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ProgramMini-courses Links to the slides: slides1, slides2, slides3.
The Farrell–Jones conjecture predicts that the algebraic K-theory of a group ring is isomorphic to a certain equivariant homology theory, and there are also versions for L-theory and Waldhausen's A-theory. In principle, this provides a way to calculate these K-groups, and has many applications. These include classifying manifolds admitting a given fundamental group and a positive resolution of the Borel conjecture. Marco Golla: Line arrangements with odd multiplicities In what ways can complex projective lines meet in the complex projective plane? For instance, do there exist infinitely many complex line arrangements where lines always meet three by three? We'll take a topological look at this question. Where nature is twisted homologies of configuration spaces of points decorated by simple roots, and quantum groups are symmetrizable quantum Kac-Moody algebras. They constitute algebraic structures extensively used in TQFT constructions, quantum invariants, etc, and we give them an initial topological flavor. (joint with S. Bigelow) The slides: memo_grenoble_aside.pdf For 3-manifolds there are various notions of topological complexity: minimal number of tetrahedra in a triangulation, Heegaard genus among others. Various results show that these are most of the time asymptotically comparable with the simplicial volume (that is the Riemannian volume of hyperbolic pieces in JSJ decompositions). In higher dimensions the picture is completely unreadable at present but similar phenomena hold for hyperbolic manifolds. I will provide various precise statements along these lines. The famous Burau representation of the braid group is known to be unfaithful for braids with at least five strands. In the early 2000s two constructions were provided to fix faithfulness: the first being the Lawrence-Krammer-Bigelow linear representation, hence proving linearity of braid groups, and the second being the Khovanov-Seidel categorical representation. In this talk, based on joint work in progress with Licata, Queffelec and Wagner, I will investigate the interplay between these two representations. Surgery diagrams provide an efficient description of smooth 3–manifolds, and Kirby’s theorem shows that any two such diagrams of the same smooth 3–manifold are related by a finite sequence of blowups, blowdowns, and handleslides. In the contact setting, contact 3–manifolds can likewise be encoded by Legendrian surgery diagrams, where the surgery coefficients are restricted to ±1 relative to the Thurston–Bennequin framing. In this talk I will present a complete set of Legendrian Kirby moves relating diagrams of the same contact 3–manifold, yielding a contact analogue of Kirby’s theorem. Our approach refines Avdek’s ribbon moves by using Gervais’ presentation of mapping class groups using only nonseparating curves, together with explicit constructions of Legendrian representatives. This is joint work with Marc Kegel, Eric Stenhende, and Daniele Zuddas.
Adrien Abgrall: Geometry of Outer Space for RAAGs Untwisted Outer Space for right-angled Artin groups (RAAGs) has been constructed by Charney, Stambaugh, and Vogtmann in 2017. It is a contractible simplicial complex, and a geometric model of the group of coarse-median-preserving automorphisms of a RAAG. In this talk, I will give structural results about the geometry of this complex and its combinatorial automorphisms, as well as many explicit examples involving interesting tessellations of Euclidean space. Achintya Dey: Bolza-like surfaces in Thurston set A surface in the Teichmüller space, where the systole function admits its maximum, is called a maximal surface. For genus two, a unique maximal surface exists, which is called the Bolza surface, whose systolic geodesics give a triangulation of the surface. We define a surface as Bolza-like if its systolic geodesics decompose the surface into (p, q, r)-triangles for some integers p, q, r. In this article, we construct a one-parameter family of hyperbolic surfaces within the Thurston set, which, in turn, yields Bolza-like surfaces for infinitely many genera g > 9. Next, we see an intriguing application of Bolza- like surfaces. In particular, we construct global maximal surfaces using these Bolza-like surfaces. Furthermore, we study a symmetric property satisfied by the systolic geodesics of our Bolza-like surfaces. We show that any simple closed geodesic intersects the systolic geodesics at an even number of points. Yohan Mandin-Hublé: Perturbative invariants of combed 3-manifolds In this talk, I will define an invariant of three-dimensional rational homology spheres equipped with a parallelization, following Witten, Kontsevich, and Kuperberg-Thurston. This invariant counts configurations of trivalent graphs in the given parallelized manifold. It can be corrected to obtain an invariant Z of the manifold using an invariant of the parallelization, which is a linear function of a Pontryaguin class. The invariant Z is a universal finite-type invariant of rational homology spheres known as the perturbative expansion of Chern-Simons theory. I will give a more flexible definition of the invariant Z using a nowhere vanishing vector field on the manifold instead of a parallelization. Deepanshi Saraf: Fundamental n-quandles are residually finite Fundamental n-quandles are canonical quotients of the fundamental quandle of an oriented link in the 3-sphere, first introduced by Joyce. While they are coarser invariants than the fundamental quandle, they are often more tractable and admit deep connections to the the n-fold cyclic branched covers of the 3-sphere branched over the link. This connection has led to striking classification results, including a complete list of links with finite fundamental n-quandles, based on Thurston's geometrisation theorem and Dunbar's classification of spherical 3-orbifolds. In earlier work, Fish and Lisitsa developed an algorithm that uses fundamental 2-quandles to detect the unknot, and they conjectured that the fundamental 2-quandle of any knot is residually finite. This property not only supports the validity of their algorithm but also implies the solvability of the word problem for these algebraic structures. In this talk, I will present recent results showing that the fundamental n-quandle of any oriented link in the 3-sphere is residually finite for all integers n greater than or equal to 2. Along the way, we'll touch on connections to branched covers and classification results for when these quandles are finite. This work contributes to a growing bridge between low-dimensional topology and the algebraic theory of quandles. Sardor Yakupov: Moduli space of curves and achiral Lefschetz fibrations Lefschetz fibrations provide a powerful tool for studying symplectic 4-manifolds and contact 3-manifolds. Achiral Lefschetz fibrations extend this framework beyond the symplectic case, allowing us to address more general questions about 4-manifolds. On the other hand, the moduli space of complex curves is one of central objects in complex and Riemannian geometry of surfaces. The goal of this talk is to introduce these two concepts separately and explore their connection through the notion of the classifying map. |
