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Number Theory Lunch Seminar
Groechenig-Wyss-Ziegler's proof of the Hausel-Thaddeus conjecture
Groechenig-Wyss-Ziegler's proof of the Hausel-Thaddeus conjecture
演讲者
时间
2025年12月04日 12:15 至 13:00
地点
A4-1
摘要
I'll present an overview of the proof by Groechenig–Wyss–Ziegler of the Hausel–Thaddeus conjecture on a form of topological mirror symmetry between the moduli spaces of semistable Higgs bundles for the groups $\mathrm{SL}_n$ and $\mathrm{PGL}_n$. After reviewing the background of mirror symmetry and the geometry of Higgs bundle moduli spaces, especially the Hitchin fibration and duality of generic fibers, I will outline how their argument reduces the conjecture to an arithmetic comparison of (stringy) Hodge numbers. Their key idea is to translate the equality of stringy E-polynomials into a comparison of point counts over finite fields, and to prove this equality using $p$-adic integration on the moduli stacks.
References:
[1] M. Groechenig, D. Wyss and P. Ziegler, Mirror symmetry for moduli spaces of Higgs bundles via $p$-adic integration. Invent. Math. 221 (2020), 505-596. arxiv:1707.06417
[2] M. Groechenig, D. Wyss and P. Ziegler, Geometric stabilisation via $p$-adic integration, J. Amer. Math. Soc. 33 (2020), 807-873. arXiv:1810.06739.
[3] Hausel and M. Thaddeus, Mirror symmetry, Langlands duality, and the Hitchin system, Invent. Math. 153 (2003), 197-229. arXiv:math/0205236.
References:
[1] M. Groechenig, D. Wyss and P. Ziegler, Mirror symmetry for moduli spaces of Higgs bundles via $p$-adic integration. Invent. Math. 221 (2020), 505-596. arxiv:1707.06417
[2] M. Groechenig, D. Wyss and P. Ziegler, Geometric stabilisation via $p$-adic integration, J. Amer. Math. Soc. 33 (2020), 807-873. arXiv:1810.06739.
[3] Hausel and M. Thaddeus, Mirror symmetry, Langlands duality, and the Hitchin system, Invent. Math. 153 (2003), 197-229. arXiv:math/0205236.
演讲者介绍
My research interests are primarily centred on Gauge theory within mathematics. Recently, my focus has been on semistable Higgs sheaves on complex projective surfaces and associated gauge-theoretic invariants, employing algebro-geometric methods. However, I also have a strong interest in working within the analytic category.