Veranstaltungen
Symplectic geometry of representation and quiver varieties

MPI für Mathematik in den Naturwissenschaften Leipzig

E1 05 (Leibniz-Saal)

conference

31.01.24 01.02.24

Symplectic geometry of representation and quiver varieties

conference

31.01.24 01.02.24

Symplectic geometry of representation and quiver varieties

Program

09:00 - 09:30	Welcome Coffee
09:30 - 10:00	Markus Reineke Floer potentials, cluster algebras and quiver representations We use cluster algebras to interpret Floer potentials of monotone Lagrangian tori in toric del Pezzo surfaces as cluster characters of quiver representations. Joint work with P. Albers, M. Bertozzi, arXiv:2309.16009
10:00 - 10:30	Lina Deschamps How can quaternions describe the magnetic geodesic flow on the 2-sphere. We will define the lift of the magnetic geodesic flow from S^2 to S^3, and then give a contact-geometric interpretation in terms of quaternionic symmetries (widely inspired by Albers-Geiges-Zehmisch arXiv:1705.08126).
10:30 - 11:00	Coffee & Tea
11:00 - 11:30	Dani Kaufman Involutive Rings and Algebras Commutative and non-commutative rings with an anti involution have been studied in many different contexts. In this talk I will give you a brief overview of some of the results and constructions from my recent and ongoing work joint with Zachary Greenberg and Anna Wienhard studying the algebraic properties of symplectic groups defined over involutive rings, including notions of traces, Caley Hamilton identities, Hopf algebra structures and some ways to realise these constructions within non-commutative cluster algebras. If time, I will also mention some new ideas about mutation of matrices with entries in an involutive algebra.
11:30 - 12:00	Zachary Greenberg Noncommutative Markov Numbers Markov numbers are a family of positive integers originally studied in relation bounds on approximating irrational numbers with rationals. They are characterized as integer solutions to the equation $x^{2} + y^{2} + z^{2} = 3 x y z$ . We will review the relationship between this equation and the cluster algebra structure of arcs on a punctured torus. Using this relationship we use the work of Berenstein and Retakh to associate a noncommutative cluster structure with values in any ring with involution. We will give examples of many such rings and the new Markov-like numbers found in them.
12:00 - 14:00	Lunch and Discussion @ MiS
14:00 - 15:00	Nonlinear Algebra Seminar: Claudia Alfes-Neumann Modular forms and their generalizations in number theory and geometry This talk is part of the Nonlinear Algebra Seminar. In this talk we introduce modular forms and harmonic weak Maass forms, real-analytic generalizations of holomorphic modular forms. We present some applications of the theory in number theory and to the theory of elliptic curves.
15:00 - 22:00	Coffee & Discussion
19:00 -	Dinner

09:00 - 09:15	Welcome Coffee
09:15 - 10:00	Pengfei Huang Quiver representations and character varieties In this talk, we will apply quiver representation theory to construct character varieties that appear in the regular nonabelian Hodge correspondence on Riemann surfaces. These character varieties are known as the moduli spaces of filtered local systems. Based on a joint work with Hao Sun.
10:00 - 11:00	Nonlinear Algebra Seminar: Arunima Ray How and why to embed surfaces in 4-dimensional spaces This talk is part of the Nonlinear Algebra Seminar. Geometric topologists like to study spaces of arbitrary dimensions. Fortunately, we at least limit ourselves to studying manifolds, which locally mimic Euclidean space. Dimension four forms a "phase transition" between low- and high-dimensional manifolds, exhibiting unique behaviour and necessitating bespoke tools. I will describe the source of this curious phenomenon, giving a few guiding examples and constructions. The key source of the problem or appeal, depending on your perspective, of 4-dimensional manifolds turns out to be the difficulty in embedding surfaces therein.
11:00 - 11:30	Coffee & Tea
11:30 - 12:00	Irina Bobrova Painlevé equations and cluster algebras In this talk, we will discuss connections between the differential and discrete Painlevé equations and cluster algebras, as well as some perspectives of their generalisations to the non-commutative case.
12:00 - 12:30	Merik Niemeyer Non-commutative cluster algebras beyond surfaces Since their introduction cluster algebras have been related to many aspects of mathematics, and notably Fock and Goncharov established a connection with higher Teichmüller theory. In trying to generalize their construction to the Theta-positive setting, one naturally encounters non-commutative cluster structures. I will give a quick overview of this, focused on the case of non-commutative surfaces as introduced by Berenstein and Retakh, before explaining ongoing work on Fock-Goncharov coordinates for representations of fundamental groups of bordered surfaces into Spin(p,q), and especially on understanding the underlying non-commutative cluster structure. This is joint work with Zack Greenberg, Dani Kaufman and Anna Wienhard.
13:00 - 14:00	Lunch @ MiS
14:00 - 16:00	Coffee & Discussion

Scientific Organizer

Anna Wienhard

Max Planck Institute for Mathematics in the Sciences, Leipzig

Administrative Contact

Antje Vandenberg

Max Planck Institute for Mathematics in the Sciences Contact via Mail