## Differential Geometry Seminars 2018

### School of Mathematical Sciences – The University of Adelaide

#### ~ Next talk ~

• Stephen Tillmann (Unversity of Sydney)
Title: Computing trisections of 4-manifolds
Friday, 23 March 2018 at 1:10pm in Barr Smith South Polygon Lec theatre

Abstract: Gay and Kirby recently generalised Heegaard splittings of 3-manifolds to trisections of 4-manifolds. A trisection describes a 4-dimensional manifold as a union of three 4–dimensional handlebodies. The complexity of the 4–manifold is captured in a collection of curves on a surface, which guide the gluing of the handelbodies. The minimal genus of such a surface is the trisection genus of the 4-manifold. After defining trisections and giving key examples and applications, I will describe an algorithm to compute trisections of 4–manifolds using arbitrary triangulations as input. This results in the first explicit complexity bounds for the trisection genus of a 4–manifold in terms of the number of pentachora (4–simplices) in a triangulation. This is joint work with Mark Bell, Joel Hass and Hyam Rubinstein. I will also describe joint work with Jonathan Spreer that determines the trisection genus for each of the standard simply connected PL 4-manifolds.

#### ~ Upcoming talks ~

• Finnur Larusson (University of Adelaide)
Title: TBA
Friday, 20 April 2018 at 1:10pm in Barr Smith South Polygon Lec theatre

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• Hao Guo (University of Adelaide)
Title: TBA
Friday, 27 April 2018 at 1:10pm in Barr Smith South Polygon Lec theatre

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• Tony Licata (Australian National University)
Title: TBA
Friday, 4 May 2018 at 1:10pm in Barr Smith South Polygon Lec theatre

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• Name David Baraglia (University of Adelaide)
Title: TBA
Friday, 11 May 2018 at 1:10pm in Barr Smith South Polygon Lec theatre

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• NO SEMINAR
Title: N/A

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• Name TBA (University of XX)
Title: TBA
Friday, 25 May 2018 at 1:10pm in Barr Smith South Polygon Lec theatre

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• Name TBA (University of XX)
Title: TBA
Friday, 1 June 2018 at 1:10pm in Barr Smith South Polygon Lec theatre

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• Name TBA (University of XX)
Title: TBA
Friday, 8 June 2018 at 1:10pm in Barr Smith South Polygon Lec theatre

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• Name TBA (University of XX)
Title: TBA
Friday, 15 June 2018 at 1:10pm in Barr Smith South Polygon Lec theatre

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• Name TBA (University of XX)
Title: TBA
Friday, 27 July 2018 at 11:10am in Barr Smith South Polygon Lec theatre

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• Name TBA (University of XX)
Title: TBA
Friday, 3 August 2018 at 11:10am in Barr Smith South Polygon Lec theatre

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• Name TBA (University of XX)
Title: TBA
Friday, 10 August 2018 at 11:10am in Barr Smith South Polygon Lec theatre

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• Name TBA (University of XX)
Title: TBA
Friday, 17 August 2018 at 11:10am in Barr Smith South Polygon Lec theatre

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• Name TBA (University of XX)
Title: TBA
Friday, 24 August 2018 at 11:10am in Barr Smith South Polygon Lec theatre

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• Name TBA (University of XX)
Title: TBA
Friday, 31 August 2018 at 11:10am in Barr Smith South Polygon Lec theatre

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• Name TBA (University of XX)
Title: TBA
Friday, 7 September 2018 at 11:10am in Barr Smith South Polygon Lec theatre

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• Name TBA (University of XX)
Title: TBA
Friday, 14 September 2018 at 11:10am in Barr Smith South Polygon Lec theatre

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• Geordie Williamson (University of Sydney)
Title: TBA
Friday, 28 September 2018 at 11:10am in Barr Smith South Polygon Lec theatre

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#### ~ Past talks ~

• Bram Mesland (Unversität Bonn)
Title: A Hecke module structure on the KK-theory of arithmetic groups
Friday, 2 March 2018 at 1:10pm in Barr Smith South Polygon Lec theatre

Abstract: Let $G$ be a locally compact group, $\Gamma$ a discrete subgroup and $C_{G}(\Gamma)$ the commensurator of $\Gamma$ in $G$. The cohomology of $\Gamma$ is a module over the Shimura Hecke ring of the pair $(\Gamma,C_G(\Gamma))$. This construction recovers the action of the Hecke operators on modular forms for $SL(2,\mathbb{Z})$ as a particular case. In this talk I will discuss how the Shimura Hecke ring of a pair $(\Gamma, C_{G}(\Gamma))$ maps into the $KK$-ring associated to an arbitrary $\Gamma$-C*-algebra. From this we obtain a variety of $K$-theoretic Hecke modules. In the case of manifolds the Chern character provides a Hecke equivariant transformation into cohomology, which is an isomorphism in low dimensions. We discuss Hecke equivariant exact sequences arising from possibly noncommutative compactifications of $\Gamma$-spaces. Examples include the Borel-Serre and geodesic compactifications of the universal cover of an arithmetic manifold, and the totally disconnected boundary of the Bruhat-Tits tree of $SL(2,\mathbb{Z})$. This is joint work with M.H. Sengun (Sheffield).

• Raul Quiroga-Barranco (CIMAT, Guanajuato, Mexico)
Title: Radial Toeplitz operators on bounded symmetric domains
Friday, 9 March 2018 at 11:10am in Lower Napier LG11

Abstract: The Bergman spaces on a complex domain are defined as the space of holomorphic square-integrable functions on the domain. These carry interesting structures both for analysis and representation theory in the case of bounded symmetric domains. On the other hand, these spaces have some bounded operators obtained as the composition of a multiplier operator and a projection. These operators are highly noncommuting between each other. However, there exist large commutative C*-algebras generated by some of these Toeplitz operators very much related to Lie groups. I will construct an example of such C*-algebras and provide a fairly explicit simultaneous diagonalization of the generating Toeplitz operators.

• Gaetan Borot (MPI Bonn)
Title: Quantum Airy structures and topological recursion
Wednesday, 14 March 2018 at 1:10pm in Ingkarni Wardli B17

Abstract: Quantum Airy structures are Lie algebras of quadratic differential operators -- their classical limit describes Lagrangian subvarieties in symplectic vector spaces which are tangent to the zero section and cut out by quadratic equations. Their partition function -- which is the function annihilated by the collection of differential operators -- can be computed by the topological recursion. I will explain how to obtain quantum Airy structures from spectral curves, and explain how we can retrieve from them correlation functions of semi-simple cohomological field theories, by exploiting the symmetries. This is based on joint work with Andersen, Chekhov and Orantin.

• Hokuto Konno (University of Tokyo)
Title: Family gauge theory and characteristic classes of bundles of 4-manifolds
Friday, 16 March 2018 at 1:10pm in Barr Smith South Polygon Lec theatre

Abstract: I will define a non-trivial characteristic class of bundles of 4-manifolds using families of Seiberg-Witten equations. The basic idea of the construction is to consider an infinite dimensional analogue of the Euler class used in the usual theory of characteristic classes. I will also explain how to prove the non-triviality of this characteristic class. If time permits, I will mention a relation between our characteristic class and positive scalar curvature metrics.