| Date | Speaker | Title | Abstract | Note |
|---|---|---|---|---|
| 23.01.25 | Loredana Lanzani (University of Bologna) | A Numerical Method for the Solution of Boundary Value Problems on Lipschitz Planar Domains | The Unified Transform Method (UTM) was pioneered in the early ’90s by A. S. Fokas and I. M. Gel’fand in their study of the numerical solution of boundary value problems for elliptic PDEs and for a large class of nonlinear partial differential equations (PDEs). The UTM provides a connection between the Fourier Transform method (FT) for linear PDEs and its nonlinear counterpart, namely the Inverse Spectral method– also known as Non Linear Fourier Transform method (NLFT). At the heart of the matter is a new derivation of the FT method for linear equations in one and two (space) variables that follows the same conceptual steps needed to implement the NLFT method for a class of nonlinear evolution equations, thus pointing to a unified approach to the numerical solution of linear and nonlinear PDEs. From the very beginning, the UTM has attracted a great deal of interest in the applied mathematics community. A multitude of versions of the original method have since been developed, each dealing with a specific family of equations. Here we focus on a 2003 result of A.S. Fokas and A.A. Kapaev pertaining to the study of boundary value problems for the Laplacian on convex polygons: their original approach relied on a variety of tools (spectral analysis of a parameter-dependent ODE; Riemann-Hilbert techniques, etc.) but it was later observed by D. Crowdy that the method can be recast within a complex function theoretic framework (the Cauchy integral) which, in turn, extends the applicability to so-called circular domains, namely domains bounded by arcs of circles (with line segments being a special case). We extend the original approach of Fokas and Kapaev for polygons, to arbitrary convex domains. It turns out that ellipses (which are not circular in the sense of Crowdy) are of particular relevance in applications to engineering because the most popular heat exchangers (the shell-and-tube exchangers) have elliptical cross section. In this talk I will describe a complex function-theory based new algorithm for convex domains, and will highlight the numerical challenges that arise when implementing it. Time permitting, I will describe some very recent new results where further tools from complex analysis (the Szegő projection and conformal mapping) allow the application of the above algorithm also to non-convex domain shapes. This is joint work with J. Hulse (Syracuse University), S. Llewellyn Smith (UCSD & Scripps Institute of Oceanography) and Elena Luca (The Cyprus Institute). |
4pm, HMI Seminar room |
| 30.01.25 | Mariam Al-Hawaj (TCD) | Generalized pseudo-Anosov Maps and Hubbard Trees | The Nielsen-Thurston classification of the mapping classes proved that every orientation preserving homeomorphism of a closed surface, up to isotopy is either periodic, reducible, or pseudo-Anosov. Pseudo-Anosov maps have particularly nice structure because they expand along one foliation by a factor of λ > 1 and contract along a transversal foliation by a factor of 1/λ. Pseudo-Anosovs have a huge role in Teichmuller theory and geometric topology. The relation between these and complex dynamics has been well studied inspired by Thurston. In this project, I develop a new connection between the dynamics of quadratic polynomials on the complex plane and the dynamics of homeomorphisms of surfaces. In particular, given a quadratic polynomial, we show that one can construct an extension of it which is generalized pseudo-Anosov homeomorphism. Generalized pseudo-Anosov means the foliations have infinite singularities that accumulate on finitely many points. We determine for which quadratic polynomials such an extension exists. My construction is related to the dynamics on the Hubbard tree which is a forward invariant subset of the filled Julia set that contains the critical orbit. |
4pm, HMI Seminar room |
| 06.02.25 | Adam Keilthy (TCD) | Reconnectads, Contractads, and extending Arnold's Trinities | In the 90s, Arnold proposed that important mathematical objects should come in "trinities" - objects existing in triples, that somehow natural go together. His initial example was the triple of the reals, the complex numbers, and the quaternions, but the example used to introduce this idea to me was the algrebraic gravity package, coming from the study of cohomological field theories. In the space of symmetric operads, there is a set of operads (HyperComm, Gerst, Grav, BV) that can be explictly related to each other. Dotsenko, Shadrin, and Vallette introduce non-symmetric analogues of all of these, satisfying the same relations, before later completing the trinity with a set of four twisted associative algebras. Remarkably, almost every result about the structure of any of these three sets holds for all of them, with no clear reason as to why. In this talk I will introduce two frameworks for explaining this similarity, based in the combinatorics of graphs. One centred around the operation of reconnection, called a reconnectad, lets us view all three of the algebraic gravity packages (modulo BV) as substructures of a corresponding reconnectad. The other comes from contraction, and we can similarly view all three of the algebraic gravity packages (modulo BV) as substructures of a corresponding contractad. In both cases, we extend this Arnold trinity to a pantheon of similar structures. We will compare the two frameworks, with a particular focus on a topological reconnectad and contractad defined in terms of compactifications of hyperplane arrangements. If time allows, we will discuss a further unifying generalisation of these structures and a potential application to the study of mixed Tate motives. |
4pm, HMI Seminar room |
| 13.02.25 | Christian Ketterer (Maynooth University) | Characterization of Gromov-Hausdorff-almost-flat tori and applications to Scalar curvature stability problems | One of the goals of this talk is to establish a nonlinear analogue of almost splitting maps into Euclidean space, as harmonic maps into a flat torus. Existence of such maps implies Gromov-Hausdorff closeness to a flat torus in any dimension. Combining these results with a Bochner-type inequality by Stern yields a new Gromov-Hausdorff stability theorem for flat 3-tori with almost nonnegative Scalar curvature bounds. Our proof uses the differential calculus on nonsmooth metric measure space that satisfy a synthetic lower Ricci curvature bound, so called RCD spaces. I will give a short introduction to this theory, and I will highlight its key feature: the geometric and functional analytic stability under measured Gromov-Hausdorff convergence. This is partly a joint project with Shouhei Honda, Ilaria Mondello, Raquel Perales and Chiara Rigoni. |
4pm, HMI Seminar room |
| 20.02.25 | Oscar Randal-Williams (University of Cambridge) | Cohomology of Moduli Spaces: A Case Study | I will explain recent work of Bergström–Diaconu–Petersen–Westerland, and of Miller–Patzt–Petersen–R-W, which uses methods developed over the last 25 years for studying the topology of certain moduli spaces in order to answer a question in arithmetic statistics (the function field analogue of a conjecture of Conrey–Farmer–Keating–Rubinstein–Snaith on moments of quadratic L-functions). My focus will be on the translation of this question to a problem in topology, and some of the modern methods which go into solving this problem. | 4pm, HMI Seminar room |
| 27.02.25 | Anne Moreau (Université Paris-Saclay, HMI Simons Visiting Professor) | Isomorphisms between W-algebras | To any vertex algebra one can attach invariants of different nature: its automorphism group, its character (a formal series), its associated variety (a Poisson variety), etc. In this talk, I will explain how to exploit the connection between these invariants to obtain nontrivial isomorphisms between W-algebras at admissible levels. To study a more general setting, one can use totally different technics developed more recently. |
4pm, HMI Seminar room |
| 13.03.25 | Viola Giovannini (University of Luxembourg) | Renormalized Volume Behaviour for Convex Co-Compact Hyperbolic 3-Manifolds with Compressible Boundary | Given a hyperbolizable 3-manifold N, the renormalized volume is a real-valued function on the space of convex co-compact hyperbolic structures on the interior of N, which always have infinite hyperbolic volume. When the boundary of N is incompressible the renormalized volume is always non-negative, otherwise it has infimum $-\infty$. Through the study of its differential, which is identified with the real part of the Schwarzian derivative assocaited to a convex co-compact structure, we present some results on the behaviour of the renormalized volume for manifolds with compressible boundary. | 11am, Research Room |
| 13-14.03.25 | 220th anniversary of W. R. Hamilton and 20th anniversary of the HMI | |||
| 20.03.25 | Patrick Dondl (Universität Freiburg) | Phase field models with connectedness constraints | Phase field models are useful tools for the approximation of geometric variational problems, the classical example being the Modica-Mortola-functional consisting of a gradient penalization and a double-well potential. This functional, with its terms suitably scaled, converges in the sense of Gamma-convergence to the perimeter, implying that the zero level sets of its minimizers approximate minimal surfaces. We consider the problem of including topological constraints in phase field models, in particular the question whether it is possible to constrain the zero level sets or sub/super-level sets to be connected by adding a suitable term to the energy. Our penalty term is based on a diffuse quantitative version of path-connectedness. As a first application, we prove convergence of the approximating penalized Modica-Mortola energies in the sense of Gamma-convergence to a connected perimeter and present numerical results and applications to image segmentation and Ohta-Kawasaki functionals modeling charged droplets, as well as the applications to fiber reinforced membranes. Furthermore, we consider a phase field model for Willmore's energy. The main problem in this case is that, to enforce connectedness of surfaces in the limit, a fairly strong convergence for sequences of bounded diffuse Willmore energy on sets of codimension 2 is required. In two space dimensions, uniform convergence (away from the limiting surface) holds, for the three dimensional case we introduce the notion of morally uniform convergence. |
4pm, HMI Seminar room |
| 27.03.25 | Thomas Blomme (Université de Neuchâtel) | A short proof of the multiple cover formula | Enumerating genus g curves passing through g points in an abelian surface is a natural problem, whose difficulty highly depends on the degree of the curves. For "primitive" degrees, we have an easy explicit answer. For "divisible" classes, such a resolution is quite demanding and often out of reach. Yet, the invariants for divisible classes easily express in terms of the invariants for primitive classes through the multiple cover formula, conjectured by G. Oberdieck a few years ago. In this talk, we'll show how tropical geometry enables to prove the formula without any kind of concrete enumeration. | 4pm, HMI Seminar room |
| 03.04.25 | Paul Breiding (University of Onsarbrück) | Probabilistic Intersection Theory in Riemannian homogeneous spaces | In this talk we will examine the intersection of randomly moved submanifolds $Y_1, \ldots, Y_s$ in a Riemannian homogeneous space $M = G/H$, where $G$ is a compact Lie group and $H$ is a closed subgroup. We will investigate the so-called probabilistic intersection ring of $M$, whose multiplication encodes the average unsigned count of intersection points when the $Y_i$ are moved by independently and uniformly chosen elements of $G$. Specifically, we will first study the zonoid algebra and then we will define the probabilistic intersection ring of a Riemannian homogeneous space $M$. Finally, we will provide details of the probabilistic intersection ring of complex projective space. | 4pm, HMI Seminar room |
| 08.04.25 | Ieke Moerdijk (Utrecht University) | Operads, configuration spaces and graphs | The notion of operad arose from an attempt to understand the stucture of iterated loop spaces in topology, and the "little disks operad" invented for this purpose still place a central role in the theory. Many more combinatorial models for this operad have been designed for various purposes. I will discuss one in particular based on graphs, and explain its relation to the original graph operad via configuration spaces. | 4pm, HMI Seminar room |
| 09.04.25 (Wednesday) | Balazs Szendroi (University of Vienna) | Polytopes, graded rings and coinvariant algebras | The coinvariant algebra, the quotient of the polynomial ring in n variables by the ideal generated by positive degree symmetric polynomials, plays a basic role in algebraic combinatorics and the representation theory of the symmetric group, equipping its regular representation with a graded algebra structure. Based on an idea from algebraic geometry, I introduce a degeneration of the coinvariant algebra, the projective coinvariant algebra. This algebra gives a bigraded structure on the regular representation of the symmetric group with interesting Frobenius character, generalising a classical result of Lusztig and Stanley. Interesting subalgebras of this algebra include finite-dimensional truncations of all Segre embeddings of products of projective spaces. There are also connections to Ehrhart theory, the theory of counting lattice points in polytopes, as well as to cohomology rings of certain symmetric varieties. Based on joint work with Praise Adeyemo, University of Ibadan, as well as my PhD student Fabian Levican. | 2:30pm, HMI Seminar room |
| 10.04.25 | Radu Balan (University of Maryland) | G-Invariant Representations using Coorbits | Consider a finite dimensional real vector space and a finite group acting unitarily on it. We study the general problem of constructing Euclidean stable embeddings of the quotient space of orbits. Our embedding is based of sorted coorbits. We obtain sufficient conditions for injective and stable embeddings. In particular, we show that, whenever such embeddings are injective, they are automatically bi-Lipschitz. Additionally, we demonstrate that stable embeddings can be achieved with reduced dimensionality, and that any continuous or Lipschitz G-invariant map can be factorized through these embeddings. This talk is based on joint works with Efstratios Tsoukanis and Matthias Wellershoff. arXiv: 2308.11784 , 2310.16365, 2410.05446. |
5pm, HMI Seminar room |
| Date | Speaker | Title | Abstract | Note |
|---|---|---|---|---|
| 12.09.24 | Wend Werner(University of Münster) | Enveloping algebras of bounded Hilbert space operators | Viewing the enveloping algebra U(g) of a Lie algebra g as an algebra of physical invariants it is of interest to see under what circumstances the elements of U(g) act as bounded operators on a Hilbert space. It turns out that this is true if and only if g is nilpotent.
We outline the basic ideas leading to this result and sketch where we would like to go from here. |
4pm, Salmon |
| 19.09.24 | Piero D'Ancona (Sapienza Università di Roma) | Dispersion estimates for Dirac equations with Aharonov–Bohm magnetic fields | We examine the dispersive properties of a two dimensional Dirac operator perturbed by a critical Aharonov--Bohm potential. The flow can be split into a dispersive part which decays like in the unperturbed case, plus a singular component with weaker decay. For a partial range of indices, we deduce sharp Strichartz estimates for the flow. This is a joint work with Federico Cacciafesta, Zhiqing Yin and Junyong Zhang. | 4pm, Salmon |
| 26.09.24 | Lev Birbrair, Universidade Federal do Ceará (Brasil) & Jagiellonian University (Poland) | Germs of Real Surfaces. Geometry and metric structure. | I am going to describe old and new results related to Inner, Outer and Ambient Lipschitz geometry of germs of Real semi-algebraic and definable surfaces. The subject is closely related to non-archimedean geometry and Knot Theory. I am going to make an overview. No preliminary knowledge is required. | 4pm, HMI Seminar room |
| 03.10.24 | Jan Steinebrunner (University of Copenhagen) | Moduli spaces of 3-manifolds with boundary are finite | In joint work with Rachael Boyd and Corey Bregman we study the classifying space BDiff(M) of the diffeomorphism group of a connected, compact, orientable 3-manifold M. I will recall the construction of this BDiff(M), which is also called the "moduli space of M", and explain how it parametrises smooth families of manifolds diffeomorphic to M. Using Milnor's prime decomposition and Thurston's geometrisation conjecture we can cut M into "geometric pieces", for which we have a better understanding of BDiff. The purpose of this talk is to explain a technique for computing the moduli space BDiff(M) in terms of the moduli spaces of the pieces. We use this to prove that if M has non-empty boundary, then BDiff(M rel boundary) has the homotopy type of a finite CW complex, as was conjectured by Kontsevich. | 4pm, HMI Seminar room |
| 10.10.24 | Markus Land (LMU Munich) | Homology Manifolds and Euclidean Bundles | I will begin by briefly explaining the role of homology manifolds in the (surgery) classification of manifolds. Then I will recall in more detail what homology manifolds and Poincare duality complexes are, and how they are distinguished by means of L-theoretic invariants, and explain Ranicki's total surgery obstruction. Finally, I will come to recent results, joint with Hebestreit-Winges-Weiss, which invalidate certain claims in the literature on homology manifolds. | 4pm, HMI Seminar room |
| 14.10.24 | Andras Szenes (University of Geneva) | Refined intersection forms and the decomposition of cohomology along the parabolic projection map | I will give a gentle introduction to the Decomposition Theorem, and describe an application of this approach worked out jointly with Camilla Felisetti and Olga Trapeznikova. | 4pm, New Seminar room |
| 17.10.24 | Anna Lachowska (EPFL) | The center and Hochschild cohomology of the small quantum group | Representation theory of quantum groups at roots of unity is connected with various problems in vertex algebras, CFT, topological invariants and modular representation theory. We have evidence of strong connection between the structure of the center of the small quantum group at a root of unity associated to the Lie algebra $g$ and the geometry of the group G. I will review the results on the center and higher Hochschild cohomologies of the small quantum group, in particular in the case of $g = sl_2$. This is a joint work with Qi You. | 4pm, HMI Seminar room |
| 06.11.24 (Wednesday) | Tamas Hausel (Institute of Science and Technology Austria) | Anatomy of big algebras | Big algebras are maximal commutative subalgebras of the Kirillov algebra attached to irreducible representations of a complex semisimple Lie group. In this talk we will visualize the anatomy of big algebras, in terms of skeletons and nerves. We show that it encodes sophisticated information of the representation, such as a ring structure on multiplicity spaces, the weight diagram and the crystal structure. | 11am, HMI Seminar room |
| 07.11.24 | Francesca Carocci (University of Rome Tor Vergata) | Correlated Gromov-Witten invariants | In this talk we introduce a geometric refinement of Gromov-Witten invariants for P1-bundles relative to the natural fiberwise boundary structure. We call these refined invariants correlated Gromov-Witten invariants. We will introduce the correlated invariants, discuss their properties and provide some computations in the case of P1-bundles over an elliptic curve. Such invariants are expected to play a role in the degeneration formula for reduced Gromov-Witten invariants for abelian and K3 surfaces. This is a joint work with Thomas Blomme. | 4pm, HMI Seminar room |
| 14.11.24 | Lars Kühne (UCD) | Unlikely intersections in algebraic and diophantine geometry | Intersections of subvarieties in a given ambient space are among the most well-studied topics in algebraic geometry. However, algebraic intersection theory is largely ineffective if the codimensions "do not add up" and an intersection should not even occur generically, for example for two curves in a three-dimensional projective space. Unfortunately, such "unlikely" intersections do occur, and controlling them has proven to be of essential importance in diophantine geometry. An argument by Pink, for example, shows that the celebrated Mordell-Lang conjecture proven by Faltings would follow from reasonable finiteness assertions on certain unlikely intersections, which are now known as the Zilber-Pink conjectures. These have been studied in various contexts, mostly in abelian and Shimura varieties, with partial results. In my talk, I will focus on some recent work by Fabrizio Barroero, Gabriel Dill, and myself in the context of abelian varieties, presenting results in a previously unknown case of the Zilber-Pink conjectures within this setting. Although the conjectures themselves are purely geometric statements, our proposed proofs rely heavily on tools from arithmetic geometry. |
4pm, HMI Seminar room |
| 21.11.24 | Nadine Große (Universität Freiburg) | Boundary value problems on singular domains | We give a geometric approach for boundary value problems of the Laplacian with Dirichlet (or mixed) boundary conditions on domains with singularities. In two dimensions these singularities also include cusps. Our approach is by blowing up the singularities via a conformal change to translate the boundary problem to one on a noncompact manifold with boundary that is of bounded geometry and of finite width. This gives a natural geometric interpretation for the weights that appear and for the additional conditions needed to obtain well-posedness results. This is joint work with Bernd Ammann and Victor Nistor. | 4pm, HMI Seminar room |
| Date | Speaker | Title | Abstract | Note |
|---|---|---|---|---|
| 01.02.24 | Vladimir Dotsenko (University of Strasbourg) | Lie algebra homology and wheeled operads | I shall recall the definition of a wheeled operad (introduced by Merkulov about 15 years ago) and explain how homotopy invariants of wheeled operads appear naturally when computing stable homology of Lie algebras of derivations of free algebras. This is a common generalization of the Loday-Quillen-Tsygan theorem on additive K-theory of an associative algebra, and the Fuchs' stability theorem for homology of the Lie algebra of vector fields. | 4pm, Synge |
| 15.02.24 | Brendan Guilfoyle (Munster Technological University) | From Euclidean 3-space to pseudo-Riemannian spaces without relativity | The relativisation of Euclidean 3-space to Minkowski 4-space is not the only way in which metrics of indefinite signature arise in a natural way in physical problems. In particular, embedding questions often reduce to under-determined hyperbolic systems (generalisations of the Codazzi-Mainardi equations) which can admit unexpected geometric reductions, even in classical settings. This talk will describe two examples: pseudo-Riemannian geometrizations of the 1-jet and 2-jet space of Euclidean 3-space. They are both of neutral signature, the former in 4 dimensions and the latter in 2 dimensions, and they arise in X-ray tomography and the erosion of a pebble on a beach, respectively. The talk will be aimed for a general mathematical audience. | 4pm, Synge |
| 29.02.24 | Markus Reineke (Ruhr University Bochum) | Expander representations | Expander graphs - graphs which are both sparse and highly connected - have numerous applications in both applied and pure mathematics. Linear algebra analogues, called dimension expanders, were constructed and studied by Bourgain, Lubotzky, Wigderson, Zelmanov. We will discuss an approach to dimension expanders via quivers yielding exact expansion rates, and present a generalization to the concept of expander representations. | 4pm, Synge |
| 14.03.24 | Tommaso Cremaschi (TCD) | Renormalised Volume of Hyperbolic 3-manifolds | I will give an overview on convex co-compact hyperbolic 3-manifolds and describe some natural notions of volume focusing on the case of renormalised volume and give some classical bounds. I will then describe some work in progress and open questions. | 4pm, Synge |
| 21.03.24 | Roberta Filippucci (Universita degli Studi di Perugia) | A priori estimates and Liouville type results for elliptic problems | In this talk, we deal with a priori estimates and Liouville type results for elliptic problems, also when a reaction involving gradient terms is included in the equation. In particular, we survey classical results for the Emden-Fowler equation, for the m-Laplacian operator and for systems of equations and inequalities, including connections between Liouville-type theorems and local properties of nonnegative solutions to elliptic problems. | 4pm, Synge |
| 21.03.24 | Martin Kolář (Masaryk University) | Equivalence and symmetries of uniformly Levi degenerate hypersurfaces in C^n | Uniformly Levi degenerate real hypersurfaces in C^n play an important role in CR-geometry and the theory of Hermitian Symmetric Domains. In this talk, I will first give an introduction to Moser's normal form approach in the study of geometry of real submanifolds in the complex n-dimensional space. This approach will then be applied to obtain a complete convergent normal form for everywhere 2-nondegenerate hypersurfaces in C^3 (joint work with Ilya Kossovskiy). The construction is based on using a rational (nonpolynomial) model for everywhere 2-nondegenerate hypersurfaces, which is the local realization due to Fels-Kaup of the well known tube over the light cone. As an application, we obtain a criterion for the local sphericity (i.e. local equivalence to the model) for a 2-nondegenerate hypersurface in terms of its normal form. As another application, we obtain an explicit description of the moduli space of everywhere 2-nondegenerate hypersurfaces. I will also discuss some new results in higher dimensions. | 5pm, Synge |
| 28.03.24 | Juan José Salazar González (Universidad de la Laguna) | Mathematics to help finding another planet to live | Astronomers are exploring the sky looking for exoplanets. This talk concerns a new optimization problem arising in the management of a multi-object spectrometer. The field of view of the spectrograph is divided into contiguous and parallel spatial bands, each one associated with two opposite sliding metal bars that can be positioned to observe one astronomical object. Thus several objects can be analyzed simultaneously within a configuration of the bars called a mask. Due to the high demand from astronomers, pointing the spectrograph’s field of view to the sky, rotating it, and selecting the objects to conform a mask is a crucial optimization problem for the efficient use of the spectrometer. We describe this optimization problem, present a Mixed Integer Linear Programming formulation for the case where the rotation angle is fixed, present a non-convex formulation for the case where the rotation angle is unfixed, describe a heuristic approach for the general problem, and discuss computational results on real-world and randomly-generated instances. | 4pm, Synge |
| 04.04.24 | Daniel Douglas (Max Planck Institute for Mathematics in the Sciences in Leipzig) | SL(n) skein algebras of surfaces | In Edward Witten’s landmark 1989 paper “Quantum Field Theory and the Jones Polynomial”, to a closed 3-manifold equipped with a colored link (a color being a finite dimensional representation of SU(n)) is assigned a numerical invariant, computed by taking the Feynman path integral of the Chern-Simons functional over all gauge equivalence classes of SU(n) connections on the manifold. This invariant also depends on the quantum level k. In the classical limit as k goes to infinity, only the stationary points of the Chern-Simons functional contribute, namely the flat connections up to gauge equivalence. In other words, the stationary points correspond to the *character variety* consisting of group homomorphisms from the fundamental group of the manifold to SU(n) up to conjugation. *Skein modules* are deformation quantizations of character varieties. A link in the 3-manifold represents an element of the skein module. In this talk, we will focus on SL(n) skein modules where the 3-manifold is a thickened surface, such as a solid torus (a thickened annulus). In this setting, skein modules acquire an algebra structure by stacking diagrams with respect to the direction of thickening. A challenging problem is to construct canonical generating sets for these *skein algebras*. It turns out that, in addition to links, it is advantageous to enlarge the class of topological objects being considered to include certain n-valent graphs called *webs*. We will discuss recent work in this direction, joint with Tommaso Cremaschi. |
4pm, Synge |
| 11.04.24 | Guillaume Chapuy (Institut de recherche en informatique fondamentale) | Random triangulations of surfaces, and the high-genus regime. | I will talk about the probabilistic behaviour of random maps on surfaces (for example, random triangulations) of given genus, when their size tends to infinity. Such questions can be asked from the viewpoint of the local behaviour (Benjamini-Schramm convergence) or global behaviour (diameter, Gromov Hausdorff convergence), and in both cases, much combinatorics is involved. I will survey the landmark results for the case of fixed genus, and state very recent results obtained jointly with Thomas Budzinski and Baptiste Louf, in which we manage to address the "high genus" regime, when the genus grows proportionally to the size -- for this regime we establish isoperimetric inequalities and prove the long-suspected fact that the diameter is logarithmic with high probability. | 4pm, Zoom with screening in Synge |
| Date | Speaker | Title | Abstract | Note |
|---|---|---|---|---|
| 28.09.23 | Lucien Hennecart (University of Edinburgh) | Cohomological integrality for 2-Calabi-Yau categories | In this talk, I will explain how to define BPS invariants of a large class of Abelian 2-Calabi-Yau (2CY) categories and their sheaf refinements (as perverse sheaves or mixed Hodge modules). Examples of relevance in geometry and representation theory are sheaves on symplectic surfaces, length-zero coherent sheaves on any surface, preprojective algebras of quivers, and fundamental groups of Riemann surfaces. These BPS invariants give cohomological integrality for the categories considered, and their categorification enjoys the richer structure of a Lie algebra, which happens to be a generalised Kac-Moody Lie algebra. This Lie algebra governs the homology of the stack of objects in the 2CY category via a PBW-type isomorphism. | 4pm, Synge |
| 02.10.23 | John Erik Fornaess (Trondheim) | Transcendental Henon maps on C^2 | A Henon map is a map of the form H(z,w)=(f(z)+aw,z). These were originally introduced as maps on R^2 by the French astronomer Michel Henon to study chaotic versus stable behaviour in Celestial Mechanics. This is joint work with Leandro Arosio, Anna Miriam Benini and Han Peters. I will discuss some of our previous work and work in progress. | 4pm, HMI Seminar room |
| 05.10.23 | Hannah Markwig (Universität Tübingen) | Counting bitangents of quartic curves - arithmetic, real, tropical | We showcase tropical geometry as a tool for geometric counting problems. A nice feature of tropical geometry is that many techniques can be applied simultaneously over various ground fields, e.g. for complex or real counting problems. Our prime example will be the count of bitangent liens to a smooth plane quartic. Already Plücker knew that a smooth complex plane quartic curve has exactly 28 bitangents. Bitangents of quartic curves are related to a variety of mathematical problems. They appear in one of Arnold's trinities, together with lines in a cubic surface and 120 tritangent planes of a sextic space curve. In this talk, we review known results about counts of bitangents under variation of the ground field. Special focus will be on counting in the tropical world, and its relations to real and arithmetic counts. We end with new results concerning the arithmetic multiplicity of tropical bitangent classes, based on joint work with Sam Payne and Kris Shaw. | 4pm, Synge |
| 12.10.23 | Mark Gross (University of Cambridge) | Intrinsic Mirror Symmetry | Mirror symmetry was a phenomenon discovered by physicists around 1989: they observed that certain kinds of six-dimensional geometric objects known as Calabi-Yau manifolds seemed to come in pairs, with a strange relationship between different kinds of geometric objects on the pairs. Since then, the subject has blossomed into a vast field, with many different approaches and philosophies. I will give a brief introduction to the subject, and explain how one of these approaches, developed with Bernd Siebert, has led to a general construction of mirror pairs. | 4pm, Synge |
| 12.10.23 | Dan Abramovich (Brown University) | The Chow ring of a weighted blowup | This is mostly a report on work of Brown PhD students Veronica Arena and Stephen Obinna. The Chow groups of a blowup of a smooth variety along a smooth subvariety are described in Fulton's book using Grothendieck's "key formula", involving the Chow groups of the blown up variety, the center of blowup, and the Chern classes of its normal bundle. If interested in weighted blowups, one expects everything to generalize directly. This is in hindsight correct, except that at every turn there is an interesting and delightful surprise, shedding light on the original formulas for usual blowups, especially when one wants to pin down the integral Chow ring of a stack theoretic weighted blowup. As an application, one obtains a quick derivation of a formula, due to Di Lorenzo-Pernice-Vistoli and Inchiostro, of the Chow ring of the moduli space $\overline{M}_{1,2}$ |
5pm, Synge |
| 19.10.23 | Emanuele Dotto (University of Warwick) | Characteristic polynomials of self-adjoint endomorphisms | The algebraic properties of the characteristic polynomial of a matrix can be efficiently packaged by expressing the characteristic polynomial as a ring homomorphism from the cyclic K-theory ring to the ring of Witt vectors. This homomorphism can moreover be interpreted as the effect in $\pi_0$ of a homotopy theoretic trace map. The talk will introduce these ideas and explain how, by extending them to an equivariant context, they give rise to a refined version of the characteristic polynomial for self-adjoint endomorphisms. | 4pm, Synge |
| 02.11.23 | Minhyong Kim (University of Edinburgh) | Diophantine Equations in Two Variables | The search for integer or rational solutions of polynomials equations in two variables has had a tendency to generate the most difficult problems in mathematics for the last few thousand years, and surprisingly little progress has been made. Nonetheless, this lecture will give a highly idiosyncratic survey of what is known, concluding with the developments of the last 20 years that make heavy use of homotopy theory. | 4pm, Synge |
| 09.11.23 | Luke Edholm (University of Vienna) | Projective invariants and the Cauchy-Leray transform | The Cauchy-Leray (or Leray) transform is a higher-dimensional analogue of the planar Cauchy transform. It can be defined on $\mathbb{C}$-convex hypersurfaces in n-dimensional projective space and when the hypersurfaces have reasonable geometric regularity, it defines a skew projection onto the Hardy space of $L^2$-boundary values of holomorphic functions. This talk will focus on important projectively invariant aspects of Cauchy-Leray theory. In particular we show how analytic quantities related to the Cauchy-Leray transform are connected to geometric invariants on the hypersurface. We will look first at interesting model cases before using the models to describe more general situations. | 4pm, Synge |
| 16.11.23 | David Sykes (Masaryk University) | Absolute parallelism constructions in CR geometry | Real hypersurfaces in complex spaces inherit a geometric structure induced by restricting Cauchy–Riemann equations to the hypersurface. The basic problem of finding biholomorphisms mapping one hypersurface onto another is only well understood for a limited class of hypersurfaces, and has a fundamental relationship to their induced CR geometries. This talk will introduce a new method (developed in joint work with Igor Zelenko) for encoding CR geometries into absolute parallelisms, applicable to a class of hypersurfaces for which this has not been achieved before. We will preview the method’s applications to CR local equivalence problems, estimating symmetry group dimensions, and classifications of homogeneous structures. | 4pm, Synge |
| 23.11.23 | David Henry (University College Cork) | Energy considerations for nonlinear water waves | The analysis of water waves is an intriguing, and theoretically challenging, research area which spans a number of scientific disciplines. From the perspective of mathematical analysis, the water wave problem is rendered highly intractable by the presence of strong nonlinearities in the governing equations, and further complicated by being an unknown free-boundary problem. The energy possessed (and transported) by water waves is a topic that has been the focus of intense multidisciplinary research over recent decades, with practical applications such as those relating to marine renewable energy, for instance. However, a major deficiency in our theoretical understanding of ocean wave energy is that the state-of-the-art is largely contingent on invoking linear approximations: there is a dearth of results pertaining to the energy of fully nonlinear (that is, large amplitude) water waves. In this talk I will present a rigorous mathematical analysis approach which has recently established results concerning the energy distribution for nonlinear water waves. It is proven, for periodic nonlinear irrotational water waves, that the excess kinetic energy density is always negative, whereas the excess potential energy density is always positive. A new characterisation of the total excess energy density as a weighted mean of the kinetic energy along the wave surface profile is also presented. | 4pm, Synge |
| Date | Speaker | Title | Abstract | Note |
|---|---|---|---|---|
| 02.02.23 | Fridrich Valach (Imperial College London) | Generalising Lie and Courant algebroids | I will introduce G-algebroids, which provide a natural generalisation of several structures appearing in mathematical physics, and show how to use them to study generalised parallelisations and dualities. | 4pm, New Seminar Room |
| 16.02.23 | Nicolas Mascot (TCD) | Computations with plane algebraic curves | Suppose you are given a curve in the form of a (more-or-less horrible) plane equation f(x,y)=0. How would you determine the genus of this curve? We will present algorithms to answer this question and more, e.g. how to test whether a curve is hyperelliptic, how to put curves of small genera in normal form, and how to compute Galois representations attached to curves. We will demonstrate these techniques by handling numerous concrete examples thanks to our own implementation. | 4pm, New Seminar Room |
| 23.02.23 | Annegret Burtscher (Radboud University) | The many faces of globally hyperbolic spacetimes | The notion of global hyperbolicity was introduced by Jean Leray in 1952 to obtain global uniqueness of solutions to nonlinear wave equations. Globally hyperbolic spacetimes subsequently turned out to be the right geometric setting not only for the well-posedness of the initial value formulation for the Einstein equations in General Relativity but also for the singularity theorems of Penrose and Hawking and several splitting results in Lorentzian geometry. We review the rich history and omnipresence of global hyperbolicity in General Relativity. Then we present a surprising new characterization of globally hyperbolic spacetimes that makes use of ideas and tools from metric geometry. | 4pm, New Seminar Room |
| 02.03.23 | Julie Rasmusen (University of Warwick) | THR of Poincaré infinity-categories | In recent years work by Calmés-Dotto-Harpaz-Hebestreit-Land-Moi-Nardin-Nikolaus-Steimle have moved the theory of Hermitian K-theory into the framework of stable infinity-categories. I will introduce the basic ideas and notions of this new theory, but as it is often the case when working with K-theory in any form, this is can be very hard to understand. I will therefore introduce a tool which might make our life a bit easier: Real Topological Hochschild Homology. I will explain the ingredients that goes into constructing in particular the geometric fixed points of this as a functor, generalising the formula for ring spectra with anti-involution of Dotto-Moi-Patchkoria-Reeh. | 4pm, New Seminar Room |
| 16.03.23 | Navid Nabijou (Queen Mary University of London) | Roots and logs in the enumerative forest | Enumerative geometry has its roots in 19th-century problems concerning counts of curves on complex manifolds. I will give an introduction to the subject, focusing on a few of these classical problems and explaining their resolution via the modern edifice of Gromov-Witten theory. I will then discuss recent work, in which we uncover a deep but complicated connection between two complimentary systems of Gromov-Witten invariants. This is joint work with Luca Battistella and Dhruv Ranganathan. | 4pm, HMI Seminar room |
| 23.03.23 | Oscar Garcia-Prada (ICMAT, Madrid) | Higgs bundles and higher Teichmüller spaces | It is well-known that the Teichmüller space of a compact real surface can be identified with a connected component of the moduli space of representations of the fundamental group of the surface in PSL(2,R). Higher Teichmüller spaces are generalizations of this, where PSL(2,R) is replaced by certain simple non-compact real Lie groups of higher rank. As for the usual Teichmüller space, these spaces consist entirely of discrete and faithful representations. Several cases have been identified over the years. First, the Hitchin components for split groups, then the maximal Toledo invariant components for Hermitian groups, and more recently certain components for SO(p,q). In this talk, I will describe a general construction of all possible higher Teichmüller spaces, and a parametrization of them using the theory of Higgs bundles, given in joint work with Bradlow, Collier, Gothen and Oliveira. | 4pm, HMI Seminar Room |
| 30.03.23 | Manuel Krannich (Karlsruher Institut für Technologie) | Pontryagin classes of Euclidean fibre bundles over spheres | Pontryagin classes were originally considered as invariants of real vector bundles, but thanks to work of Novikov, Sullivan, and Kirby--Siebenmann it was realised in the 60s that these invariants in fact do not depend on the linear structure on the fibres and can be defined more generally for Euclidean fibre bundles, that is, fibre bundles whose fibres are homeomorphic to Euclidean space. This led to the question whether the well-known fact that the k th Pontryagin class of a d-dimensional vector bundle vanishes for k > d/2 continues to hold in the setting of Euclidean fibre bundles. Surprisingly, Michael Weiss proved a few years ago that this often fails, even for bundles over spheres. I will explain a strengthening of Weiss’ result resulting from recent joint work with A. Kupers: For every k > 0, there exists a 6-dimensional Euclidean fibre bundle over a sphere whose k th Pontryagin class is nontrivial. | 4pm, HMI Seminar Room |
| 06.04.23 | Rhiannon Savage (Oxford) | Derived Geometry Relative to Monoidal Quasi-abelian Categories | In the theory of relative algebraic geometry, we work relative to a symmetric monoidal category C. The affines are now objects in the opposite category of commutative algebra objects in C. The derived setting is obtained by working with a symmetric monoidal model or ∞-category C, with derived algebraic geometry corresponding to the case when we take C to be the category of simplicial modules over a simplicial commutative ring k. Kremnizer et al. propose that derived analytic geometry can be recovered when we replace k with a simplicial commutative complete bornological ring. We work more generally with simplicial commutative algebra objects in certain quasi-abelian categories. In this talk, I will introduce these ideas and briefly discuss how we can obtain interesting results, such as a representability theorem for derived stacks. | 4pm, HMI Seminar Room |
| 13.04.23 | Francis Brown (Oxford and HMI, TCD) | Elementary irrationality proofs and recent developments | I will give an introduction to irrationality proofs for classical numbers, including Beukers and Apery’s proofs of the irrationality of zeta(2) and zeta(3). If time permits I will discuss some geometric interpretations and recent partial progress on zeta(5). | 5pm, HMI Seminar Room |
| Date | Speaker | Title | Abstract | Note |
|---|---|---|---|---|
| 22.09.22 | Jack Kelly (TCD) | How (and why) to derive analytic geometry | In this talk I will present a model of derived analytic geometry (over either the complex numbers or a non-Archimedean Banach field) as derived geometry, in the sense of Toën-Vezzosi, relative to the monoidal model category of simplicial complete bornological vector spaces. In particular I will explain how this setup provides a solution to the long-standing issue of defining a well-behaved category of quasi-coherent sheaves on an analytic space which satisfies descent. I will also explain how these constructions permit a definition of analytic geometry over any Banach ring, including `universal' analytic geometry over the Banach ring of integers (the integers equipped with the absolute value norm). This is joint work with Oren Ben-Bassat and Kobi Kremnizer, and also uses work of Federico Bambozzi. | 4pm, New Seminar Room |
| 06.10.22 | Anton Alekseev (University of Geneva) | Virasoro Hamiltonian spaces | We develop a theory of Hamiltonian actions of the canonical central extension of the group of diffeomorphisms of the circle. It turns out that Virasoro Hamiltonian spaces (this is another name for such Hamiltonian actions) are in bijective correspondence with group valued Hamiltonian spaces with moment map taking values in (a certain part of) the universal cover of the group SL(2, R). Among other things, this correspondence allows to recover the classical result of Lazutkin-Pankratova, Kirillov, Segal, Witten (and others) on classification of coadjoint orbits of the Virasoro algebra. Interesting examples of Virasoro Hamiltonian spaces arise as moduli spaces of conformally compact hyperbolic metrics on oriented surfaces with boundary. The talk is based on a joint work in progress with Eckhard Meinrenken. | 4pm, New Seminar Room |
| 13.10.22 | Jan Pulmann (University of Edinburgh) | On quantizations of moduli spaces of flat connections on marked surfaces | The moduli space of flat G-connections on a surface S, being the phase space of the Chern-Simons theory on S×Ɪ, has a natural symplectic structure. With some boundary conditions, we get a so-called quasi-Poisson structure, which was deformation quantized by Li-Bland and Ševera. I will describe the combinatorial data necessary to specify their quantization uniquely and the relations between all such quantizations. The answer will use the Konstevich integral and the Drinfeld associator, relating the pentagon equation of the associator with the pentagon equation for triangulations of the surface S. Finally, I will describe some simple consequences of this description of the quantization. | 4pm, New Seminar Room |
| 20.10.22 | Konstantin Wernli (University of Southern Denmark) | Generalized Hamilton-Jacobi actions and Chern-Simons partition functions | This talk is based on the papers 2012.13270 and 2012.13983 which are joint with A. Cattaneo and P. Mnev. To any constrained system, we associate a generalized Hamilton-Jacobi (HJ) action, which is a generating function for the evolution relation. One can quantize this constrained system in the BV-BFV formalism, and we show that the generalized Hamilton-Jacobi action is the leading order of the corresponding effective action. Applied to Chern-Simons theory, this gives a precise effective action explanation of the CS-WZW correspondence (in 3d) and the CS-BCOV correspondence (in 7d). These results suggest that holography is at heart an effective phenomenon. | 4pm, New Seminar Room |
| 3.11.22 | Veronica Fantini (IHES) | Wall-Crossing structures in mirror symmetry | Mirror symmetry is one of the most fascinating conjectures in mathematical physics, in a nutshell it is a duality between the complex and the symplectic geometry of Calabi-Yau varieties. Since it was first conjectured in string theory, it has been explained within different geometric approaches such as the Strominger-Yau-Zaslow conjecture, Kontsevich homological mirror symmetry, etc. In addition, mathematicians got interested in the so called "reconstructing problem" aimed to understand how to reconstruct the mirror of a given Calabi-Yau. Among the main contributions, I will briefly recall ideas from Fukaya's multivalued Morse theory, Kontsevich-Soibleman non-Archimedean approach and the Gross-Siebert program, whose common point is to use "scattering diagrams". The latter encode geometric data (such as Gromov-Witten invariants) through a precise algebraic structure. In particular, the algebraic structure of scattering diagrams which conjecturally govern the reconstruction of the mirror of holomorphic pairs (a holomorphic vector bundle over a complex manifold) is the extended tropical vertex group (introduced in arXiv:1912.09956). I will then discuss a possible approach to understand the reconstruction of the mirror of holomorphic pairs based on the relationship between scattering diagrams in the extended tropical vertex group and 2d-4d wall-crossing structures. | 4pm, New Seminar Room |
| 10.11.22 | Alexander Shapiro (University of Edinburgh) | Cluster algebra and quantum Teichmüller theory | Given a surface and a group one can consider the corresponding character variety. Quantum Teichmüller theory uses cluster algebra to quantise the character variety, and construct its representations, equivariant under the action of the mapping class group of the surface. Moreover, it is conjectured that the resulting construction respects cutting and gluing of surfaces. In this talk, I will discuss this conjecture, and its relation to representation theory of quantum groups and 3-manifold invariants. | 4pm, New Seminar Room |
| 17.11.22 | Kasia Rejzner (University of York) | Algebraic structures in quantization of gauge theories | In this talk I will present the framework of perturbative algebraic quantum field theory and explain how it applies to quantization of physically realistic gauge theories, including QED and the Standard model of particle physics. The main mathematical tools are in homological algebra, where one uses a version of the BV (Batalin-Vilkovisky) formalism, combined with some functional analysis. | 4pm, New Seminar Room |
| 24.11.22 | Manuel Araújo (University of Cambridge) | String diagrams for semistrict n-categories | String diagrams are a powerful computational tool, most commonly used in the context of monoidal categories and bicategories. I will talk about extending this to higher dimensions. The natural setting for n-dimensional string diagrams should be some form of semistrict n-category, where composition operations are strictly associative and unital, but the interchange laws hold only up to coherent equivalence. One idea is to define a semistrict n-category as something which admits composites for labelled string diagrams, much as one can define a strict n-category as something that admits composites for labelled pasting diagrams. The first step in this program, which we have carried out, is to develop a theory of n-sesquicategories based on simple string diagrams. These are like strict n-categories without the interchange laws. The second step, which is still work in progress, is to add operations implementing weak versions of the interchange laws, obtaining the desired notion of semistrict n-category. In dimension 3, this recovers the notion of Gray 3-category. (I will start with a gentle introduction to strict n-categories and the idea of weak n-categories). | 4pm, New Seminar Room |
| Date | Speaker | Title | Abstract | Note |
|---|---|---|---|---|
| 3.12.20 | Jack Kelly (TCD) | Koszul Duality in Exact Categories | Toën and Vezzosi’s model of algebraic geometry as derived geometry relative to the monoidal model category of complexes of modules over a ring allows one to apply tools from homological algebra to study the geometry of derived stacks. A result of homological algebra which is of fundamental importance to derived algebraic geometry, particularly in the context of deformation theory and formal moduli problems, is Koszul Duality. Due to, amongst others, Quillen, Hinich, and Vallette, Koszul duality says that there is a Quillen equivalence between the category of coaugmented cocommutative differential graded coalgebras and the category of differential graded Lie algebras. The geometric interpretation is that a Lie algebra determines a unique formal space (i.e. a coalgebra) of which it is the shifted tangent Lie aglebra. Recent work with Bambozzi, Ben-Bassat, and Kremnitzer suggests that derived analytic geometry over a Banach field k of characteristic 0 can be modelled as derived geometry relative to the monoidal model category of chain complexes of bornological k-vector spaces. As is the case for derived algebraic geometry, one could in principle use techniques of homological algebra (internal to the category of bornological spaces) to understand analytic geometry. In this talk I will present a generalisation of Koszul duality to so-called monoidal elementary exact categories. The category of bornological spaces over k is such a category, so this result has implications for derived analytic geometry. | 3pm, Zoom |
| Date | Speaker | Title | Abstract | Note |
|---|---|---|---|---|
| 20.02.20 | Balazs Szendroi (Oxford University) | Hilbert schemes of points on singular surfaces: geometry, combinatorics and representation theory | Given a smooth algebraic surface S over the complex numbers, the Hilbert scheme of points of S is the starting point for many investigations, leading in particular to generating functions with modular behaviour and Heisenberg algebra representations. I will explain aspects of a similar story for surfaces with rational double points, with links to algebraic combinatorics and the representation theory of affine Lie algebras. | 4pm, Synge |
| Date | Speaker | Title | Abstract | Note |
|---|---|---|---|---|
| 25.09.19 | Yoshiaki Goto (Otaru University of Commerce) | Contiguity relations for hypergeometric integrals of type (k,n) | There are several generalizations of the Gauss hypergeometric function. The Aomoto-Gelfand hypergeometric function is one of them, which are introduced in the view point of integral representations. To study such hypergeometric integrals, twisted homology and cohomology groups are useful. In this talk, I would like to talk about these hypergeometric integrals and twisted cohomology groups mainly. I will give a short introduction to twisted cohomology groups and the intersection form on them, and explain derivations of contiguity relations (difference equation) as their application. Slides. | 2pm, Seminar Room |
| 25.09.19 | Saiei-Jaeyeong Matsubara-Heo (Kobe University) | Evaluating cohomology intersection numbers from twisted period relations | After the pioneering work of Kazuhiko Aomoto in 70's, the study of hypergeometric integrals is now regarded as a twisted analogue of that of period integrals. Through Poincare duality, it is straightforward to define (co)homology intersection numbers, but their exact evaluation is not an easy problem to solve except for some special integrals such as Aomoto-Gelfand hypergeometric integral. In this talk, we investigate the possibilities of evaluating cohomology intersection numbers (c.i.n.) through the twisted analogue of period relations. One way is to characterize the c.i.n. as a solution of differential equations. This part is the joint work with Nobuki Takayama. The other way is to expand the c.i.n. at toric infinity which can be obtained by using GKZ hypergeometric systems. | 3pm, Seminar Room |
| 27.09.19 | Maxim Kontsevich Andrei Okounkov Yuri Tschinkel |
HMI mini-conference | ||
| 30.10.19 | Francis Brown (HMI Simons Visiting Professor) | Motivic hypergeometric functions | 2:15pm, Seminar Room |
| Date | Speaker | Title | Abstract | Note |
|---|---|---|---|---|
| 24.01.19 | Adam Keilthy (Oxford University) | Relations and Filtration on Multiple Zeta Values | Multiple zeta values are a class of numbers arising naturally in algebra, geometry and even QFT. They have a rich algebraic structure, and satisfy many relations, such as the double shuffle relations, and the associated relations. We will discuss these relations and their connection to the motivic theory of $\mathbb P^1-\{0,1,\infty\}$ along side two naturally arising filtrations on the algebra of MZVs. We shall then introduce a new filtration and derive two new sets of relations, including Charlton's cyclic insertion, that hold in the associated graded algebra with respect to this filtration | 4pm, Synge |
| 20.03.19 | Francis Brown (HMI Simons Visiting Professor) | New perspectives on periods and integration, with applications to physics | Mini-course | 2pm, Seminar Room |
| 21.03.19 | Marton Hablicsek (University of Copenhagen) | Derived intersections and interactions with deformation theory | One purpose of derived algebraic geometry is to deal with bad situations in geometry. In intersection theory, bad situations arise when the intersection is not of the expected dimension or when the intersection is highly singular. In my talk, I will use the theory of derived intersections to compare intersections of Lagrangians in symplectic varieties (and higher analogues) with their deformation quantizations. As applications, I will show how one can obtain the Hodge theorem and how one can obtain extra structures on deformation quantizations in positive characteristics. | 4pm, Synge |
| 22.03.19 | Marton Hablicsek (University of Copenhagen) | Extra structures on deformation quantizations of shifted symplectic derived stacks in positive characteristics | Bezrukavnikov and Kaledin showed that convergent deformation quantizations of symplectic varieties over a perfect field of positive characteristic give rise to sheaves of Azumaya algebras. We investigate the structure of the convergent quantization of shifted symplectic derived stacks, especially the 1-shifted cotangent bundle $S$ of a smooth scheme $X$ over a perfect field of positive characteristic. We show that the quantization is an $E_2$-algebra over the Frobenius twist $S'$ of the 1-shifted cotangent bundle which restricted to the zero section $X'\to S'$ is weakly $E_2$-Morita equivalent to the structure sheaf of the Frobenius twist $X'$ of $X$. | 4pm, Seminar Room |
| 28.03.19 | Alex Fink (QMU London) | Some tropical geometry | Tropical mathematics is the study of the tropical semifield, $\mathbb R\cup\{\infty\}$ with minimum as addition and plus as multiplication. It congealed as a field of study in the mid-1990 as several existing lines of research came to be recognised as related, especially in optimisation and in algebraic geometry. There are ways to "change base" to the tropical semifield from the real numbers or from fields with nonarchimedean valuation; the tropical varieties that result retain a surprising amount of the properties of the originals. I'll give a general introduction to the area, perhaps making it to some of my own contributions regarding tropical linear spaces. | 4pm, Synge |
| 3.04.19 | Matija Tapuskovic (Oxford University) | Motivic Galois coaction on 1-loop motivic Feyman amplitudes | I will present some algebro-geometric tools and examples for computing the coaction of a Hopf algebra corresponding to the action of the motivic Galois group on 1-loop motivic Feynman amplitudes following Francis Brown. We will remain in the relam of 1-loop Feyman graphs because their associated motives can be proved, using the same techniques we will see during the talk, to be of Mixed-Tate type which is understood quite well in algebraic geometry. This will enable us to go into the subtleties of the computations, and of relating the Galois conjugates of a Feynman amplitude associated to a graph to periods of motives of its sub-quotient graphs. | 11am, Seminar Room |
| 4.04.19 | Johan Leray (University Paris 13) | Double Poisson algebras up to homotopy | The notion of double Poisson algebra, introduced by Van den Bergh in 2008, is the noncommutative analogous of Poisson structure. After recalling what is a Poisson structure and presenting and motiving this new algebraic structure, I will define the notion of (pr)operads, which are algebraic objects used to encode many algebraic structures. I will show how we can use this notion to understand what is double Poisson algebras up to homotopy. | 4pm, Synge |
| 8-10.05.19 | Analysis, Geometry and Algebra Conference | |||
| 13-17.05.19 | HMI Workshop Gauge theory and virtual invariants | |||
| 27-31.05.19 | HMI Workshop Homotopy meets homology |
| Date | Speaker | Title | Abstract | Note |
|---|---|---|---|---|
| 20.9.18 | James Griffin (Coventry) | The space of circles | To define a Euclidean circle in three dimensional space you need six parameters, three for the coordinates of its centre, one for its radius and two for its spatial orientation. As such the space of circles, Circ is a six dimensional manifold; we investigate its algebra and geometry. For example the set of osculating circles to a three dimensional path is a path in Circ. Such paths may be characterised as those that are "doubly light-like" using a complexified version of Minkowski space. The configuration space of k pairwise unlinked circles has fundamental group isomorphic to a subgroup of automorphisms of the free group with k generators. The homology groups of this subgroup was computed by Jensen, McCammond, and Meier, and their ranks suggest there may be an topological operad structure on the configuration space of unlinked circles. It turns out there is and in joint work with Allen Hatcher we describe the operad and give a presentation. | 5pm, Synge |
| 10.10.18 | Timothy Logvinenko (Cardiff) | Bar category of modules and homotopy adjunction | I will describe joint work with Rina Anno in which we introduce the bar category of modules Modbar(A) over a DG category A. It is a DG-enhancement of the derived category D(A) of A which is isomorphic to the category of DG A-modules with A-infinity morphisms between them. However, it is defined intrinsically in the language of DG-categories and requires no complex machinery or sign conventions of A-infinity categories. We define for these bar categories Tensor and Hom bifunctors, dualisation functors, and a convolution of twisted complexes. The intended application is to working with DG-bimodules as enhancements of exact functors between triangulated categories. As a demonstration, we develop homotopy adjunction theory for tensor functors between derived categories of DG-categories. | 3pm, NSR |
| 11.10.18 | Timothy Logvinenko (Cardiff) | Generalised braid category | Ordinary braid group $Br_n$ is a well-known algebraic structure which encodes configurations of n non-touching strands (“braids”) up to continious transformations (“isotopies”). A classical result of Khovanov and Thomas states that this group acts categorically on the space $Fl_n$ of complete flags in $C^n$. I will begin by reviewing the basics on braid group and flag varieties, and then give a sketch of the geometry involved in the Khovanon-Thomas construction of the categorical action of $Br_n$ on $T^* Fl_n$. I will then describe a longstanding work-in-progress with Rina Anno: the categorification of generalised braids. These are the braids whose strands are allowed to touch in a certain way. They have multiple endpoint configurations and can be non-invertible, thus forming a category rather than a group. A decade old conjecture states that generalised braids act categorically on the spaces of full and partial flags in $C^n$. I will describe our present progress towards it and future expectations. | 5pm, Synge |
| 25.10.18 | Francis Brown (HMI Simons Visiting Professor) | Single-valued integration | Single-valued functions are ubiquitous in mathematics and physics, since a well-defined problem has a well-defined answer. On the other hand, the solution to such a problem is often given by an integral, which is usually a multi-valued function of its parameters. The reason is that integration is a pairing between differential forms and chains of integration, and the latter are ambiguously defined. In this talk, which is joint work with Clément Dupont, I will describe a way to pair differential forms with `duals of differential forms'. This defines a theory of integration which satisfies the usual rules, but is automatically single-valued. Many well-known constructions in mathematics and physics are examples of such objects, and I will illustrate the theory with examples depending on the interests of the audience. | 5pm, Synge |
| 8.11.18 | David Carchedi (MPIM Bonn) | The Universal Property of Derived Manifolds | Given two smooth maps of manifolds $f:M \to L$ and $g:N \to L$, if they are not transverse, the fibered product $M \times_L N$ may not exist, or may not have the correct cohomological properties. Thus lack of transversality obstructs many natural constructions in topology and differential geometry. Derived manifolds generalize the concept of smooth manifolds to allow arbitrary (iterative) intersections to exist as smooth objects, regardless of transversality. We will discuss the universal property of derived manifolds, and how this naturally gives rise to concrete models. | 5pm, Synge |
| 29.11.18 | Caner Nazaroglu (University of Cologne) | Squashed Toric Manifolds and Higher Depth Mock Modular Forms | In 1980’s, a three-way relation between compact Calabi-Yau manifolds, two dimensional (2,2) superconformal field theories and modular forms was established. These objects can be tied together through the corresponding Gauged Linear Sigma Models and the computation of elliptic genera via such models. In this talk, we will focus on a class of sigma models that describe toric Calabi-Yau manifolds which are squashed. In the simplest one-dimensional example previously studied, the supersymmetric partition function that computes the elliptic genus is known to produce a mixed mock Jacobi form including its modular completion. I will describe the automorphic nature for the general case of squashed toric sigma models and show that they yield higher-depth mock modular forms that have been recently formulated. Finally, I will discuss further refinements and possible applications to physics and geometry. | 5pm, Synge |
| Date | Speaker | Title | Abstract | Note |
|---|---|---|---|---|
| 9.2.18 | L. Takhtajan N. Nekrasov Yu. Tschinkel |
HMI mini-conference | ||
| 22.2.18 | Mikhail Kapranov (HMI Simons Visiting Professor) | Triangulated categorification, perverse sheaves and perverse schobers | The idea of categorification, that is replacing vector spaces by (triangulated) categories, has been an important guiding tool in mathematics and physics. For example, the Fukaya category of a symplectic manifold can be seen as categorification of the middle homology. The talk will explain applications of categorification to the theory of perverse sheaves, a central tool in topology of algebraic varieties. The corresponding categorified concept, perverse schobers, turns out to be related to the program of introducing coefficients into Fukaya categories as well as to the homological minimal model program in birational algebraic geometry. | 4pm, Synge |
| 22.2.18 | Estanislao Herscovich (Grenoble) | Double monoidal categorical structures appearing in Quantum Field Theory | R. Borcherds has introduced a different point of view to formalise perturbative Quantum Field Theory (pQFT), making use of several objects that behave somehow like bialgebras and comodules over them, and which are essential in his definition of Feynman measure. The former objects don’t seem however to be bialgebras in the classical sense, for their product and coproduct are with respect to two different tensor products, and similarly for comodules. Moreover, following physical motivations, these objects are given as some symmetric constructions of geometric nature. The aim of this talk is on the one hand to show that the “bialgebras” and “comodules” introduced by Borcherds cannot “naturally” exist, and on the other side to provide a background where a modified version of the so-called “bialgebras” and “comodules” do exist. This involves a category provided with two monoidal structures satisfying some compatibility conditions. As expected, the modified version of the mentioned “bialgebras” and “comodules” are not so far from the original one, considered by Borcherds. Moreover, we remark that these new candidates allowed us to prove the main results stated by Borcherds in his article (see my manuscript "Renormalization in Quantum Field Theory (after R. Borcherds)"). | 5pm, Synge |
| 22.3.18 | Herve Gaussier (Grenoble) | About curvature of domains in the complex Euclidean space | Different generalizations of the Gauss curvature were introduced on manifolds, in different contexts. We will try to explain, on simple examples, the links between them for domains in $\mathbb C^n$. | 5pm, Synge |
| Date | Speaker | Title | Abstract | Note |
|---|---|---|---|---|
| 19.10.17 | Yuuji Tanaka (Oxford University) | On the singular sets of solutions to the Kapustin-Witten equations and the Vafa-Witten ones on compact Kahler surfaces | I'll talk about some observations on singular sets, which appear in some asymptotic analysis of solutions to the Kapustin-Witten equations and the Vafa-Witten ones on closed four-manifolds. Rather recently, Cliff Taubes made a great breakthrough about the analysis of them by proving that the real two-dimensional Hausdorff measures of these singular sets are finite. In this talk, we analyse the singular sets more in some cases, in particular, we figure out that they have the structure of an analytic subvariety when the underlying manifold is a compact Kahler surface. | Seminar, 3pm, SNIAM |
| 19.10.17 | Yuuji Tanaka (Oxford University) | Vafa-Witten invariants for projective surfaces | This talk describes studies of the Vafa-Witten theory on projective surfaces using techniques in algebraic geometry. These are joint work with Richard Thomas. After mentioning some backgrounds and motivation, we define invariants by (virtual) $\mathbb C^*$ localisations from the moduli space of Higgs pairs, which is the algebraic counterpart of the moduli space of solutions to the Vafa-Witten equations. We then describe calculations of them in examples, which match with Vafa-Witten's original conjectures raised more than 20 years ago. | Colloquium, 5pm, Synge |
| Oct 23-27 | HMI Workshop Geometry and Combinatorics of Associativity | |||
| 2.11.17 | Yan Soibelman (Kansas State University) | Riemann-Hilbert correspondence, Fukaya categories and periodic monopoles | Classical Riemann-Hilbert correspondence gives a categorical equivalence of connections with regular singularities on curves and representations of the fundamental groups of these curves (via the monodromy). One can state the same problem in the case of connections with irregular singularities, or more generally, in the cases of q-difference and elliptic difference equations. It turns out that the underlying geometry in all cases is the geometry of complex symplectic manifolds (surfaces in one-dimensional case), which are partially compactified to Poisson manifolds. The structure which replaces the representations of the fundamental groups is the one of the Fukaya category of those symplectic manifolds (category of A-branes in the language of physics). I plan to discuss these ideas in the one-dimensional case. If time permits, I will discuss the relationship of doubly and triply periodic monopoles on $\mathbb R^3$ with twistor families, which generalize the twistor family giving the hyperkahler structure on the moduli space of stable Higgs bundles on a projective curve. | Colloquium, 5pm, Synge |
| 9.11.17 | Thomas Poguntke (Bonn University) | Higher Segal structures in algebraic K-theory | One of the main results of Dyckerhoff-Kapranov's work on higher Segal spaces concerns the fibrancy properties of Waldhausen's simplicial construction of algebraic K-theory, which are in particular responsible for the associativity of various Hall algebras. We will explain their result and generalize it to a certain higher dimensional analogue, replacing short exact sequences by longer extensions, in the case of an abelian category. Finally, if time permits, we will indicate how to relate it back to (an iteration of) Waldhausen's construction, in particular showing how it recovers the whole K-theory spectrum. | Colloquium, 5pm, Synge |
| 16.11.17 | Tobias Dyckerhoff (Bonn University) | A categorified Dold-Kan correspondence | Various recent developments, in particular in the context of topological Fukaya categories, seem to be glimpses of an emerging theory of categorified homotopical and homological algebra. The increasing number of meaningful examples and constructions make it desirable to develop such a theory systematically. In this talk, we discuss a step towards this goal: a categorification of the classical Dold–Kan correspondence. | Colloquium, 5pm, Synge |
| 23.11.17 | Natalia Iyudu (University of Edinburgh) | Sklyanin algebras via Groebner bases and finiteness conditions for potential algebras | I will discuss how some questions on Sklyanin algebras can be solved using combinatorial techniques, namely, the theory of Groebner bases, and elements of homological algebra. We calculate the Poincaré series, prove Koszulity, PBW, Calabi-Yau, etc., depending on the parameters of the Sklyanin algebras. There was a gap in the Artin-Schelter classification of algebras of global dimension 3, where Koszulity and the Poincaré series for Sklyanin algebras were proved only generically. It was filled in the Grothendieck Festschrift paper of Artin, Tate and Van den Bergh, using the geometry of elliptic curves. Our point is that we recover these results by purely algebraic, combinatorial means. We use similar methods for other potential algebras as well, including homology of moduli of pointed curves given by Keel relations, and contraction algebras arising in noncommutative resolution of singularities. |
Colloquium, 5pm, Synge |
| Nov 27-Dec 1 | HMI Workshop Hall algebras and related topics | |||
| 7.12.17 | Wiesław Pawłucki (Jagiellonian University) | Hironaka Rectilinearization in general O-minimal structures |
| Date | Speaker | Title | Abstract | Note |
|---|---|---|---|---|
| 26.1.17 | Hans Franzen (Bochum University) | Donaldson-Thomas invariants of quivers via Chow groups of quiver moduli | We use a presentation of Chow rings of (semi-)stable quiver moduli to show that the primitive part of the Cohomological Hall algebra of a symmetric quiver is given by Chow groups of moduli of simple representations. This implies that the DT invariants are determined by these Chow groups. | Seminar, 3pm, OSR |
| 26.1.17 | Hans Franzen (Bochum University) | Geometric properties of quiver Grassmannians | Quiver Grassmannians are geometric objects which parametrize subrepresentations of quiver representations. They can ban be regarded as analogs of (the usual) Grassmannians. Caldero and Chapoton have shown that the cohomology of quiver Grassmannians plays an important role in the theory of cluster algebras. After giving a basic introduction to representations of quivers and quiver Grassmannians, we exhibit classes of representations for which the quiver Grassmannian has a very simple geometric structure. | Colloquium, 5pm, Synge |
| 2.2.17 | Norbert Hoffmann (Mary Immaculate College, Limerick) | Del Pezzo surfaces and universal torsors | A homogenuous polynomial equation of degree three in four variables defines a cubic surface S in complex projective threespace. Manin's Conjecture relates the geometry of this complex surface S to integer solutions of the equation. A main tool to study such integer solutions are the universal torsors over S introduced by Colliot-Thelene and Sansuc. All this generalizes from cubic surfaces to more general del Pezzo surfaces. After explaining this, I will speak about recent joint work with Ulrich Derenthal on universal torsors over degenerating del Pezzo surfaces. | Colloquium, 5pm, Synge |
| 9.2.17 | G. Gabadadze Yu. Tschinkel A. Okounkov |
HMI mini-conference | ||
| 10.2.17 | M. Kontsevich N. Nekrasov |
HMI mini-conference | ||
| 16.2.17 | Antonio Giambruno (Palermo University) | Polynomial identities and their growth | A polynomial identity satisfied by an algebra A is a polynomial in non commuting indeterminates vanishing identically when evaluated in A. These polynomials form a T-ideal of the free algebra, i.e., an ideal invariant under the endomorphisms of the free algebra. In general one can attach to such a T-ideal a numerical sequence, called the sequence of codimensions of A. In characterstic zero this sequence gives a quantitative measure of the polynomial identities satisfied by the algebra A. The aim of this talk is to present some of the results obtained in recent years on the asymptotic computation of such sequence. As a by-product I shall describe a method based on the representation theory of the symmetric group leading to some classification results. | Colloquium, 5pm, Synge |
| 23.2.17 | Sung Yeon Kim (CMC, KIAS) | CR maps between boundaries of flag domains | Colloquium, 5pm, Synge | |
| 16.3.17 | Leandro Vendramin (University of Buenos Aires) | Nichols algebras and applications | Nichols algebras appear in several branches of mathematics going from Hopf algebras and quantum groups, to Schubert calculus and conformal field theories. In this talk we review the main problems related to Nichols algebras and we discuss some classification theorems and some applications. | Seminar, 2pm, OSR |
| 16.3.17 | Brent Pym (University of Edinburgh) | Divergent series and the Stokes groupoids | Near a point where an ordinary differential equation has a pole of order two or more, its solutions may have very complicated behaviour, including essential singularities and divergent power series expansions. Nevertheless, using natural objects of differential geometry known as Lie groupoids, one can construct a canonical domain on which the solutions are globally defined and holomorphic. I will describe this construction and explain how it gives a new geometric interpretation of the classical techniques for "resumming" divergent series. This talk is based on joint work with Marco Gualtieri and Songhao Li. | Colloquium, 5pm, Synge |
| 30.3.17 | Franc Forstnerič (University of Ljubljana) | Complete bounded submanifolds in different geometries | In this lecture I will survey recent results on the construction of bounded metrically complete submanifolds in several different geometries: holomorphic submanifolds of complex Euclidean spaces (the problem of Paul Yang), null holomorphic curves and conformal minimal surfaces in Euclidean spaces (the Calabi-Yau problem), and complete Legendrian curves in contact complex manifolds. | Colloquium, 5pm, Synge |
| 7.4.17 | Kobi Kremnitzer (Oxford University) | Constructing geometries | A possible approach to solving certain mathematical problems is to construct a new geometry which is suitable to the problem. The main motivating example is the attempt to construct a geometry in which the integers become a "continuous" object (over the field with one element) as an approach to solving the Riemann Hypothesis. In this talk I will explain what do I mean by geometry and different ways of constructing new geometries. I will also give some examples of applications to number theory, analytic geometry and quantum field theory. | Colloquium, Friday 3pm, Salmon |
| Date | Speaker | Title | Abstract | Note |
|---|---|---|---|---|
| 29.9.16 | Emilio Franco (Unicamp, Brazil) | Brane involutions and irreducible holomorphic symplectic manifolds | We study natural brane-involutions on moduli space of sheaves over symplectic surfaces and their behaviour under Fourier-Mukai transform and lattice Mirror symmetry. This is joint work with M. Jardim (Campinas State University) and G. Menet (Campinas State University). | |
| 6.10.16 | Victoria Lebed (TCD) | Unexpected facets of the Yang-Baxter equation | In this talk I propose to look at the good old Yang-Baxter equation from an unexpected viewpoint. We shall see that this equation generalizes basic algebraic laws: associativity, the Jacobi identity, self-distributivity, the axioms of a lattice. In spite of this generality, one can say non-trivial things valid for all solutions to the YBE - namely, concerning their representation and (co)homology theories. Moreover, any solution comes with a quadratic universal enveloping algebra, which for certain solution classes enjoys particularly nice algebraic properties. As an application, we shall outline how to construct resolutions of some algebras by interpreting them as the universal enveloping algebras of easily manipulable solutions to the YBE. Knot-theoretic applications will also be sketched. | Mathematics Colloquium Synge 5pm |
| 13.10.16 | Jonas Kaszian (University of Cologne) | Indefinite Theta Functions arising in Gromov-Witten Theory | In this talk, we consider a function occuring naturally in the open Gromov-Witten potential of an elliptic orbifold. To help illuminate its mysterious nature, we connect it with higher-depth mock modular forms by studying related indefinite theta functions of signature (1,3). The study of these functions was recently opened up by work of Alexandrov, Banerjee, Manschot, and Pioline discussing the case of signature (n,2). | |
| 20.10.16 | Raf Bocklandt (University of Amsterdam) | Moduli of Matrix factorizations | We discuss how to construct moduli spaces of matrix factorizations using concepts from mirror symmetry and illustrate these ideas with examples coming from dimer models. | |
| From the Freezer to the Tropics | Following Goncharov and Kenyon one can define a dynamical system from a consistent dimer model. On the other hand one can also use this dimer model to resolve a singularity. We will explain how these two become related if we go to the absolute zero temperature in the dynanical system and to the tropical limit of the resolution. | Mathematics Colloquium Synge 5pm |
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| 27.10.16 | Emmanuel Letellier (Paris Diderot-Paris 7 University) |
Higgs bundles and indecomposable parabolic vector bundles over the projective line | In this talk we will count the number of isomorphism classes of geometrically indecomposable parabolic bundles over the projective line over a finite field. We will explain the relation between this counting and the moduli space of Higgs bundles with prescribed residues over the complex projective line. | |
| Character varieties and representation theory | The aim of this talk is to explain the relation between two problems in mathematics: the first one is about the representation theory of finite Lie groups and the second one is about the geometry of the so-called character varieties, namely the moduli space of representations of the fundamental groups of punctured compact Riemann surfaces into complex Lie groups. | Mathematics Colloquium Synge 5pm |
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| 17.11.16 | Herve Gaussier (Grenoble) | How can metrics explain some geometric properties of complex manifolds? | The Poincaré distance is an example of a hyperbolic distance on the unit disk in C. It admits different generalizations in the context of complex geometry, such as the Bergman metric, example of a Kähler-Einstein metric, or the Kobayashi metric, example of a Finsler metric, defined on complex manifolds. We will explain how the properties of such metrics restrict the geometry of the manifold. | Mathematics Colloquium Synge 5pm |
| 24.11.16 | Tyler L. Kelly (University of Cambridge) | Unifying Mirror Constructions | Mirror symmetry uses dualities in string theory to predict that, given a symplectic variety, there should exist an algebraic variety known as the mirror such that various geometric and physical data are exchanged. Over the past 25 years, there have been many recipes that have been proposed to construct the mirror for certain symplectic varieties. However, in certain cases, for the same symplectic variety, different recipes can give different algebraic varieties as the mirror. In this talk, we will talk about how one can fix this discrepancy in the context of Kontsevich's Homological Mirror Symmetry. | Mathematics Colloquium Synge 5pm |
| 1.12.16 | Arnaud Mortier (DCU) | Finite-type cohomology of the space of long knots | The aim of this talk is to give an overview of Vassiliev's cohomology of the space of knots. Topics discussed will include the origins in singularity theory, several algebraic reformulations, the Kontsevich integral (a universal Vassiliev knot invariant), a systematic way to construct cocycles, and some examples. | |
| Knots, virtual knots, and finite-type invariants | In the classical sense, a knot is an embedding of a circle in R^3. This notion has known a multitude of variations, with as many purposes: trading the circle for another 1-manifold, increasing dimensions... Virtual knot theory is such a variation. It came up during the 90's after a period of enthusiasm towards a new kind of knot invariants, today widely understood except for the fact that it is still open whether they classify all knots: Vassiliev's 'finite-type' invariants. I will discuss the main ideas behind these notions. | Mathematics Colloquium New sem. room 4:30pm |
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| 9.12.16 | Nikos Diamantis (University of Nottingham) |
Values of L-functions and shifted convolutions | A characterisation of the field containing the values of L-functions is proposed. This is described in terms of shifted convolution series of divisor sums and to establish it we use a double Eisenstein series we previously studied with C. O'Sullivan and a kernel function used by F. Brown in his study of multiple modular values. | Mathematics Colloquium Synge 4pm |