Book of abstract here
Anton Alekseev
Central extensions and Teichmueller spaces
In this talk, we explain that (infinite dimensional) Teichmueller spaces associated to hyperbolic surfaces with absolute boundary carry Hamiltonian actions of the Virasoro algebra. If time permits, we will also state some open problems for surfaces with marked points. Our study is motivated on the one hand by the work of Saad–Shenker–Stanford on Jackiw–Teitelboim gravity, and on the other hand by the Meinrenken–Woodward theory of Hamiltonian loop group actions.
The talk is based on a joint work with Eckhard Meinrenken, see arXiv:2401.03029.
Changha Choi
Supersymmetric localization and Selberg trace formula
In this talk, I will explain two joint works with Prof. Leon, which led to the discovery of a path integral derivation of the Selberg trace formula using the novel concept of supersymmetric localization.
Giordano Cotti
Gromov-Witten theory, integral transforms, special functions
The quantum differential equations (qDEs) define a class of ordinary differential equations in the complex domain, whose study represents a challenging and active area in both contemporary geometry and mathematical physics. The qDEs define rich invariants attached to smooth projective varieties.
These equations encapsulate information not only about the enumerative geometry of varieties but also, conjecturally, about their topology and complex geometry. The way to disclose such a huge amount of data is through the study of the asymptotics and monodromy of their solutions.
In this talk, the speaker will address the problem of explicitly integrating the quantum differential equations of varieties and will report on his progress in a long-term project on this topic. Focusing on the case of projectivizations of vector bundles, he will first introduce a family of integral transforms and special functions (the integral kernels). Then, he will show how to use these tools to find explicit integral representations of solutions. Based on arXiv:2005.08262 (Memoirs of the EMS, 2022) and arXiv:2210.05445 (Journal Math. Pures et Appl. 2024).
Rukmini Dey
Interpolation by Minimal and Maximal surfaces and related problems (online)
Minimal surfaces are zero mean curvature surfaces in Euclidean 3-space and maximal surfaces are zero mean curvature surfaces in Lorentz–Minkowski 3-space.
We will talk of interpolation problems of two types. First type of interpolation problem we talk of is that given two real analytic curves can one interpolate them with a minimal or maximal surface? — a type of a Plateau's problem. For minimal surfaces this problem was solved by Douglas and Rado in great generality. We show that indeed, if the curves are "close" enough in a certain sense, then interpolation is possible. We will also talk of existence of a maximal surface containing a given real analytic curve and a special singularity, under certain conditions.
The second type of interpolation we will talk about is given a array of surfaces placed at some periodic intervals, can one interpolate them by a minimal/maximal surface, in the sense that the height functions of surfaces at these arrays sum up to a height function of a minimal/maximal surface.
This uses some Euler–Ramanujan identities. The latter problem was inspired by the work of Randall Kamien and his collaborators, in the context of liquid crystals.
(These works are in collaboration with Dr. Pradip Kumar, SNU and Dr. Rahul K Singh, IIT Patna).
Peter Gothen
Moduli of Higgs bundles and higher Teichmüller spaces
Let M be the moduli space of surface group representations in a real reductive Lie group G. The most basic topological question one can ask about M is the determination of its connected components. For example, when G=PSL(2,R), Goldman proved that M has 4g-3 components, two of which can be identified with Teichmüller space. Much progress has been made on this problem over the last 30-40 years but it is, in general, still open. A conjectural answer to the question — based on Hitchin's approach via non-abelian Hodge Theory — can be stated in terms of the generalised Cayley correspondence for moduli spaces of G-Higgs bundles, and is closely related to higher Teichmüller theory.
The talk is mainly based on joint work with Steve Bradlow, Brian Collier, Oscar Garcia-Prada, and André Oliveira.
Alexander Its
On some Hamiltonian properties of the isomonodromic tau functions
We will discuss some new aspects of the theory of the Jimbo–Miwa–Ueno tau function which have come to light within the recent developments in the global asymptotic analysis of the tau functions related to the Painlevé equations. Specifically, we will show that up to the total differentials the logarithmic derivatives of the Painlevé tau functions coincide with the corresponding classical action differential. This fact simplifies considerably the evaluation of the constant factors in the asymptotics of tau-functions, which has been a long-standing problem of the asymptotic theory of Painlevé equations. The talk is based on the joint work with O. Lisovyy and A. Prokhorov.
Vladimir Korepin
Simplex equations and Clifford algebras (online)
The Yang–Baxter equations were generalized to higher dimensions by Zamolodchikiv and Bazhanov. The generalizations are called simplex equations. We use Clifford algebra to find solutions.
Ari Laptev
A sharp Lieb–Thirring inequality for functional difference operators (online)
We prove sharp Lieb–Thirring type inequalities for the eigenvalues of a class of one-dimensional functional difference operators associated to mirror curves. We furthermore prove that the bottom of the essential spectrum of these operators is a resonance state.
Marcos Mariño
From string theory to quantum mechanics, and back
Simplified models of string theory or conformal field theory can be sometimes related to quantum mechanical problems, and this leads to a fruitful interaction between the two disciplines. In this talk I will consider a family of quantum mechanical operators which arises in mirror symmetry and is related to enumerative geometry and to relativistic integrable models. I will review some conjectural results for their spectrum and explain how they can be used to provide exact descriptions of string theory models.
Claudio Meneses
Stability variations and analytic invariants on moduli of parabolic bundles
A characteristic of moduli spaces of parabolic bundles on Riemann surfaces is their dependence on a set of real parameters, which gives rise to wall-crossing phenomena in the study of their birational geometry. In a less explicit way, this dependance also occurs in their natural Kähler metrics, and it is compelling to express this dependence as a suitable "Torelli theorem" answering the question: to what extent does the Kähler metric (an analytic invariant) determine the stability conditions that define the moduli problem?
In this talk I will describe a strategy to address this variation problem in terms of the complex-analytic techniques in the study of the spectral geometry of resolvents, championed by Takhtajan–Zograf. I will also explain how these techniques can be applied to the study and classification of gravitational instantons of ALG type. This is work in progress, joint with Hartmut Weiss.
Jinsung Park
Maximal discs of Weil–Petersson class in the anti-de Sitter 3-space
The Weil-Petersson Teichmuller space was introduced by Takhtajan and Teo in their work to introduce the universal Liouville action on this space. An element in the Weil–Petersson Teichmuller space can be represented by a quasisymmetric S¹-homeomorphism of Weil–Petersson class. In this talk, I will explain some basic geometric properties of maximal discs with the corresponding Weil–Petersson condition. In particular, I will explain variational formulae of geometric quantities of maximal discs of Weil-Petersson class and their relation with the symplectic property of the Mess map.
Vamsi Pingali
Gravitating vortices and cosmic strings
The gravitating vortex (GV) equations on a compact Riemann surface arise as a dimensional reduction of the Kaehler–Yang–Mills equations. A special case of the GV equations includes the equations governing the (as of now, hypothetical) cosmic strings. I shall describe existence, uniqueness, and algebro-geometric obstructions to existence to these equations. This talk is based on joint work with M. Garcia-Fernandez, L. Alvarez-Consul, O. Garcia-Prada, and Chengjian Yao.
Evgeny Sklyanin
Wavefunction reduction in the quantum integrable systems
We discuss a peculiar phenomenon observed for a number of quantum integrable systems. A restriction of a multivariate eigenfunction of the commuting Hamiltonians on a certain curve produces a univariate function whose zeroes satisfy Bethe Ansatz equations. The list of examples includes Calogero–Moser system, Benjamin–Ono, Korteweg–de Vries, Intermediate Long Waves equations.
Marko Stošić
Knots, quivers and billiards
The knots-quivers correspondence relates colored HOMFLY-PT invariants of a knot, and motivic Donaldson-Thomas invariants of a corresponding knot. This correspondence has implications on the properties of the (super-)A-polynomials of knots, and also on enumerative combinatorics of lattice paths and Schröder paths. Surprisingly, such quiver generating series are also related to the count of billiard partitions — the combinatorial objects that arise from the description of periodic trajectories of ellipsoidal billiards in n-dimension Euclidean space. In this talk I will gave an overview of all these relationships.
Dennis Sullivan
Three dimensional universal geometry for closed 3-manifolds (online)
For studying our 3D physical - spatial reality it is perhaps useful to determine coordinate changes that are sufficient to describe all closed three manifolds. This goal is encouraged by the theory of Thurston, Hamilton and Perelman using hyperbolic geometry, the Ricci flow and the synthetic geometry of Alexandrov and the St. Petersburg school. Using this result we show how projective geometry is universal by naturally embedding conformal geometry into projective geometry.
Alexander Veselov
Delay Painleve-I equation and Masur–Veech volumes
The delay Painleve-I equation can be obtained as the simplest delay periodic reduction of Shabat's dressing chain. We use it to generate a new family of Bernoulli-Catalan polynomials and apply them to the calculation of the Masur–Veech volumes of the moduli spaces of meromorphic quadratic differentials. The talk is based on a joint work with John Gibbons and Alex Stokes.
Lin Weng
Reductive moduli spaces of semi-stable arithmetic $G$-torsors
Stability plays a central role in modern mathematics. For example, as a natural generalization of Dedekind zeta function for a number field $F$, the rank $n$ zeta function $\widehat \zeta_{F,n}(s)\ (n=1,2,3,...)$ of $F$ is defined as an integration over the moduli space $\mathcal M_{F,n}^{ss}$ of rank $n$ semi-stable $\mathcal O_F$-lattices. In this talk, motivated by Zucker's reductive Borel-Serre compactifications
$\overline {\Gamma\backslash G/K}^{\mathrm {rBS}}$ of locally symmetric spaces, we introduce what we call the {\it reductive moduli spaces}
$\mathcal M_{F,G}^{\mathrm{rBS}}$ of semi-stable arithmetic $G$-torsors over $\overline{\mathrm {Spec}}\,\mathcal O_F$ and explore their geo-ari structures, say, the corresponding Zucker conjecture, i.e. the Looijenga–Saper–Stein Theorem, and Borel's fundamental works on arithmetics of $F$, among others. If time allows, we will also explain a conjectural abelianization theory for these non-commutative spaces, motivated by the works of Donagi-Gaitsgory on Higgs bundles in geometry and our own works on the rank $n$ zeta $\widehat\zeta_{F,n}(s)$ of $F$.
While not that much traditional, I feel very much regretted to mention that this work has not been supported by any Kakenhi (research fund) from JSPS.
Paul Wiegmann
Wess–Zumino–Novikov multivalued functional in fluid mechanics (online)
We discuss a deformation of hydrodynamics by the Novikov multivalued functional generated by combined actions of gauge transformations and spacetime diffeomorphisms. This deformation is structured analogously to the Wess–Zumino functional, commonly used as an effective description of field theories with the chiral anomaly.