Speaker: Oscar Bendix Harr (University of Copenhagen) Abstract: Not much is known about the cohomology of the moduli space of handlebodies (equivalently, the cohomology of handlebody mapping class groups) except what it stabilizes to, by Hatcher's analogue of the Madsen--Weiss theorem. Hatcher and Wahl have shown that the degree d cohomology of the moduli space of genus g handlebodies is stable for d less than roughly g/2. I will discuss some work from my PhD thesis (in preparation) which expands this range to g roughly less than 2g/3, using the methods of Galatius, Kupers, and Randal-Williams. By a computation of Morita, this is the optimal range for homological stability for moduli spaces of handlebody groups. A similar argument shows that the cohomology of automorphism groups of free groups Aut(F_n) is stable in degrees roughly less than 2n/3, which improves the best known range for integer coefficients. Time provided, I will also say something about ongoing work on the cohomology outside the stable range.

Speaker: Chris Grossack (UC Riverside) Abstract:Often in math it's easier to solve a problem "locally", and then one studies how to glue these local solutions into a global solution of the whole problem. In recent years there's been a lot of progress in our understanding of skein algebras and how they glue, culminating in the work of Ben-Zvi--Brochier--Jordan and Cooke on Skein Categories, which can be computed with Factorization Homology. In a separate line of development, there has been progress in the gluing of (topological) Fukaya Categories and Hall Algebras, using the Dyckerhoff--Kapranov machinery of 2-Segal Spaces. In the two halves of this talk we'll discuss Factorization Homology and 2-Segal Spaces. In any remaining time we'll speculate on how these techniques might be used to globalize certain results in Skein Theory.

Speaker: Peter Samuelson (UC Riverside) Abstract:The Homflypt skein relations were originally used in the 90's to define polynomial invariants of knots in S^3. In this talk we describe several other ways they arise in algebra and geometry: intertwiners for certain quantum group representations, commutator relations in quantum groups, deformation quantization of trace functions on character varieties, commutator relations in Hall algebras of Fukaya categories of surfaces, and higher dimensional Heegaard-Floer homology of configuration spaces of points in cotangent bundles to surfaces. This will be a survey style talk; in particular, no significant prior knowledge of these topics will be assumed, and the precision of the statements will decrease monotonically as the talk progresses. Our main goal is to describe a nonobvious common theme in these topics.

Speaker: Frederic Fauvet (IRMA, University of Strasbourg) Abstract:Paralogarithms constitute a family of resurgent functions which have been introduced by J. Ecalle, in order to solve, in a very general and systematic way, inverse problems of Riemann--Hilbert type that are expressed in the formalism of alien derivations. I will present "paralogarithmic resurgence monomials" and describe how they are implemented to yield differential equations with given sets of Stokes data, focussing on a simple non--linear example.

Speaker: Sergey I. Bozhevolnyi Center for Nano Optics University of Southern Denmark Abstract:Manipulation of single-photon emission from quantum emitters (QEs) has attracted a considerable attention in recent years due to its importance for quantum information technologies in quantum communication, computation, sensing and metrology. Here, recent progress in on-chip manipulation of the polarization, directionality and phase distribution in single-photon emission by making use of planar holographic QE-coupled metasurfaces is presented and discussed. The underlying idea is related to the concept of meta-atom, in which a QE is efficiently and non-radiatively coupled to surface modes, such as surface plasmon polaritons (SPPs), that are subsequently outcoupled into free propagating waves. An innovative metasurface design approach, vectorial scattering (computer-generated) holography, is introduced for the purpose of designing hybrid SPP-QE coupled metasurfaces suitable for generation of well-collimated beams of single photons with desirable polarization characteristics propagating along given directions. Latest results include its extension for realizing single-photon sources with radiation channels that exhibit diverse (including vectorial with spin and orbital angular momenta) wavefronts and polarization characteristics, opening thereby a way to generating quantum structured light in high dimensions.Location: D-IAS Aud. (V24-501a-0), Danish Institute for Advanced Study - DIAS. The event is open to all.

Speaker: Mykola Dedushenko (Simons Center for Geometry and Physics, Stony Brook) Abstract:I will review my recent work with Nekrasov on the aspects of relation between supersymmetric gauge theories and quantum integrable models. An important role is played by the supersymmetric defects (especially interfaces) in gauge theories, which provide a QFT realization of the infinite-dimensional quantum groups underlying the integrability structures. These defects realize, and in many ways are inspired by, the constructions of Maulik with Okounkov, and Aganagic with Okounkov, of stable envelopes and chamber R-matrices in geometry.

Speaker: Thibault Langlais (University of Oxford) Abstract:The Lie group G2 is one of the two exceptional cases in Berger's list of possible Riemannian holonomy groups. I will start by introducing the basics of G2-geometry, emphasizing its relation to Calabi-Yau geometry in (real) dimension 4 and 6. The moduli spaces of G2-manifolds are smooth and carry a natural Riemannian metric, analogous to the Weil-Petersson metric, whose properties are poorly understood. I will show that certain singular G2-manifolds correspond to finite-distance limits in the moduli spaces, which proves that G2-moduli spaces are generally not complete. Time permitting, I will also say a few words about the computation of curvatures.

Speaker: Manuel Meyer Department of Physics, Chemistry and PharmacyUniversity of Southern Denmark Abstract:The nature of dark matter continues to elude us, even after almost one century after first evidence for this non-luminous substance appeared. A leading hypothesis is that dark matter, which makes up more than 80% of all matter in the universe, consists of yet undiscovered fundamental particles. Such dark matter particles might interact feebly with known particles of the Standard Model, making them potentially detectable in the laboratory. The predicted low interaction rates require extremely sensitive detectors and ultra-low background levels. In this talk, I will discuss our recent progress in characterizing and improving the performance of transition edge sensors (TESs). Such TESs are planned to be used in the Any Light Particle (ALPS) II experiment, which searches for axions and axion-like particles. I will also review how we can use our TES to search for weakly interacting massive particles. With our ongoing research, we hope to achieve a new record for background suppression employing both optical filtering inside a cryostat as well as machine learning techniques while maintaining a quantum efficiency close to unity. Location: D-IAS Aud. (V24-501a-0), Danish Institute for Advanced Study - DIAS. The event is open to all.

Speaker: Ron Donagi (University of Pennsylvania) Abstract:After a brief review of the background on super geometry and of Super Riemann surfaces and their punctures, I will discuss the proof (with Witten) that the supermoduli space of super Riemann surfaces is not projected for g ≥5, and the more recent proof, with Ott, that the supermoduli space Mg,0,2r of super Riemann Surfaces with Ramond punctures is also not projected, for all g≥5r+1 and r≥6.

Speaker: Nadia Ott (University of Southern Denmark) Abstract:D’Hoker and Phong’s calculation of the genus g = 2 superstring amplitude uses, in a crucial way, a projection from genus g = 2 supermoduli space to its underlying reduced space. They define this projection using a formula for the genus g = 2 super period matrix. Witten generalized their formula for the super period matrix to higher genus g and found that the super period matrix may develop a pole along a particular divisor in supermoduli space if g ≥ 11. This divisor is commonly called the bad divisor. Witten also considered super period matrices on super Riemann surfaces with a nonzero number of Ramond punctures (note: the word puncture is a bit of a misnomer). He found that in the presence of Ramond punctures, a closed one form has, in addition to the usual 2g ”even” periods (defined by integrals over one cycle in homology), 2r fermionic periods. The fermionic periods of one form w are certain constants appearing in the restriction of w to the Ramond divisor. In joint work with Ron Donagi, we identify the 2r fermionic periods of w with the residues of a particular global section of the twisted spin structure on the underlying curve. As in the unpunctured case, the super period matrix with Ramond punctures may develop a singularities as we vary over supermoduli space. Using this identification of the fermionic periods in terms of residues, we explicitly describe this bad locus in supermoduli space.

Speaker: Nadia Ott (University of Southern Denmark) Abstract:I will give an introductory talk on supermanifolds, super Riemann surfaces, and supermoduli space, starting with the definitions of these objects. I will then introduce a current area of research in supergeometry, super period matrices, and state two famous results in the field: Donagi and Witten’s proof that supermoduli space is not split for genus g ≥ 5, and D’Hoker and Phong’s calculation of the genus g = 2 superstring amplitude.

Speaker: Erik Donovan Hedegård Department of Physics, Chemistry and Pharmacy University of Southern Denmark Abstract:Transition metals in biological systems pose a formidable challenge in modern quantum chemistry. Unfortunately, these metals are close to everywhere in biological systems. For instance, about one-third of all enzymes contain a transition metal. The main issues when dealing with transition metals are (i) the metal typically demands so-called complete active space (CAS) methods with billions of electron configurations included. This quickly becomes computationally costly. (ii) Relativistic effects can be sizable. Addressing these effects also becomes computationally demanding. (iii) The relevant chemistry usually occurs in a surrounding solvent or within a protein environment that also needs to be taken into account, both in terms of the nuclear dynamics as well as the electronic interactions between the metal and the environment.In this talk, we demonstrate how challenges (i)–(iii) can be tackled efficiently and accurately. We discuss how this allows us to understand a part of the global carbon cycle, enzymes that can boost the production of biofuel, and new drugs against cancer.Location: D-IAS Aud. (V24-501a-0), Danish Institute for Advanced Study - DIAS. The event is open to all.

Speaker: Ying-Hsuan Lin (Harvard University) Abstract:We numerically studied an anyon chain based on the Haagerup fusion category, and found evidence that it leads in the long-distance limit to a conformal field theory (CFT) whose central charge is ~2. Our findings were in agreement with concurrent research by Vanhove et al., who explored a closely related statistical lattice model. Subsequent analyses of the operator product expansion by Liu, Zou, Ryu, however, revealed a deeper complexity than our data had initially suggested. In this talk, I will discuss these developments and contextualize them within the broader program of classifying CFTs.

Danish Quantum Community invites to participate in our annual Scientific Quantum Conference 2024 This year’s conference is hosted by Centre for Quantum Mathematics (QM) with support from the Novo Nordisk Foundation. Join us for a two-day deep dive into the Danish quantum research community - across universities and across disciplines. About the event:The conference will present recent results from Danish quantum research with a special focus on younger researchers at Danish universities. Serving as a meeting point for the research community, the conference aims to strengthen the Danish quantum ecosystem and promote industry-academia collaborations.Across two days, the conference will provide a deep dive into four key research tracks: quantum hardware, quantum software/algorithms, components for quantum systems, and early quantum applications in chemistry and finance.The conference is hosted by Centre for Quantum Mathematics (QM) at SDU in Odense with support from the Novo Nordisk Foundation.Participation is free of charges, but requires registration.

Speaker: Merlin Christ (Institut de mathématiques de Jussieu – Paris Rive Gauche) Abstract:A complex of stable infinity-categories is a categorification of a chain complex, meaning a sequence of stable infinity-categories together with a differential that squares to the zero functor. To produce examples of such categorical complexes we introduce a categorification of the totalization construction, which produces a categorical complex from a categorical multi-complex, such as a commuting cube of stable infinity-categories. We will then explain how categorified perverse sheaves, also known as perverse schobers, on C^n (with a certain stratification) can be described in terms of categorical cubes and categorical complexes of spherical functors. We will also see what categorical totalization means in this case geometrically. This talk is based on joint work with T. Dyckerhoff and T. Walde.

Speaker: Edmund Heng (Institut des Hautes Études Scientifiques) Abstract:Classically, finite symmetries are captured by the action of a finite group. Moving to the quantum world, one has to allow for (possibly non-invertible) quantum symmetries — these are instead captured by the action of a more general algebraic structure, known as a fusion category. Such quantum symmetries are actually ubiquitous in mathematics; for example, given a category with an action of a finite group G (e.g. rep(Q), Coh(X) etc.), its G-equivariant category has instead the action of the category of representations rep(G), where rep(G) has the structure of a fusion category (and is not just a group when G is non-abelian). The aim of this talk is to introduce fusion categories and discuss their role as “quantum symmetries” in relation to Bridgeland’s stability conditions. We first introduce a generalised notion replacing “G-invariant stability conditions” in the setting of a triangulated category equipped an action of a fusion category C, which we will “C-equivariant stability conditions”. The first result is that these stability conditions form a closed submanifold of the stability manifold, just as the G-invariant stability conditions do. Moreover, given a triangulated D with a G-action, so that its G-equivariant category D^G has a rep(G)-action, we will see the following (Morita) duality result for stability conditions: the complex manifold of G-invariant stability conditions (associated to D) is homeomorphic to the complex manifold of rep(G)-equivariant stability conditions (associated to D^G).If time allows, I will discuss other more “exotic” actions of fusion categories on triangulated categories, and possibly its relation to Coxeter theory.This is part of joint work with Hannah Dell and Anthony Licata.

DIAS lecture by Professor Francesco Sannino, DIAS Chair of Physics, Head of the Quantum Field Theory Center, Founder of the Centre for Cosmology and Particle Physics Phenomenology (CP³-Origins) at SDU. The lecture takes place in the DIAS Auditorium and is open to all.Abstract We live in an era marked by the LIGO Laboratory discovery of gravitational waves, emitted when two black holes coalesce, and the Event Horizon Telescope imaging of black holes. While these amazing discoveries crystallise Einstein’s theory of general relativity they also beg for a fresh look at the problem of unifying gravity and quantum field theory. These two theories constitute our current understanding of Nature but are notoriously at odds with each other. I will first review basic facts about black holes and then argue in favour of a recent approach that aims at shortcutting the problem of the absence of a theory of quantum gravity. This will be achieved by introducing a quantum gravity model independent approach focussed on effective metric descriptions of quantum black holes. We believe that our findings will herald novel ways to explore quantum corrections to black hole dynamics with impact for our understanding of the fabric of space time.About Francesco SanninoFrancesco Sannino is the Head of the Quantum Field Theory Center, the Founder of the Centre for Cosmology and Particle Physics Phenomenology (CP³-Origins) at SDU. He is also one of the Founders of the Danish IAS and Professor of Theoretical Physics at the Federico II University in Italy.Professor Francesco Sannino is widely recognized for having pioneered the analytical and numerical investigations of the conformal structure of gauge theories of fundamental interactions, for the construction of minimal composite extensions of the standard model, and for the recent discovery of four-dimensional asymptotically safe theories.Recently he has also applied and developed mathematical tools stemming from theoretical physics to describe the evolution of infectious diseases at human and viral level.His work crosses several realms of particle physics and cosmology from bright and dark extensions of the standard model and inflationary cosmology to the mathematical underpinning of theories of fundamental interactions.

The seminar is cancelled.Speaker: Astrid Eichhorn Department of Physics, Chemistry and Pharmacy University of Southern Denmark Abstract:I will present a tentative answer to this question, consisting in a theory of quantum spacetime and matter, to which I will give a pedagogical introduction. I will argue that the main challenge for such a theory is the question of how to probe it experimentally, given that we expect its effects to manifest at tiny distances below the Planck length of 10^-35 m. I will challenge this expectation and show that not all effects are untestably small. Concretely, I will show how consequences of the microscopic physics leave their imprints on macroscopic scales in particle physics and black holes.Location: D-IAS Aud. (V24-501a-0), Danish Institute for Advanced Study - DIAS. The event is open to all.

Speaker: Apoorv Tiwari (Niels Bohr International Academy, University of Copenhagen) Abstract:In this talk, I'll describe how finite symmetries of quantum systems are naturally encoded in fusion higher categories, presenting a universal and systematic construction of quantum systems with such higher categorical symmetries starting from quantum systems with conventional group-like symmetries. I'll discuss the Symmetry Topological Field Theory (SymTFT), a conceptual tool that facilitates the study of the generalized charges and gapped phases realized in quantum systems with categorical symmetries. Utilizing the SymTFT, I'll demonstrate a systematic construction of lattice models realizing categorical symmetry-protected gapped phases and phase transitions between them. Finally, I'll outline ongoing efforts toward realizing these novel phenomena in experimental cold-atom platforms.

Speaker: Kazuki Ikeda (Stony Brook University) Abstract:In this presentation, I will discuss the crucial role of quantum mathematics in understanding quantum many-body systems such as nuclear physics and condensed matter physics. I will also introduce the frontiers of quantum software development and system integration necessary to realize efficient quantum simulations. In particular, I will discuss quantum language development and the category theoretic construction of quantum compilers, and highlight the important and specific role that quantum mathematics plays in this field. Moreover I will introduce new interactions between algebraic geometry, quantum physics and quantum computation. This includes the novel construction of band theory on general Riemann surfaces, topological quantum computation, quantum error correction. Through my presentation, the audience will grasp the diverse applications and progress in Quantum Mathematics to solve various problems in modern QIST.

Speaker: Nate Bottman (Max Planck Institute for Mathematics) Abstract:I will present the 2-associahedra, which I constructed in 2017 in the context of symplectic geometry, and explain some current developments, focusing on symplectic aspects. First, I will explain how 2-associahedra form the right operadic structure for endowing the Fukaya category with functoriality properties. Specifically, the 2-associahedra lead to the notion of the symplectic (A-infinity,2)-category Symp, which is the natural setting for building functors, associated to Lagrangian correspondences, between Fukaya categories. Second, I will describe a related family of posets called constrainahedra, which Daria Poliakova and I constructed in 2022. The constrainahedra also translate into an algebraic structure in symplectic geometry: in ongoing work with Mohammed Abouzaid and Yunpeng Niu, we aim to show that Fukaya categories of Lagrangian torus fibrations are monoidal A-infinity categories, where the latter notion is constructed using of constrainahedra. This will fulfill a longstanding expectation in mirror symmetry.

Speaker: Kaiwen Sun (Uppsala University) Abstract:Nahm sums, also called fermionic sums are some special q-series which are closely related to 2d CFTs and SCFTs. For example, the renowned Rogers-Ramanujan functions are Nahm sums and also the characters of Lee-Yang model M(5,2). It is an open question to classify the Nahm sums with modularity. In rank 1, it was proved there only exist 7 Nahm sums. In rank 2 and 3, a conjectural list has been given by Don Zagier. I will discuss some recent progress on Nahm sums and generalized Nahm sums. This is based on a joint work with Haowu Wang.

Speaker: Matthias Wilhelm (Niels Bohr Institute) Abstract:In this talk, I will give an overview of the numbers and functions that occur in Quantum Field Theory, which are relevant for the calculation of precision predictions for collider and gravitational-wave experiments. In the simplest cases, the functions that occur are multiple polylogarithms, generalizations of logarithms. They are by now very well understood and allow us to reach precision that would be impossible using any other method; in one case up to the eighth subleading order in perturbation theory! In the next cases, we encounter elliptic integrals, where much progress was made in recent years. Beyond these, we encounter an infinite zoo of further functions involving integrals over Calabi-Yau manifolds, which are currently being explored.

Speaker: N. Asger Mortensen POLIMA – Center for Polariton-driven Light-Matter Interactions Danish Institute for Advanced Study University of Southern Denmark Abstract:Polaritons are hybrid particles – quasiparticles – that result from the strong interaction between light (photons) and matter (such as electrons, excitons or phonons). These quasiparticles exhibit mixed characteristics, combining features of both photons and elementary particles like electrons. Colloquially, they constitute half-light-half-matter excitations. The interaction between light and matter in polaritons leads to unique physical properties and behaviors that are now also being explored in sheets of atomically thin two-dimensional materials and in artificial nanostructures. The electrodynamics of matter and optical phenomena are commonly explored within the framework of classical electrodynamics and semiclassical models for the interactions of light with matter. Materials are commonly assumed homogenous, and light-matter interactions are treated in an intuitive local manner. The plasmonic response of metal nanostructures is one such example, where the understanding of mesoscopic electrodynamics at metal surfaces is, however, becoming increasingly important for both fundamental developments in quantum plasmonics and potential applications in emerging light-based quantum technologies. The seminar will discuss recent examples of quantum nonlocal effects that emerge in surface-polaritonic systems, including metal surfaces, 2D materials, and combinations thereof. Biography: Mortensen is a full professor in the Center for Polariton-driven Light-Matter Interactions (POLIMA) and a Chair of Physics in the Danish Institute for Advanced Study, both at the University of Southern Denmark. Previously, he held a full professorship (faculty since 2004) at the Technical University of Denmark (DTU). In addition to his MSc (1998) and PhD (2001) degrees from DTU, he holds higher doctoral degrees from University of Copenhagen (Dr. Scient., 2021) and DTU (Dr. Tech., 2006). He is a fellow of APS, OSA, SPIE, IOP, and European Academy of Sciences. Currently, he serves the AAAS as an associate editor for Science Advances.

Speaker: Jesse Cohen (University of Hamburg) Abstract:Bordered Floer homology, due to Lipshitz, Ozsváth, and Thurston [LOT], is a generalization of Heegaard Floer homology to 3-manifolds with parametrized boundary. The simplest incarnation of this invariant can be regarded as a differential module CFD(Y) and a pairing theorem of [LOT] tells us that the complex of module homomorphisms between two such modules is homotopy equivalent to the Heegaard Floer complex of the 3-manifold obtained by gluing. We will discuss a topological interpretation of composition of module homomorphisms in this context, and applications thereof, including forthcoming work on deformations of arc algebras and a spectral sequence for links in S^1xS^2.

Speaker: Jian Qiu (Uppsala University) Abstract:I want to explain in this talk the framework of quantization using branes, as developed by Gukov-Witten and later refined by Gaiotto-Witten. This approach embeds the symplectic manifold to be quantized into a bigger space (of double the dimension) as a brane, and the open strings ending on the brane play the role of the operators acting on the Hilbert space.The open strings are expected to be quantized via the A-model, while in practice the open string wave functions are identifiable as holomorphic functions on the larger space and therefore many algebraic approach can be applied for its quantization and bypass the A-model.I will apply Kontsevich's deformation quantization to deform the algebra of operators. Contrary to what one expects from deformation quantisation, it is very computable, thanks to some recently developed technical results due to Barmeier and Wang.I will run you through some examples, the simplest being the Kleinian singularity. In the A_1 case, the quantization gives the Verma module of Usu(2) including the integrality condition for hbar, which Is unusual for deformation quantization.

Speaker: Shuhan Jiang (Max Planck Institute, Leipzig) Abstract:In this talk, I will introduce a geometric framework for classical cohomological field theories (CohFTs) using G^{\star} algebras and variational bicomplexes. This framework enables a mathematical interpretation of scalar and vector supersymmetries, along with a systematic classification of solutions to the descent equations for CohFTs. Furthermore, I will provide a BV-BFV extension of this framework, illustrating its application through the (fully extended) Donaldson-Witten theory.

Speaker: Ce Ji (Peking University) Abstract:Over decades of development of the Witten conjecture, many enumerative geometries are related to integrable hierarchies. On the other hand, many of such theories also admit mirror symmetry in the sense of the remodeling conjecture with an underlying object called the spectral curve. In this talk, we propose a generalization of the Witten conjecture from spectral curve, which produce descendent potential functions related to certain reductions of (multi-component) KP integrable hierarchy. Proof for genus zero spectral curve with one boundary will be sketched, which can be applied to deduce the rKdV integrability of deformed negative r-spin theory, conjectured by Chidambaram--Garcia-Falide--Giacchetto. This talk is based on the joint work with Shuai Guo and Qingsheng Zhang.

Speaker: Cris Kuo (University of Southern California) Abstract:For complex manifolds, the Riemann-Hilbert correspondence generalizes the classical correspondence between finite dimensional local systems and vector bundles with connections to perverse sheaves and regular holonomic D-modules. These later objects are in fact microlocal in nature that they can be regarded as living on the cotangent bundles, and the correspondence admits a microlocalization as well. Continuing from an earlier joint work with Côté, Nadler, and Shende, we will globalize this correspondence to general complex contact manifolds in an upcoming work.

Speaker: Alexander Stottmeister (University of Hannover) Abstract:I will discuss a generalized notion of inductive systems of Banach spaces, coined soft inductive systems, that can serve as a flexible tool to describe the limits of physical theories, e.g., in the context of the thermodynamic limit and phase transitions or renormalization of quantum field theories.General criteria for the convergence of dynamics can be obtained in this setting, which I will illustrate using the example of renormalization of free fermion theories on a one-dimensional spatial lattice. Specifically, this example can be used to prove the conformal invariance of various correlation functions of the classical Ising model using operator algebraic methods.This talk is based on work with L. van Luijk, T.J. Osborne, and R.F. Werner.

Speaker: Guillem Cassasus (QM, SDU) Abstract:We will talk about algebraic structures arising in Lagrangian Floer homology in the presence of a Hamiltonian action of a compact Lie group. First, we will show how the Lagrangian Floer complex can be equipped with an A-infinity module structure over the Morse complex of the group, and how this action permits to define equivariant versions of Floer homology. We will then explain how this structure interacts with the structure of the Fukaya category: both can be packaged into (our version of) an A-infinity bialgebra action, giving an alternative answer to a conjecture of Teleman. This should enable one to build an extended topological field theory corresponding to Donaldson-Floer theory. This is based on two joint work in progress, one with Paul Kirk, Mike Miller-Eismeier and Wai-Kit Yeung, and another with Alex Hock and Thibaut Mazuir.

Speaker: Ingmar Saberi (Ludwig-Maximilians-Universität München) Lectures: Tuesday, Wednesday and Thursday 15:00-16:00. Q&A session: Friday 15:00-16:00.Abstract: In this mini-course, I will discuss some approaches to the geometry of supersymmetric field theory that offer promising new perspectives on issues related to M-theory, in particular on eleven-dimensional supergravity and six-dimensional theories with (2,0) supersymmetry. One key ingredient will be a structural analogy between superspaces and almost-complex manifolds, which allows for a unified formulation of supersymmetric theories and their twists. Another will be the notion of factorization current algebra and the factorization Noether theorem of Costello and Gwilliam. By the end, I hope to have reviewed both some recent progress and some outstanding questions related to twisted eleven-dimensional supergravity.

Speaker: David O'Connell (University of Okinawa) Abstract:Topology change is typically modelled by a Lorentz cobordism, which can be seen as a maximal spacelike hypersurface that globally splits into multiple pieces. In this talk we will take the absolute opposite approach, and consider models of topology change in which individual points are duplicated and then allowed to propagate into an eventual cobordism. As we will see, such models are non-Hausdorff, with the Hausdorff-violation occurring along the future nullcones of the duplicated points. Using recent developments in non-Hausdorff differential geometry, we will evaluate the Einstein-Hilbert action on topology changing spacetimes in two dimensions. Interestingly, even in the case of seemingly-flat geometries there will be non-zero curvature contributions coming from Hausdorff-violating submanifolds that sit inside the spacetime. These observations suggest that such spacetimes will generally be suppressed in a path integral that sums over topologies, with more elaborate branching receiving a stronger suppression.

Speaker: Guglielmo Lockhart (Universität Bonn) Abstract:String theory compactification on elliptic Calabi-Yau manifolds provides a bridge between enumerative geometry on one side, and vertex operator algebras and a modular or Jacobi forms of various kinds on the other. This comes about because compactification leads to supersymmetric quantum field or quantum gravity theories that include two-dimensional objects, the noncritical strings, among their degrees of freedom. Exploiting this connection has led to a deeper understanding both of the spectrum of supersymmetric theories and of the modular properties of generating functions of enumerative invariants. In the first part of the talk I will give a broad overview of this subject, while in the second part I will focus on a recent work with Michele Del Zotto, where we find a link between Donaldson-Thomas invariants of higher rank, vector-valued Jacobi forms, and a two-dimensional analogue of Kronheimer-Nakajima quivers.

Speaker: Muyang Liu (QM, SDU) Abstract:F-theory is remarked by its powerful model building potential due to feasible geometric descriptions of string compactifications. It translates physics concepts in the theory concerned to mathematical objects extracted from the beautiful structures of elliptic fibrations. In particular, the resolved Calabi-Yau manifolds are determined within the framework of toric geometry. This gives a combinatorically simple description of the geometry and can be algorithmized with many computer algebra programs. In this talk, I will explore the searching of explicit models in the language of F-theory geometry, ranging from those admitting exact 4D Minimal Supersymmetric Standard Model (MSSM) matter spectra to novel 6D (1,0) heterotic theories and their T-dual network.

Speaker: Paolo Ghiggini (Université Grenoble Alpes) Abstract:Vincent Colin, Ko Honda, Michael Hutchings and I defined embedded contact homology groups for sutured three-manifolds as a slight modification of Hutching's embedded contact homology for closed three manifolds. I will sketch a strategy to prove that those groups are isomorphic to Juhász's sutured Floer homology, and therefore are topological invariants. The strategy is to extend the knot complement to a larger closed manifold, and then apply the isomorphism between Heegaard Floer homology and embedded contact homology to the closed manifold. In the talk I will focus on the effect of that extension on embedded contact homology, and therefore no knowledge of suitured Floer homology will be necessary beyond the fact that it exists and is interesting. On the other hand the definition of embedded contact homology for sutured three-manifolds will be sketched. This is a joint work in progress with Vincent Colin and Ko Honda.

Speaker: Saebyeok Jeong (Department of Theoretical Physics, CERN) Abstract:The Coulomb branch of four-dimensional N=2 supersymmetric gauge theory of class S is well-known to be identical to the moduli space of Hitchin's equations. I will consider the case of genus-0 with regular marked points, where the associated integrable system is the Gaudin model. I'll explain how the quantization of the Gaudin model can be formulated in the N=2 supersymmetric gauge theory with the help of half-BPS surface defects. We consider two types of surface defects: the "canonical" surface defect and the "regular monodromy" surface defect, inserted on top of each other. The action of the former on the latter is diagonal due to cluster decomposition, interpreted as the Hecke eigensheaf property with the eigenvalue given by the oper. In this way, we formulate the geometric Langlands correspondence in the N=2 gauge theory setting. I will also explain that the separation of variables and the bispectral duality of the quantum Hitchin integrable system can be accounted for by surface defect transitions.

Speaker: Greg Stevenson (AU) Abstract:Differential graded algebras, which consist of a graded algebra together with a square-zero derivation, are an algebraic way of modelling multiplication up to homotopy. They are well adapted to, and arise naturally from, the study of derived categories in algebraic geometry. In general, the presence of the derivation makes these very complicated objects which are not necessarily amenable to usual techniques from ring theory.In my talk I'll explain some recent results, by my student Isambard Goodbody and others, in the setting where the underlying algebra is finite dimensional over some field. In this case, a surprising number of results from the theory of finite dimensional algebras can be generalized to the differential graded setting.

Location D-IAS Auditorium, Fioniavej 34, 5230 OdenseProgramme: 09:00 Welcome by DIREC Managing Director Thomas Riisgaard Hansen09:10 Topological Quantum Computing, Prof. & QM Centre Director Jørgen Ellegaard Andersen10:20 Coffee break10:35 Gaussian Boson Sampling, Assistant Professor Shan Shan11:15 Coffee break11:30 Measurement-based Quantum Computing, PhD Student Santiago Q. RiosAll are welcomeCo-organised by DIREC Note: This event will be followed by the SDU Quantum Hub Opening Event – same location from 13:00.

Speaker: Greyson Potter (SDU) Abstract:We develop a computational approach to analyzing the generalized volume conjecture, which relates topological recursion and Chern-Simons theory for hyperbolic 3-manifolds with torus boundary. The conjecture states that there is an equality between two series in a formal parameter hbar: the non-perturbative wave function from topological recursion and a state-integral for SL(2,C) Chern-Simons theory. Both series are expressible as sums over graphs. We develop algorithms for efficiently computing the graph sum for the non-perturbative wave function via quantum Airy structures and a software package, called qairy, which implements our algorithms. We further develop tools and techniques which are widely applicable to the calculation and analysis of the non-perturbative wave functions associated to genus one spectral curves. Using these tools, we are able to verify the generalized volume conjecture in several cases up to much higher order in hbar than was previously accessible as well as analyze the arithmetic aspects of the conjecture.

Speaker: Changlong Zhong (State University of New York at Albany) Abstract:Equivariant elliptic cohomology of symplectic resolutions was recently studied by Okounkov and his collaborators. For example, elliptic stable envelop is defined and it is closely related to geometric representation theory, mathematical physics and 3d mirror symmetry. For the cotangent bundle T^*G/B, it basically says that the restriction to torus fixed points of elliptic stable envelops are related with that for the Langlands dual. This is proved by Rimanyi-Weber. In this talk, I will focus on the elliptic Demazure-Lusztig operators that generate the elliptic classes corresponding to elliptic stable envelop. The (sheaf of) modules spanned by these classes are called the periodic module. The main result of this talk will show that the elliptic Demazure-Lusztig operators can be assembled together to obtain a canonical isomorphism between the periodic module and that for the Langlands dual system. In particular, it recovers Rimanyi-Weber's result. This is joint work with C. Lenart and G. Zhao.

Speaker: Andres Ibanez Nunes (University of Oxford) Abstract:In work of Haiden-Katzarkov-Konsevich-Pandit (HKKP), a canonical filtration, labeled by sequences of real numbers, of a semistable quiver representation or vector bundle on a curve is defined. The HKKP filtration is a purely algebraic object, yet it governs the asymptotic behaviour of a natural gradient flow on the space of metrics of the object.In this talk, we show that the HKKP filtration can be recovered from the stack of semistable objects and a so called norm on graded points. This approach gives a generalisation of the HKKP filtration to other moduli problems of non-linear origin.

Speaker: Guillaume Laplante-Anfossi (University of Melbourne) Abstract:On the one hand, simplex equations are higher dimensional generalizations of the Yang—Baxter (aka triangle) equation. The tetrahedron equation was introduced by Zamolodchikov in 1980, and the equations for higher dimensional simplices can be defined via higher Bruhat orders, a family of posets introduced by Manin and Schechtman in 1989. On the other hand, Steenrod squares are operations acting on the mod 2 cohomology of a topological space, introduced by Steenrod in 1947. They arise from correcting homotopically the lack of cocommutativity of the Alexander—Whitney diagonal at the cochain level. Together with Nicholas Williams, we found a surprising bijection between the terms of these two constructions: each side of a higher Yang—Baxter equation defines a Steenrod cup-i coproduct, and vice-versa. A conceptual explanation for, as well as consequences of this result remain unknown to us, and I hope to stimulate discussions about what could grow out of this new dictionary between pieces of mathematical physics and classical algebraic topology.

Speaker: Matthias Christandl (University of Copenhagen) Abstract:Molecular biology and biochemistry interpret microscopic processes in the living world in terms of molecular structures and their interactions, which are quantum mechanical by their very nature. Whereas the theoretical foundations of these interactions are very well established, the computational solution of the relevant quantum mechanical equations is very hard. However, much of molecular function in biology can be understood in terms of classical mechanics, where the interactions of electrons and nuclei have been mapped onto effective classical surrogate potentials that model the interaction of atoms or even larger entities. The simple mathematical structure of these potentials offers huge computational advantages; however, this comes at the cost that all quantum correlations and the rigorous many-particle nature of the interactions are omitted. In this work, we discuss how quantum computation may advance the practical usefulness of the quantum foundations of molecular biology by offering computational advantages for simulations of biomolecules. We not only discuss typical quantum mechanical problems of the electronic structure of biomolecules in this context, but also consider the dominating classical problems (such as protein folding and drug design) as well as data-driven approaches of bioinformatics and the degree to which they might become amenable to quantum simulation and quantum computation.

Speaker: Robin Kaarsgaard Sales (University of Southern Denmark) Abstract:We provide a universal construction of the category offinite-dimensional C*-algebras and completely positivetrace-nonincreasing maps from the rig category offinite-dimensional Hilbert spaces and unitaries. Thisconstruction, which can be applied to any dagger rig category, isdescribed in three steps, each associated with their ownuniversal property, and draws on results from dilation theory infinite dimension. In this way, we explicitly construct thecategory that captures hybrid quantum/classical computation withpossible nontermination from the category of its reversiblefoundations. We discuss how this construction can be used in thedesign and semantics of quantum programming languages.This talk is based on joint work with Pablo Andrés-Martínez (Quantinuum) and Chris Heunen (University of Edinburgh).

Speaker: Noémie Legout (University of Uppsala) Abstract:To a Legendrian submanifold in the contactisation of a Liouville domain, one can associate a differential graded algebra (DGA) called the Chekanov-Eliashberg algebra (C-E algebra), and from which many Legendrian isotopy invariants are defined. Given any DGA, it is a hard question to know if it can be the C-E algebra of a Legendrian or not. In this talk, we will give a characterization of the C-E algebra, namely we show that for a displaceable Legendrian sphere this algebra is a Calabi-Yau DGA. This means that there is a quasi-isomorphism of DG-bimodules between the diagonal bimodule and the inverse dualizing bimodule. In order to prove this, we define another complex associated to a Legendrian, the Rabinowitz complex, and show that the acyclicity of this complex implies the existence of a Calabi-Yau structure. If time permits we will briefly talk about a work in progress with Georgios Dimitroglou-Rizell for the case when the Legendrian is not necessarily a sphere. In this case, and under certain hypothesis, the C-E algebra admits a relative Calabi-Yau structure.

Speaker: William Elbæk Mistegård (University of Southern Denmark) Abstract:In his seminal work on quantum Chern-Simons field theory and the Jones polynomial, Witten envisioned a 2+1 dimensional TQFT. Further, he presented two procedures for constructing the underlying modular functor, namely by conformal field theory (CFT) and by geometric Kähler quantization of the moduli spaces of flat principal bundles on two-manifolds. Subsequently, Reshetikhin and Turaev constructed a TQFT using surgery of three-manifolds and representation theory of quantum groups, and this is now known as the WRT-TQFT. The CFT approach was initially developed by Tsuchiya, Ueno and Yamada, and then further refined to a full modular functor by Andersen and Ueno, who also established that this is equivalent to the WRT-TQFT. The geometric Kähler quantization approach was initialized by Axelrod, Witten and Pietra, and Hitchin. The quantization approach results in a projective representation of the mapping class group of a two-manifold, which is known due to Laszlo to be projectively equivalent to the representation of the WRT-TQFT. However, the full TQFT is not yet described from the point of view of quantization. In particular, the so-called TQFT factorization axiom has not been proved using only geometric quantization with respect to Kähler polarizations. However, Andersen have understood factorization from the point of view of geometric quantization with respect to reducible non-negative polarizations. Jeffrey and Weitsman introduced a different quantization procedure. Given a trinion decomposition of a closed two-manifold S, one can obtain a system of functions on the moduli space of flat bundles on the two-manifold S. This is given by taking the trace of the holonomy representations of the boundary circles of the trinions. This results in a fibration of the moduli space, the fibres of this map being Lagrangian tori. The vertical distribution of this fibration is a polarization in the sense of geometric quantization. Jeffrey and Weitsman considered the so-called Bohr-Sommerfeld quantization of this system (i.e. this polarization), and proved that it results in a finite dimensional Hilbert space, which we denote by V(S). Further, they showed that V(S) has dimension equal to the dimension of the Hilbert space of the WRT-TQFT (this dimension is the famous Verlinde formula). In this talk, we present aspects of an ongoing project on quantization and TQFT, which is complementary to the factorization construction of Andersen. We consider the two-manifold S' obtained by cutting along a circle of the trinion decomposition. This two-manifold have two boundaries and an induced trinion decomposition. Following Jeffrey and Weitsman, we consider for each element j in the gauge group, the associated Lagrangian torus fibration of the moduli space of flat bundles on S' with boundary holonomy conjugate to j. We consider the Bohr-Sommerfeld quantization of this system, which we denote by V(S',j). We show that in accordance with the TQFT factorization axiom, the Hilbert space of the closed surface, denoted above by V(S), splits as a direct sum indexed by elements of the WRT-TQFT label set, the summand indexed by j being equal to V(S',j). Further, we show that this direct sum decomposition is conjugate to factorization isomorphism of the WRT-TQFT. Thus these results gives a geometric illumination of the factorization axiom of the WRT-TQFT by means of the Bohr-Sommerfeld quantization of Chern-Simons theory.

Speaker: William Elbæk Mistegård (University of Southern Denmark) Abstract:It was envisioned by Witten that quantum Chern-Simons field theory is a 2+1 dimensional topological quantum field theory (TQFT) in the sense of Atiyah. Further, Witten argued that the link invariant known as the Jones polynomial, can be computed in this TQFT as the expectation value of Wilson loop operators. This gave an intrinsic definition of the Jones polynomial, as opposed to the known combinatorial definition, and thus solved an important problem in topology. Subsequently, a full TQFT was constructed mathematically rigorously by Reshetikhin and Turaev, using surgery presentations of three-manifolds and modular tensor categories. This is now known as the Witten-Reshetikhin-Turaev TQFT. To a closed oriented three-manifold M, the WRT-TQFT associates a topological invariant in the form of a complex number. This is called the quantum invariant of M. These invariants are essentially the building blocks of the WRT-TQFT. The goals of this introductory talk are to introduce the quantum invariants and to present the axioms of TQFT (or more precisely, the axioms of a modular functor).

Speaker: Tobias Dyckerhoff (University of Hamburg) Abstract:Derived categories have come to play a decisive role in a wide range of topics. Several recent developments, in particular in the context of topological Fukaya categories, arouse the desire to study not just single categories, but rather complexes of categories. In this talk, we will discuss examples of such complexes in algebra, topology, algebraic geometry, and symplectic geometry, along with some results and conjectures involving them.

Speaker: Robin Kaarsgaard Sales (University of Southern Denmark) Abstract:We provide a universal construction of the category offinite-dimensional C*-algebras and completely positivetrace-nonincreasing maps from the rig category offinite-dimensional Hilbert spaces and unitaries. Thisconstruction, which can be applied to any dagger rig category, isdescribed in three steps, each associated with their ownuniversal property, and draws on results from dilation theory infinite dimension. In this way, we explicitly construct thecategory that captures hybrid quantum/classical computation withpossible nontermination from the category of its reversiblefoundations. We discuss how this construction can be used in thedesign and semantics of quantum programming languages.This talk is based on joint work with Pablo Andrés-Martínez (Quantinuum) and Chris Heunen (University of Edinburgh).

Seminar by Mikhail Gorsky (University of Vienna) Abstract: A braid variety is a certain affine algebraic variety associated with a simple algebraic group G and a positive braid of the corresponding type. These varieties generalise open Richardson varieties and appear in the context of symplectic topology and in the study of link invariants such as HOMFLY-PT polynomials and Khovanov-Rozansky homology. In this talk, I will give a proof of the existence of cluster A-structures and cluster Poisson structures on any braid variety. I will sketch an explicit construction of cluster seeds involving the diagrammatic calculus of weaves and explain certain properties of these cluster algebras such as local acyclicity. The talk is based on joint work with Roger Casals, Eugene Gorsky, Ian Le, Linhui Shen, and José Simental (arXiv:2207.11607).

Seminar by Côme Dattin (Uppsala University) Abstract: Starting with a Riemannian manifold M, its unit bundle UM is a contact manifold, whose Reeb vector field lifts the geodesic flow. Given a submanifold N, its unit conormal is a Legendrian submanifold in UM. Thus this construction takes us from the smooth world to the contact world, and one may use Legendrian invariants to study smooth (sub)manifolds.In this talk we will show that, if the unit conormals of two surface braids are Legendrian isotopic (relatively to their boundary), then the braids are equivalent. The main tool will be an invariant of Legendrians with boundary, called the stopped sutured homology. Using its associated exact sequence, we recover a classical invariant of braids, namely an automorphism of the punctured surface.

Seminar by Ingmar Saberi (University of Munich) Abstract:In recent years, there has been a great deal of progress on ideas related to twisted supergravity, building on the definition given by Costello and Li. Much of what is explicitly known about these theories comes from the topological B- model, whose string field theory conjecturally produces the holomorphic twist of type IIB supergravity. Progress on eleven-dimensional supergravity has been hindered, in part, by the lack of such a worldsheet approach. I will discuss a rigorous computation of the twist of the free eleven-dimensional supergravity multiplet, as well as an interacting BV theory with this field content that passes a large number of consistency checks. Surprisingly, the resulting holomorphic theory on flat space is closely related to the infinite-dimensional exceptional simple Lie superalgebra E(5|10), and another such exceptional simple algebra - E(3|6) - appears in the context of M5-branes. This is joint work with Surya Raghavendran and Brian Williams.

Seminar by Bingyu Zhang (SDU, QM) Abstract: To bounded open sets in a cotangent bundle, we can associate a pair of sheaves, the microlocal kernels, following the idea of Tamarkin and Chiu. Using microlocal kernels for the open set, we can define the Chiu-Tamarkin complex, which is a cohomology invariant associated with the open set. In this talk, I would like to explain the construction of these cohomology theories and some computations. Then I would like to explain how to construct a sequence of capacities based on the Chiu-Tamarkin complex.

By H2 we denote the Lie algebra of polynomial hamiltonian vector fields on the plane. We consider the moduli space of stable twisted Higgs bundles on an algebraic curve of given coprime rank and degree. De Cataldo, Hausel and Migliorini proved in the case of rank 2 and conjectured in arbitrary rank that two natural filtrations on the cohomology of the moduli space coincide. One is the weight filtration W coming from the Betti realization, and the other one is the perverse filtration P induced by the Hitchin map. Motivated by computations of the Khovanov-Rozansky homology of links by Gorsky, Hogancamp and myself, we look for an action of H2 on the cohomology of the moduli space. We find it in the algebra generated by two kinds of natural operations: on the one hand we have the operations of cup product by tautological classes, and on the other hand we have the Hecke operators acting via certain correspondences. We then show that both P and W coincide with the filtration canonically associated to the sl2 subalgebra of H2. Based on joint work in progress with Hausel, Minets and Schiffmann.Read more about Assistant Professor Anton Mellit from University of Vienna.