@inproceedings{18847, abstract = {Machine Learning and AI have the potential to transform data-driven scientific discovery, enabling accurate predictions for several scientific phenomena. As many scientific questions are inherently causal, this paper looks at the causal inference task of treatment effect estimation, where the outcome of interest is recorded in high-dimensional observations in a Randomized Controlled Trial (RCT). Despite being the simplest possible causal setting and a perfect fit for deep learning, we theoretically find that many common choices in the literature may lead to biased estimates. To test the practical impact of these considerations, we recorded ISTAnt, the first real-world benchmark for causal inference downstream tasks on high-dimensional observations as an RCT studying how garden ants (Lasius neglectus) respond to microparticles applied onto their colony members by hygienic grooming. Comparing 6 480 models fine-tuned from state-of-the-art visual backbones, we find that the sampling and modeling choices significantly affect the accuracy of the causal estimate, and that classification accuracy is not a proxy thereof. We further validated the analysis, repeating it on a synthetically generated visual data set controlling the causal model. Our results suggest that future benchmarks should carefully consider real downstream scientific questions, especially causal ones. Further, we highlight guidelines for representation learning methods to help answer causal questions in the sciences.}, author = {Cadei, Riccardo and Lindorfer, Lukas and Cremer, Sylvia and Schmid, Cordelia and Locatello, Francesco}, booktitle = {ICML 2024 Workshop AI4Science}, publisher = {Curran Associates}, title = {{Smoke and mirrors in causal downstream tasks}}, volume = {38}, year = {2024}, } @inproceedings{19509, abstract = {Machine Learning and AI have the potential to transform data-driven scientific discovery, enabling accurate predictions for several scientific phenomena. As many scientific questions are inherently causal, this paper looks at the causal inference task of treatment effect estimation, where the outcome of interest is recorded in high-dimensional observations in a Randomized Controlled Trial (RCT). Despite being the simplest possible causal setting and a perfect fit for deep learning, we theoretically find that many common choices in the literature may lead to biased estimates. To test the practical impact of these considerations, we recorded ISTAnt, the first real-world benchmark for causal inference downstream tasks on high-dimensional observations as an RCT studying how garden ants (Lasius neglectus) respond to microparticles applied onto their colony members by hygienic grooming. Comparing 6 480 models fine-tuned from state-of-the-art visual backbones, we find that the sampling and modeling choices significantly affect the accuracy of the causal estimate, and that classification accuracy is not a proxy thereof. We further validated the analysis, repeating it on a synthetically generated visual data set controlling the causal model. Our results suggest that future benchmarks should carefully consider real downstream scientific questions, especially causal ones. Further, we highlight guidelines for representation learning methods to help answer causal questions in the sciences. Code: https://github.com/CausalLearningAI/ISTAnt Data: https://doi.org/10.6084/m9.figshare.26484934.v2}, author = {Cadei, Riccardo and Lindorfer, Lukas and Cremer, Sylvia and Schmid, Cordelia and Locatello, Francesco}, booktitle = {38th Conference on Neural Information Processing Systems}, issn = {1049-5258}, location = {Vancouver, Canada}, publisher = {Neural Information Processing Systems Foundation}, title = {{Smoke and mirrors in causal downstream tasks}}, doi = {10.52202/079017-0820}, volume = {37}, year = {2024}, } @article{17461, abstract = {Socially living animals can counteract disease through cooperative defences, leading to social immunity that collectively exceeds the sum of individual defences. In superorganismal colonies of social insects with permanent caste separation between reproductive queen(s) and nonreproducing workers, workers are obligate altruists and thus engage in unconditional social immunity, including highly specialised and self-sacrificial hygiene behaviours. Contrastingly, cooperation is facultative in cooperatively breeding families, where all members are reproductively totipotent but offspring transiently forgo reproduction to help their parents rear more siblings. Here, helpers should either express condition-dependent social immunity or disperse to pursue independent reproduction. We advocate inclusive fitness theory as a framework to predict when and how indirect fitness gains may outweigh direct fitness costs, thus favouring conditional social immunity.}, author = {Cremer, Sylvia and Pull, Christopher}, issn = {1471-5007}, journal = {Trends in Parasitology}, number = {9}, pages = {780--787}, publisher = {Elsevier}, title = {{Unconditional versus condition-dependent social immunity}}, doi = {10.1016/j.pt.2024.07.014}, volume = {40}, year = {2024}, } @article{14479, abstract = {In animals, parasitic infections impose significant fitness costs.1,2,3,4,5,6 Infected animals can alter their feeding behavior to resist infection,7,8,9,10,11,12 but parasites can manipulate animal foraging behavior to their own benefits.13,14,15,16 How nutrition influences host-parasite interactions is not well understood, as studies have mainly focused on the host and less on the parasite.9,12,17,18,19,20,21,22,23 We used the nutritional geometry framework24 to investigate the role of amino acids (AA) and carbohydrates (C) in a host-parasite system: the Argentine ant, Linepithema humile, and the entomopathogenic fungus, Metarhizium brunneum. First, using 18 diets varying in AA:C composition, we established that the fungus performed best on the high-amino-acid diet 1:4. Second, we found that the fungus reached this optimal diet when given various diet pairings, revealing its ability to cope with nutritional challenges. Third, we showed that the optimal fungal diet reduced the lifespan of healthy ants when compared with a high-carbohydrate diet but had no effect on infected ants. Fourth, we revealed that infected ant colonies, given a choice between the optimal fungal diet and a high-carbohydrate diet, chose the optimal fungal diet, whereas healthy colonies avoided it. Lastly, by disentangling fungal infection from host immune response, we demonstrated that infected ants foraged on the optimal fungal diet in response to immune activation and not as a result of parasite manipulation. Therefore, we revealed that infected ant colonies chose a diet that is costly for survival in the long term but beneficial in the short term—a form of collective self-medication.}, author = {Csata, Eniko and Perez-Escudero, Alfonso and Laury, Emmanuel and Leitner, Hanna and Latil, Gerard and Heinze, Juerge and Simpson, Stephen and Cremer, Sylvia and Dussutour, Audrey}, issn = {1879-0445}, journal = {Current Biology}, number = {4}, pages = {902--909.e6}, publisher = {Elsevier}, title = {{Fungal infection alters collective nutritional intake of ant colonies}}, doi = {10.1016/j.cub.2024.01.017}, volume = {34}, year = {2024}, } @misc{18895, abstract = {ISTAnt is a new ecological dataset for social immunity and represents the first real-world benchmark for causal inference downstream tasks on high-dimensional observations. It analyzes grooming behavior in the ant Lasius neglectus in groups of three worker ants. The workers for the experiment were obtained from their laboratory stock colony, which had been collected from the field in 2022 in the Botanical Garden Jena, Germany. Ant collection and all experimental work were performed in compliance with international, national and institutional regulations and ethical guidelines. For the experiment, the body surface of one of the three ants was treated with a suspension of either of two microparticle types (diameter ~5 µm) to induce grooming by the two nestmates, which were individually color-coded by application of a dot of blue or orange paint, respectively. The three ants were housed in small plastic containers (diameter 28mm, height 30mm) with moistened, plastered ground and the interior walls covered with PTFE (polytetrafluoroethane) to hamper climbing by the ants. Filming occurred in a temperature- and humidity-controlled room at 23°C within a custom-made filming box with controlled lighting and ventilation conditions. We set up nine ant groups at a time (always containing both treatments) and placed them randomly on positions 1-9 marked on the floor in a 3x3 grid, about 3mm from each other. The experiment was performed on two consecutive days. Videos were acquired using a USB camera (FLIR blackfly S BFS-U3-120S4C, Teledyne FLIR) with a high-performance lens (HP Series 25mm Focal Length, Edmund optics 86-572) in OBS studio 29.0.0 \citep{bailey2017obs} at a framerate of 30 FPS and a resolution of 2500x2500 pixels. From each original video (105x105 mm), we generated nine individual videos .mkv (each ~32x32 mm, 770x770 pixels) by determining exact coordinates per container from one frame in GIMP 2.10.36 and cropping of the videos with FFmpeg 6.1.1. Annotation was performed over two consecutive days by three observers who had not been involved in the experimental setup or recording and were unaware of the treatment assignments to ensure bias-free behavioral annotation. They annotated the behavior of the ants during video observations, using custom-made software that saves the start and end frames of behaviors marked in a .csv file (see 'annotations' folder). In one of the videos, one of the nestmates' legs got inadvertently stuck to its body surface during the color-coding, interfering with its behavior, so the video was discarded. This left 44 videos from 5 independent setups (n=24 of treatment 1 and n=20 of treatment 2) of 10 minutes each for a total of 792 000 annotated frames (see 'video' folder). For each video, we provide the following information: the number of the set to which it belongs (1-5); the number of the position within the set reflecting the position of the ant group under the camera (1-9), for which we also provide ‘coordinates’ in the 3x3 grid (taking values -1/0/1 for both X and Y axis); treatment (1 or 2); the hour of the day when the recording was started (in 24h CEST); experimental day (A or B); the top left coordinate of the cropping square from the original video (CropX/CropY); the person annotating the video (given as A, B, C); the date of annotation (1: first day, 2: second day) and in which order the videos were annotated by each person, both reflecting a possible training effect of the person (see 'experiments_settings.csv' file).}, author = {Cadei, Riccardo and Locatello, Francesco and Cremer, Sylvia M and Lindorfer, Lukas and Schmid, Cordelia}, publisher = {Institute of Science and Technology Austria}, title = {{ISTAnt}}, doi = {10.6084/M9.FIGSHARE.26484934.V2}, year = {2024}, } @article{18171, abstract = {Defense against pathogens and parasites requires substantial investment of energy and resources on part of the host. This makes the host immune function dependent on availability and accessibility of resources. A resource deprived host is therefore expected to be more susceptible to infections, although empirical results do not always align with this prediction. Limiting host access to resources can additionally impact within-host pathogen numbers, either directly by altering the amount of resources available to the pathogens for proliferation or indirectly by altering the efficiency of the host immune system. We tested for the effects of host starvation (complete deprivation of resources) on susceptibility to bacterial pathogens, and within-host pathogen proliferation, in Drosophila melanogaster females. Our results show that starvation increases post-infection mortality of the host, but in a pathogen-specific manner. This increase in mortality is always accompanied by increased within-host pathogen proliferation. We therefore propose that starvation compromises host resistance to bacterial infections in Drosophila melanogaster females thereby increasing susceptibility to infections.}, author = {Basu, Aabeer and Singh, Aparajita and Sehgal, Suhaas and Madaan, Tanvi and Prasad, Nagaraj Guru}, issn = {1096-0805}, journal = {Journal of Invertebrate Pathology}, number = {11}, publisher = {Elsevier}, title = {{Starvation increases susceptibility to bacterial infection and promotes systemic pathogen proliferation in Drosophila melanogaster females}}, doi = {10.1016/j.jip.2024.108209}, volume = {207}, year = {2024}, } @article{14986, abstract = {We prove a version of the tamely ramified geometric Langlands correspondence in positive characteristic for GLn(k). Let k be an algebraically closed field of characteristic p>n. Let X be a smooth projective curve over k with marked points, and fix a parabolic subgroup of GLn(k) at each marked point. We denote by Bunn,P the moduli stack of (quasi-)parabolic vector bundles on X, and by Locn,P the moduli stack of parabolic flat connections such that the residue is nilpotent with respect to the parabolic reduction at each marked point. We construct an equivalence between the bounded derived category Db(Qcoh(Loc0n,P)) of quasi-coherent sheaves on an open substack Loc0n,P⊂Locn,P, and the bounded derived category Db(D0Bunn,P-mod) of D0Bunn,P-modules, where D0Bunn,P is a localization of DBunn,P the sheaf of crystalline differential operators on Bunn,P. Thus we extend the work of Bezrukavnikov-Braverman to the tamely ramified case. We also prove a correspondence between flat connections on X with regular singularities and meromorphic Higgs bundles on the Frobenius twist X(1) of X with first order poles .}, author = {Shen, Shiyu}, issn = {1687-0247}, journal = {International Mathematics Research Notices}, keywords = {General Mathematics}, number = {7}, pages = {6176--6208}, publisher = {Oxford University Press}, title = {{Tamely ramified geometric Langlands correspondence in positive characteristic}}, doi = {10.1093/imrn/rnae005}, volume = {2024}, year = {2024}, } @article{18108, abstract = {Here we announce the construction and properties of a big commutative subalgebra of the Kirillov algebra attached to a finite dimensional irreducible representation of a complex semisimple Lie group. They are commutative finite flat algebras over the cohomology of the classifying space of the group. They are isomorphic with the equivariant intersection cohomology of affine Schubert varieties, endowing the latter with a new ring structure. Study of the finer aspects of the structure of the big algebras will also furnish the stalks of the intersection cohomology with ring structure, thus ringifying Lusztig’s q-weight multiplicity polynomials i.e., certain affine Kazhdan–Lusztig polynomials.}, author = {Hausel, Tamás}, issn = {1091-6490}, journal = {Proceedings of the National Academy of Sciences of the United States of America}, number = {38}, publisher = {National Academy of Sciences}, title = {{Commutative avatars of representations of semisimple Lie groups}}, doi = {10.1073/pnas.2319341121}, volume = {121}, year = {2024}, } @article{15248, abstract = {Applying the technique of p-adic integration, we prove the topological mirror symmetry conjecture of Hausel-Thaddeus for the moduli spaces of (strongly) parabolic Higgs bundles for the structure groups SLn and PGLn, building on previous work of Groechenig-Wyss-Ziegler on the non-parabolic case. We also prove the E-polynomial of the smooth moduli space of parabolic GLn-Higgs bundles is independent of the degree of the underlying vector bundles.}, author = {Shen, Shiyu}, issn = {1090-2082}, journal = {Advances in Mathematics}, number = {5}, publisher = {Elsevier}, title = {{Mirror symmetry for parabolic Higgs bundles via p-adic integration}}, doi = {10.1016/j.aim.2024.109616}, volume = {443}, year = {2024}, } @article{14930, abstract = {In this paper we investigate locally free representations of a quiver Q over a commutative Frobenius algebra R by arithmetic Fourier transform. When the base field is finite we prove that the number of isomorphism classes of absolutely indecomposable locally free representations of fixed rank is independent of the orientation of Q. We also prove that the number of isomorphism classes of locally free absolutely indecomposable representations of the preprojective algebra of Q over R equals the number of isomorphism classes of locally free absolutely indecomposable representations of Q over R[t]/(t2). Using these results together with results of Geiss, Leclerc and Schröer we give, when k is algebraically closed, a classification of pairs (Q, R) such that the set of isomorphism classes of indecomposable locally free representations of Q over R is finite. Finally when the representation is free of rank 1 at each vertex of Q, we study the function that counts the number of isomorphism classes of absolutely indecomposable locally free representations of Q over the Frobenius algebra Fq[t]/(tr). We prove that they are polynomial in q and their generating function is rational and satisfies a functional equation.}, author = {Hausel, Tamás and Letellier, Emmanuel and Rodriguez-Villegas, Fernando}, issn = {1420-9020}, journal = {Selecta Mathematica}, number = {2}, publisher = {Springer Nature}, title = {{Locally free representations of quivers over commutative Frobenius algebras}}, doi = {10.1007/s00029-023-00914-2}, volume = {30}, year = {2024}, } @article{15339, abstract = {We define even very stable Higgs bundles and study the Hitchin map restricted to their upward flows. In the GLn case, we classify the type (1,…,1) examples, and find that they are governed by a root system formed by the roots of even height. We discuss how the spectrum of equivariant cohomology of real and quaternionic Grassmannians, 4n-spheres and the real Cayley plane appear to describe the Hitchin map on even cominuscule upward flows. The even upward flows in question are the same as upward flows in Higgs bundle moduli spaces for quasi-split inner real forms. The latter spaces have been pioneered by Oscar García-Prada and his collaborators.}, author = {González, Miguel and Hausel, Tamás}, issn = {1793-6519}, journal = {International Journal of Mathematics}, number = {09}, publisher = {World Scientific Publishing}, title = {{Hitchin map on even very stable upward flows}}, doi = {10.1142/S0129167X2441009X}, volume = {35}, year = {2024}, } @article{18970, abstract = {Given a smooth projective curve C, nonabelian Hodge theory gives a diffeomorphism between two different moduli spaces associated to C. The first is the moduli space of Higgs bundles on C of rank n, which is equipped with the structure of an algebraic completely integrable Hamiltonian system. The second is the character variety of representations of the fundamental group of C into GL(n). In 2012, de Cataldo, Hausel, and Migliorini [1] proposed the P=W conjecture which identifies the perverse filtration on the cohomology of the Higgs moduli space with the weight filtration on the cohomology of the character variety. Recently, in 2022, two independent proofs of the P=W Conjecture appeared, in work of Maulik &Shen [2] and Hausel, Mellit, Minets &Schiffmann [6]. The aim of the Arbeitsgemeinschaft was to understand the P=W Conjecture and these two recent proofs.}, author = {Hausel, Tamás and Maulik, Davesh and Mellit, Anton and Schiffmann, Olivier and Shen, Junliang}, issn = {1660-8941}, journal = {Oberwolfach Reports}, number = {2}, pages = {949--1004}, publisher = {EMS Press}, title = {{Arbeitsgemeinschaft: Geometry and representation theory around the P=W conjecture}}, doi = {10.4171/owr/2024/16}, volume = {21}, year = {2024}, } @article{17292, abstract = {The Gibbons-Hawking ansatz provides a large family of circle-invariant hyperkähler 4-manifolds, and thus Calabi-Yau 2-folds. In this setting, we prove versions of the Thomas conjecture on existence of special Lagrangian representatives of Hamiltonian isotopy classes of Lagrangians, and the Thomas-Yau conjecture on longtime existence of the Lagrangian mean curvature ow. We also make observations concerning closed geodesics, curve shortening flow and minimal surfaces.}, author = {Lotay, Jason D. and Oliveira, Goncalo}, issn = {0022-040X}, journal = {Journal of Differential Geometry}, number = {3}, pages = {1121--1184}, publisher = {International Press of Boston}, title = {{Special Lagrangians, Lagrangian mean curvature flow and the Gibbons-Hawking ansatz}}, doi = {10.4310/jdg/1717348872}, volume = {126}, year = {2024}, } @phdthesis{17156, abstract = {This dissertation is the summary of the author’s work, concerning the relations between cohomology rings of algebraic varieties and rings of functions on zero schemes and fixed point schemes. For most of the thesis, the focus is on smooth complex varieties with an action of a principally paired group, e.g. a parabolic subgroup of a reductive group. The fundamental theorem 5.2.11 from co-authored article [66] says that if the principal nilpotent has a unique zero, then the zero scheme over the Kostant section is isomorphic to the spectrum of the equivariant cohomology ring, remembering the grading in terms of a C^* action. A similar statement is proved also for the G-invariant functions on the total zero scheme over the whole Lie algebra. Additionally, we are able to prove an analogous result for the GKM spaces, which poses the question on a joint generalisation. We also tackle the situation of a singular variety. As long as it is embedded in a smooth variety with regular action, we are able to study its cohomology as well by means of the zero scheme. In case of e.g. Schubert varieties this determines the cohomology ring completely. In largest generality, this allows us to see a significant part of the cohomology ring. We also show (Theorem 6.2.1) that the cohomology ring of spherical varieties appears as the ring of functions on the zero scheme. The computational aspect is not easy, but one can hope that this can bring some concrete information about such cohomology rings. Lastly, the K-theory conjecture 6.3.1 is studied, with some results attained for GKM spaces. The thesis includes also an introduction to group actions on algebraic varieties. In particular, the vector fields associated to the actions are extensively studied. We also provide a version of the Kostant section for arbitrary principally paired group, which parametrises the regular orbits in the Lie algebra of an algebraic group. Before proving the main theorem, we also include a historical overview of the field. In particular we bring together the results of Akyildiz, Carrell and Lieberman on non-equivariant cohomology rings.}, author = {Rychlewicz, Kamil P}, issn = {2663-337X}, keywords = {equivariant cohomology, zero schemes, algebraic groups, Lie algebras}, pages = {117}, publisher = {Institute of Science and Technology Austria}, title = {{Equivariant cohomology and rings of functions}}, doi = {10.15479/at:ista:17156}, year = {2024}, } @article{17127, abstract = {Let P(x)∈Z[x] be a polynomial with at least two distinct complex roots. We prove that the number of solutions (x1,…,xk,y1,…,yk)∈[N]2k to the equation ∏1≤i≤kP(xi)=∏1≤j≤kP(yj)≠0 (for any k≥1 ) is asymptotically k!Nk as N→+∞. This solves a question first proposed and studied by Najnudel. The result can also be interpreted as saying that all even moments of random partial sums 1N√∑n≤Nf(P(n)) match standard complex Gaussian moments as N→+∞ , where f is the Steinhaus random multiplicative function.}, author = {Wang, Victor and Xu, Max Wenqiang}, issn = {1469-2120}, journal = {Bulletin of the London Mathematical Society}, number = {8}, pages = {2718--2726}, publisher = {London Mathematical Society}, title = {{Paucity phenomena for polynomial products}}, doi = {10.1112/blms.13095}, volume = {56}, year = {2024}, } @article{10018, abstract = {In order to study integral points of bounded log-anticanonical height on weak del Pezzo surfaces, we classify weak del Pezzo pairs. As a representative example, we consider a quartic del Pezzo surface of singularity type A1 + A3 and prove an analogue of Manin's conjecture for integral points with respect to its singularities and its lines.}, author = {Derenthal, Ulrich and Wilsch, Florian Alexander}, issn = {1475-3030 }, journal = {Journal of the Institute of Mathematics of Jussieu}, keywords = {Integral points, del Pezzo surface, universal torsor, Manin’s conjecture}, number = {3}, pages = {1259--1294}, publisher = {Cambridge University Press}, title = {{Integral points on singular del Pezzo surfaces}}, doi = {10.1017/S1474748022000482}, volume = {23}, year = {2024}, } @article{12312, abstract = {Let $\ell$ be a prime number. We classify the subgroups $G$ of $\operatorname{Sp}_4(\mathbb{F}_\ell)$ and $\operatorname{GSp}_4(\mathbb{F}_\ell)$ that act irreducibly on $\mathbb{F}_\ell^4$, but such that every element of $G$ fixes an $\mathbb{F}_\ell$-vector subspace of dimension 1. We use this classification to prove that the local-global principle for isogenies of degree $\ell$ between abelian surfaces over number fields holds in many cases -- in particular, whenever the abelian surface has non-trivial endomorphisms and $\ell$ is large enough with respect to the field of definition. Finally, we prove that there exist arbitrarily large primes $\ell$ for which some abelian surface $A/\mathbb{Q}$ fails the local-global principle for isogenies of degree $\ell$.}, author = {Lombardo, Davide and Verzobio, Matteo}, issn = {1420-9020}, journal = {Selecta Mathematica}, number = {2}, publisher = {Springer Nature}, title = {{On the local-global principle for isogenies of abelian surfaces}}, doi = {10.1007/s00029-023-00908-0}, volume = {30}, year = {2024}, } @article{17447, abstract = {Let F be a diagonal cubic form over Z in six variables. From the dual variety in the delta method of Duke–Friedlander–Iwaniec and Heath‐Brown, we unconditionally extract a weighted count of certain special integral zeros of F in regions of diameter X - 8 . Heath‐Brown did the same in four variables, but our analysis differs and captures some novel features. We also put forth an axiomatic framework for more general F.}, author = {Wang, Victor}, issn = {1469-7750}, journal = {Journal of the London Mathematical Society}, number = {3}, publisher = {Wiley}, title = {{Special cubic zeros and the dual variety}}, doi = {10.1112/jlms.12975}, volume = {110}, year = {2024}, } @article{18173, abstract = {Using a two-dimensional version of the delta method, we establish an asymptotic formula for the number of rational points of bounded height on non-singular complete intersections of cubic and quadric hypersurfaces of dimension at least 23 over Fq(t), provided char (Fq)>3. Under the same hypotheses, we also verify weak approximation.}, author = {Glas, Jakob}, issn = {1469-7750}, journal = {Journal of the London Mathematical Society}, number = {4}, publisher = {London Mathematical Society}, title = {{Rational points on complete intersections of cubic and quadric hypersurfaces over Fq(t)}}, doi = {10.1112/jlms.12991}, volume = {110}, year = {2024}, } @phdthesis{18132, abstract = {In this thesis, we are dealing with both arithmetic and geometric problems coming from the study of rational points with a particular focus on function fields over finite fields: (1) Using the circle method we produce upper bounds for the number of rational points of bounded height on diagonal cubic surfaces and fourfolds over Fq(t). This is based on joint work with Leonhard Hochfilzer. (2) We study rational points on smooth complete intersections X defined by cubic and quadratic hypersurfaces over Fq(t). We refine the Farey dissection of the “unit square” developed by Vishe [202] and use the circle method with a Kloosterman refinement to establish an asymptotic formula for the number of rational points of bounded height on X when dim(X) ≥ 23. Under the same hypotheses, we also verify weak approximation. (3) In joint work with Hochfilzer, we obtain upper bounds for the number of rational points of bounded height on del Pezzo surfaces of low degree over any global field. Our approach is to take hyperplane sections, which reduces the problem to uniform estimates for the number of rational points on curves. (4) We develop a version of the circle method capable of counting Fq-points on jet schemes of moduli spaces of rational curves on hypersurfaces. Combining this with a spreading out argument and a result of Mustaţă [150], this allows us to show that these moduli spaces only have canonical singularities under suitable assumptions on the degree and the dimension. In addition, we give an overview of guiding questions and conjectures in the field of rational points and explain the basic mechanism underlying the circle method. }, author = {Glas, Jakob}, issn = {2663-337X}, pages = {195}, publisher = {Institute of Science and Technology Austria}, title = {{Counting rational points over function fields}}, doi = {10.15479/at:ista:18132}, year = {2024}, }