Venue:
SISSA Main Building,
Wednesday 16:00-17:00, Room 136.
Organized by:
Matteo Gallone
Davide Guzzetti
22/5/2024 - Changyu Zhou (University of Tohoku, Japan) - Kovalevskaya exponents on quasi-homogeneous integrable vector field
Abstract: Regular weight, a tuple of positive satisfying certain conditions related to singularity theory, are classified and proved to be associated with differential equations satisfying Painlevé property. For 2 and 4-dim cases, Painlevé equations defined by semi-quasihomogeneous Hamiltonian functions can be classified by their weights and Kovalevskaya exponents. Motivated by the observation of patterns in 4-dim Painlevé equation's Kovalevskaya exponents, further research is conducted to construct a lower indicial locus from the principle one.
21/3/2024 - Pavlo Gavrylenko (SISSA) - Isomonodromic tau functions and Fredholm determinants.
Abstract: Isomonodromic deformation problems are the most natural non-autonomous generalizations of the integrable systems. Unlike the latter ones, the former were believed to be highly transcendental. However, after the work of Gamayun, Iorgov, and Lisovyy it became clear that solutions of the isomonodromic systems can have explicit combinatorial expansions in terms of conformal blocks/Nekrasov functions. The most elementary way to the proof of such formulas goes through the block Fredholm determinants of a special form. Representation in terms of such determinants also allows one to find the connection constants between the isomonodromic tau functions in different domains of the complex moduli space. In the talk I will give the general overview of different classes of isomonodromic deformations, and then focus on the 2*2 problem on a torus with one puncture.
NOTE: Room 136 h 16:00
20/3/2024 - Guilherme Feitosa de Almeida (University of Mannheim) - 1D Landau Ginzburg Superpotential of Big Quantum Cohomology of CP2
Abstract: Using the inverse period map of the Gauss-Manin connection associated with $QH^{*}(\mathbb{CP}^2)$ and the Dubrovin construction of Landau-Ginzburg superpotential for Dubrovin Frobenius manifolds, we construct a one-dimensional Landau-Ginzburg superpotential for the quantum cohomology of $\mathbb{CP}^2$. In the case of small quantum cohomology, the Landau-Ginzburg superpotential is expressed in terms of the cubic root of the j-invariant function. For big quantum cohomology, the one-dimensional Landau-Ginzburg superpotential is given by Taylor series expansions whose coefficients are expressed in terms of quasi-modular forms. Furthermore, we express the Landau-Ginzburg superpotential for both small and big quantum cohomology of $QH^{*}(\mathbb{CP}^2)$ in closed form as the composition of the Weierstrass $\wp$-function and the universal coverings of $\mathbb{C} \setminus (\mathbb{Z} \oplus e^{\frac{\pi i}{3}}\mathbb{Z})$ and $\mathbb{C} \setminus (\mathbb{Z} \oplus z\mathbb{Z})$ respectively. This seminar is based on the results of arxiv.org/abs/2402.09574.
19/3/2024 - Luisa Andreis (Politecnico di Milano) - Spatial particle processes with coagulation: Gibbs-measure approach, gelation and Smoluchowski equation.
Abstract: We consider a spatial Markovian particle system with pairwise coagulation: after independent exponential random times, particle pairs merge into a single particle, and their masses are summed. We derive an explicit formula for the joint distribution of the particle configuration at a given fixed time, which involves the binary trees describing the history of how each of the particles has been formed via coagulations. While usually these processes are studied with the help of PDE (generalisation of the well-known Smoluchowski equation), our approach comes from statistical mechanics. The description is indeed in terms of a reference process, a Poisson point process of point group distributions, where each of the histories is an independent tree, and the non-coagulation between any two of them induces an exponential pair-interaction. Based on this formula, we can give a (conditional) large-deviation principle for the joint distribution of the particle histories in the limit of many particles with explicit identification of the rate function. We characterise its minimizer(s) and give criteria for the occurrence of a gelation phase transition, i.e., a loss of mass in the limiting configuration.
This talk is based on a joint work with W. Konig, H. Langhammer and R.I.A. Patterson (WIAS Berlin).
NOTE: Room 136 h 16:00
6/3/2024 - Gabriele Rembado (University of Montpellier) - Moduli spaces of wild connections: deformations and quantisation
Abstract: Moduli spaces of logarithmic connections on Riemann surfaces have a rich geometric structure, and in particular can be made into symplectic spaces isomorphic to (complex) character varieties. The general wisdom is that the moduli spaces are obtained by gluing local pieces at each simple pole, involving choices of coadjoint orbits for a (dual) complex reductive Lie algebra: in particular quantising such orbits is a preliminary step towards quantising the moduli spaces themselves.
In this talk we will aim at a review of this story, and then describe a recent extension for irregular singular (= wild) meromorphic
connections. The quantisation of the corresponding orbits is based upon a result of Alekseev--Lachowska, and is joint work with D. Calaque, G. Felder, and R. Wentworth. (The main ingredient are the Shapovalov form for certain representations of truncated current Lie algebras, generalising the (generalised) Verma modules.)
If time allows, we will also recall how the moduli spaces can be deformed, and describe the universal space of local deformations: this is joint work with P. Boalch, J. Douçot, and M. Tamiozzo.
NOTE: Online - meeting in room 136
28/2/2024 - Mikhail Bershtein (The University of Edinburgh) - Blowup relations, vertex algebras, and local systems
Abstract: The subject of the talk lies on the interface of algebraic geometry, representation theory, and differential equations. The blowup equations provide a very effective method for studying invariants of complex surfaces. They can be proven using representation theory, namely they follow from the isomorphisms between vertex algebras. Similar isomorphisms can be used to derive formulas for tau functions of isomonodromic deformations of linear differential equations and to obtain new Selberg type integrals. Furthermore, these isomorphisms can be lifted to the level of quantum groups (toroidal algebras), where connections with cluster varieties arise.
NOTE: Online - meeting in room 136
31/1/2024 - Veronica Fantini (IHES, Paris) - Solving regular singular Volterra equations
Abstract: Solving ODEs in the complex domain with irregular singularities can be done either formally or analytically. At the formal level, a method by Poincaré allows to build a frame of formal (and divergent) solutions. Then, it's possible to use Borel summability methods to "re-sum" the formal solutions and find an analytic frame. In this talk I'll discuss a different approach showing how to build a frame of analytic solutions by turning the ODE into an integral Volterra type equation. This is based on a joint work with A. Fenyes.
NOTE: Unusual time! The seminar will start at 16:30.
30/1/2024 - Pierpaolo Mastrolia (Università di Padova) - Algebraic Structures of Feynman Integrals and Scattering Amplitudes
Abstract: Scattering Amplitudes play a pivotal role in the context of Fundamental Physics, describing some of the most fascinating events taking place from the tiniest to the largest regions of our Universe, from the quantum interactions of elementary particles to the merging of black-holes giving rise to gravitational waves. The evaluation of the scattering amplitudes requires the calculation of Feynman Integrals. The latter can be considered as regulated period integrals, like Euler-Mellin Integrals and GKZ-hypergeometric functions, and have been found to admit a vector space structure, recently discovered by applying De Rham co-homology theory. The analytic-algebraic properties of these classes of integrals offer a unique playground for the interplay of Differential and Algebraic Geometry, Topology, Number Theory and Theoretical Physics. Overdetermined system of polynomial equations, ideal decomposition, D-module decomposition, difference equations, Pfaffian systems of differential equations, global residues, Macaulay matrices, Morse theory, Groebner bases, finite fields arithmetics, intersection numbers are just a few of many mathematical concepts that emerge during the evaluation of Feynman integrals. In this seminar, I will elaborate on a few of those mathematical aspects that could offer potential paths for developing common research initiatives between Mathematics and Theoretical Physics.
NOTE: Unusual day, time and room! Seminar will take place in room 134 at 15:30.
11/12/2023 - Jonathan Husson (University of Michigan) - Random matrices, large deviations and spherical integral
Abstract: Random matrix theory was pioneered in the 1920s by the statistician John Wishart to tackle empirical covariance matrices and in the 1950s by Eugene Wigner to study heavy nuclei. It has since expanded to multiple branches of mathematics from number theory to statistical physics. In this talk we will consider recent advances regarding the large deviations of spectra of random matrices whose dimension goes to infinity. In other words we want to quantify how fast the probability that those spectra (and in particular the largest eigenvalue) being close to a value different from their limit vanishes. Linked to this question, we also consider the asymptotics of spherical integrals, which in addition to being rich mathematical objects, have played an instrumental role in the previously mentioned advances.
NOTE: Unusual day! Seminar will be online at 16:00-17:00 on Monday 11/12.
22/11/2023 - Pierre Karim Emile Lazag (SISSA) - Hyperuniform determinantal point processes
Abstract: An infinite point process (= an infinite discrete random set of points) is said to be hyperuniform along an exhaustion by bounded sets if the ratio (variance of the number of points)/(expectation of the number of points) goes to zero along the exhaustion. In this talk, I will present geometric methods that allow to derive hyperuniformity or non-hyperuniformity for determinantal point processes that are invariant by isometries. In particular, we obtain that translation invariant determinantal point processes on R^d are hyperuniform along any exhaustion by open sets, while determinantal point processes on Gromov hyperbolic metric spaces - which include the known examples on Cayley trees and standard real or complex hyperbolic spaces H^d - are never hyperuniform. If time permits, I will discuss applications to accumulated spectrograms together with some analogy with quantum ergodicity.