendental Brauer groups of generalized Kummer surfaces |
Abstracts
Carolina Tamborini: Non-tautological double cover cycles.
After an introduction on moduli spaces of curves and their tautological rings, I will discuss joint works together with V. Arena, S. Canning, E. Clader, R. Haburcak, A.Q. Li, and S.C. Mok and with D. Faro on the construction of many new non-tautological algebraic cohomology classes arising from double cover cycles, generalising previous work by Graber-Pandharipande and van Zelm.
Andreas Pieper: Newton meets Torelli.
The Newton stratification is a natural refinement of the $f$-number stratification of the moduli space $\mathcal{A}_g$ in characteristic $p>0$. In the beginning of the talk I will define the stratification and discuss its properties. The main part will be about the restriction of the Newton-stratification to the Torelli locus. Here much less is known; despite numerous contributions there is no complete (even conjectural) picture of questions regarding non-emptiness, dimensions, or the closure relation.
I will report on work in progress that shows that all the Newton strata on $\mathcal{M}_4$ are not empty and have the expected dimension.
Sebastian Bartling: Some remarks on Brauer groups of proper and smooth Deligne-Mumford stacks over the integers.
Class field theory implies that $Br(Spec(\mathbb{Z}))=0$. One might wonder whether $Br(X)=0$ for any DM-stack $X\rightarrow Spec(\mathbb{Z})$ that is proper and smooth. I want to explain that $Br(X)$ is always finite, give conditions when $Br(X)=0$ and explain that for the moduli stack of stable genus $g$ curves with $n$ marked points, we have $Br(\overline{\mathcal{M}}_{g,n,\mathbb{Z}})=0$ for $(g,n)=(1,1),(1,2),(2,0),(3,0)$, any $g$ greater or equal than 4. This is joint work with Kazuhiro Ito.
Robert Franz: Barcodes and the Witten Deformation on a Singular Space.
The Witten deformation is a powerful tool for understanding the topology of smooth manifolds. By studying the exponentially small eigenvalues of the Witten Laplacian, one obtains an analytical proof of the celebrated Morse inequalities. In this talk, I will discuss a recent result by Le Peutrec-Nier-Viterbo that provides accurate decay rates for these small eigenvalues. Furthermore, I will report on current work in progress aimed at generalizing their findings to spaces with isolated conical singularities.
Pietro Gigli: Generators of the algebraic symplectic cobordism ring.
Algebraic Symplectic cobordism, denoted by MSp, is a cohomology theory which is universal for carrying a system of characteristic classes for symplectic vector bundles. The ring of coefficients of MSp is bigraded, and we call its diagonal the “Symplectic cobordism ring”. This ring has no known presentation. Through a symplectic version of the Pontryagin-Thom construction, one can associate any symplectic variety with an element in the Symplectic cobordism ring. Still, the problem in using this construction to study the Symplectic cobordism ring is the paucity of non-trivial examples of symplectic varieties. We modify this construction to obtain elements in the Symplectic cobordism ring from a large family of varieties that are not symplectic but carry a certain “symplectic twist”. Then, using a strategy based on the Adams Spectral sequence for MSp, we find a criterion to select generators among these elements, after appropriate completions.
Thiago Solovera e Nery: A stratification on the special fiber of ramified local models
Given a local Shimura data $(G,\mu)$ one constructs a scheme $M^{loc}$, projective over some local ring of integers $\mathcal{O}_E$, whose singularities mimic that of the completion of Shimura varieties associated to G at the given prime $p$. When $G$ is unramified, X. He related the singularities of the special fiber of $M^{loc}$ to those of a subvariety of the wonderful compactification of (a char. $p$ deformation of) $G$. In this talk we will explain what to expect in the case where $G$ becomes ramified.
Maximilian Hauck: Syntomic cohomology and stacks in p-adic geometry
I will briefly review the stacky approach to p-adic cohomology theories due to Bhatt—Lurie and Drinfeld using a new view on the filtered prismatisation recently found by Gardner—Madapusi. Then I will show how to use these tools to obtain a new proof of the Beilinson fibre square of Antieau—Mathew—Morrow—Nikolaus and a generalisation of a comparison theorem between rational arithmetic étale cohomology and crystalline cohomology of Colmez—Nizioł.
Louisa Bröring: $\mathbb{A}^1$-Euler Characteristic of Low Symmetric Powers
The $n$-th symmetric power of a smooth, quasi-projective scheme $X$ is the quotient $X^n/S_n$ where S_n acts on $X^n$ by permuting the factors.
For a smooth, projective scheme $X$, we compute the compactly supported $\mathbb{A}^1$-Euler characteristic of $\operatorname{Sym}^2(X)$ and $\operatorname{Sym}^3(X)$ over fields of characteristic not two or three. For this, we will also present some ideas for computing the compactly supported $\mathbb{A}^1$-Euler characteristic of quotients in general.
With this computation we will partially answer a question of Pajwani-Pál affirmatively.
Giorgio Navone: Transcendental Brauer groups of generalized Kummer surfaces
The Brauer-Manin obstruction to the Hasse Principle is an important tool to determine the existence of rational points on a variety. After introducing this technique from its foundations, we will present ongoing work computing the transcendental Brauer group for a family of K3 surfaces, constructed from a planar cubic curve in a similar fashion to Kummer surfaces.