The Stark conjecture over the field of rational numbers: A new approach
Contact: Pieter Moree (moree @ mpim-bonn.mpg.de)
We present a new proof for the algebraicity of the cyclotomic Stark units. These units are defined in a transcendental way, via the derivative at s=0 of a partial zeta function. The proof links these units to the special values of the Riemann zeta function at positive even integers. The rationality of these special values (modulo a power of pi) implies the algebraicity of the cyclotomic Stark units.
On the natural stratification of the space of differentiable functions and the space of Morse functions
Hybrid. Contact: Christian Kaiser (kaiser @ mpim-bonn.mpg.de)
Margulis Lemma and RCD spaces
Contact: Christian Kaiser (kaiser @ mpim-bonn.mpg.de)
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Contact: Aru Ray (aruray @ mpim-bonn.mpg.de), Steven Sivek (sivek @ mpim-bonn.mpg.de)
4-genus bounds from the 10/8+4 theorem
Contact: Aru Ray (aruray @ mpim-bonn.mpg.de)
Donald and Vafaee described a way to use Furuta’s 10/8 theorem to obstruct sliceness in the 4-ball and Linh Truong used a refinement, the 10/8 +4 theorem of Hopkins-Lin-Shi and Hu, to strengthen this sliceness obstruction. We will show how to expand on this technique to obtain lower bounds for four-ball genus and present some calculations for satellite knots. This is joint work in progress with Sashka Kjuchukova and Linh Truong.
The Nielsen realization for non-spin 4-manifolds
Contact: Aru Ray (aruray @ mpim-bonn.mpg.de)
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Contact: Nicolas Addington
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Contact: Nicolas Addington
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Contact: Nicolas Addington
On opers and complex lagrangians
Contact: Christian Kaiser
In 2014 Gaiotto conjectured a correspondence between a holomorphic lagrangian (Hitchin section) in the Dolbeault moduli space of Higgs bundles on a curve and the holomorphic lagrangian of opers in the de Rham moduli space of holomorphic connections. This conjecture was established in 2016 for holomorphic opers with an arbitrary complex simple Lie group; the construction is known today as the conformal limit.
Algebraic structures in symplectic geometry I
Contact: Christian Kaiser
Ever since seminal work of Gromov, Floer, Donaldson, and Fukaya in the late 80s and early 90s, a major theme in symplectic geometry has been the construction of invariants defined in terms of moduli spaces of J-holomorphic curves. In this minicourse, I will explain some of these invariants, and describe how creative use of auxiliary moduli spaces of J-curves can equip these invariants with extra structure. The main reference will be my survey article with Abouzaid, https://arxiv.org/abs/2210.11159 . I will not assume any prior knowledge of symplectic geometry.
Algebraic structures in symplectic geometry II
Contact: Christian Kaiser
Ever since seminal work of Gromov, Floer, Donaldson, and Fukaya in the late 80s and early 90s, a major theme in symplectic geometry has been the construction of invariants defined in terms of moduli spaces of J-holomorphic curves. In this minicourse, I will explain some of these invariants, and describe how creative use of auxiliary moduli spaces of J-curves can equip these invariants with extra structure. The main reference will be my survey article with Abouzaid, https://arxiv.org/abs/2210.11159 . I will not assume any prior knowledge of symplectic geometry.
On Chowla's non-vanishing conjecture over function fields
Chowla's conjecture postulates that the L-function of a quadratic character over Q should never vanish at s=1/2. At present we know that this holds for a positive proportion of characters, thanks to Soundararajan. We also know that over function fields Fq(T) the naive analogue of Chowla's conjecture is false: Li has constructed infinitely many quadratic characters with L-function vanishing at 1/2. The refined form of Chowla's conjecture postulates that the non-vanishing should hold with probability 1. This statement is grounded in the Katz--Sarnak random matrix heuristics and has seen partial evidence thanks to the works of Florea,Florea--David--Lalin, Ellenberg--Li--Shusterman, where it was established for each q a positive proportion of non-vanishing (with proportions getting better as q goes to infinity). In this talk I will discuss some of the ingredients of an upcoming joint work with Koymans and Shusterman, where we establish that the refined form of Chowla's conjecture holds for each fixed q congruent to 3 modulo 4.
The Kodaira classification of the moduli of hyperelliptic curves
We study the birational geometry of the moduli spaces of hyperelliptic curves with marked points. We show that these moduli spaces are much worst from the point of view of singularities than the moduli space of pointed stable curves M_{g,n}, but much better from the point of view of birational complexity. We provide a full Kadaira classification with the surprising feature that for n large fixed, the Kodaira dimension decreases as the genus grows. In other words, the space becomes "simpler" as the genus grows. Further, we study the polyhedrality of the effective cone. This is joint work with Scott Mullane.
Course on slice knots and knot concordance
Contact: Aru Ray (aruray @ mpim-bonn.mpg.de)
Dirichlet eigenvalues of regular polygons
Contact: Pieter Moree (moree @ mpim-bonn.mpg.de)
For N>4 the first Dirichlet eigenvalue of a regular N-gon does not seem to have any closed form expression in terms of other known mathematical constants. However, it has been previously observed that it has an asymptotic expansion in powers of 1/N whose low order coefficients can be expressed in terms of special values of the Riemann zeta function at positive integers. It turns out that all coefficients of this expansion can be computed in closed form, but in general they involve more general multiple zeta values that (conjecturally) cannot be expressed in terms of the usual zeta values. I will discuss the proof of this result as well as some curious results and identities that arise in the process. The talk is based on a joint work with David Berghaus, Bogdan Georgiev, and Hartmut Monien.
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Course on slice knots and knot concordance
Contact: Aru Ray (aruray @ mpim-bonn.mpg.de)
Strongly quasipositive knots are concordant to infinitely many strongly quasipositive knots
Contact: Aru Ray (aruray @ mpim-bonn.mpg.de)
We show that every non-trivial strongly quasipositive knot is smoothly concordant to infinitely many pairwise non-isotopic strongly quasipositive knots. In contrast to our result, Baader, Dehornoy and Liechti showed that every (topologically locally-flat) concordance class contains at most finitely many positive knots. Moreover, it was conjectured by Baker that smoothly concordant strongly quasipositive fibered knots are isotopic. Our construction uses a satellite operation with companion a slice knot with maximal Thurston-Bennequin number -1. If time permits, we will say a few words about how the construction extends to links.
Differentiation of higher groupoids in tangent categories
Hybrid.
Contact: Christian Kaiser
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