**“30+45” seminars**

**Anna Medvekovsky (MPI Bonn)**- Title:
*Modular forms mod 3 of level one*

Abstract: We discuss modular forms mod 3 of level one from the beginning: the space of forms of all weights, the Hecke algebra that acts on this space, the Galois representation over this Hecke algebra, and explicit matrices realizing this representation. We follow with applications to “densities” of mod-3 modular forms: that is, the distribution of their prime Fourier coefficients. This work owes a great debt of inspiration to published and unpublished work of Bellaïche, Serre, and Nicolas on modular forms modulo 2.

- Title:

**Fumiaki Suzuki (Hannover)**- Title:
*Rationality questions for Fano schemes of intersections of two quadrics*

Abstract: We study rationality questions for the Fano schemes of non-maximal linear spaces on a smooth complete intersection X of two quadrics, especially over non-closed fields. We start by showing that they are all geometrically rational. We then ask their rationality over k and analyze in details the case of second maximal linear spaces. In particular, we generalize results of Hassett-Tschinkel and Benoist-Wittenberg when X has odd dimension, and extend work of Hassett-Kollár-Tschinkel when X has even dimension and k = R. This is joint work with Lena Ji.

- Title:

**Kamil Rychlewicz (ISTA)**- Title:
*Equivariant cohomology and rings of functions*

Abstract: In topology of manifolds, there is a long history of theorems which relate global topological invariants to local behavior of vector fields, functions or group actions. This dates back to Poincare-Hopf, but many important results have been proved since, with Morse theory, Bott residue formula for complex manifolds and Białynicki-Birula decomposition for algebraic varieties being notable examples. The classical work of Carrell–Liebermann and Akyildiz goes further and sees the whole cohomology ring as a ring of functions on a thick point, zero of a vector field. In a joint work with Tamas Hausel we generalize the work of Brion and Carrell (for Borel of SL_2) and show how to see the equivariant cohomology as a ring of function for quite general setups. I will start the talk by recalling the classical results, and present the newest developments in the field. This allows us to see geometrically the spectra of equivariant cohomology of classical spaces like flag or Schubert varieties. I will also show how we might be able to tackle the cohomology rings of spherical varieties, extract some information for singular varieties, and how much can be said when we replace cohomology with K-theory.

- Title:

**Francesco Denisi (IMJ-PRG)**- Title:
*The pseudo-effective cone of a hyper-Kähler manifold and polygons of Newton-Okounkov type*

Abstract: Hyper-Kähler (HK) manifolds form one of the three building blocks of compact Kähler manifolds with vanishing first (real) Chern class. In recent years, there has been a considerable effort by many mathematicians to understand as much as possible about the geometry of these varieties. Despite everything, the positivity of divisors is one of the least understood things in HK geometry. The purpose of this talk is to discuss some results about the positivity of divisors on HK manifolds, which, in some sense, generalise some results from the theory of smooth projective surfaces and provide further credibility to the heuristic that linear series on HK manifolds behave very much like in the surface case.

- Title:

**Shiyu Shen**()- Title:
*Examples of relative Langlands duality for D-modules in positive characteristic*

Abstract: Using the Azumaya property of (crystalline) differential operators in positive characteristic, Bezrukavnikov-Braverman and Chen-Zhu established a generic version of the geometric Langlands correspondence, which is realized as a twisted version of the Fourier-Mukai transform on dual Hitchin systems. In the recent work of Ben-Zvi–Sakellaridis–Venkatesh, they proposed the relative Langlands duality: Given a Hamiltonian G-space and its dual, they constructed the so-called period sheaf and L-sheaf that conjecturally correspond with each other under the geometric Langlands correspondence. In the setting of D-modules in positive characteristic, I will discuss work in progress to establish a generic version of this correspondence for Hamiltonian G-spaces that arise from quasi-split real groups.

- Title:

**Mathias Schuett (Université de Hannover)**- Title:
*Real multiplication for K3 surfaces*

Abstract: Real multiplication (RM) not only occurs on abelian varieties (in terms of endomorphisms), but also on K3 surfaces and other Calabi-Yau varieties via Hodge structures. Contrary to the CM case, RM remains rather mysterious, largely evading explicit examples. I will report on joint work with Eva Bayer and Bert van Geemen which engineers both explicit new constructions of K3 surfaces with RM (and CM) and abstract existence results.

- Title:

()**Annette Huber**- Title:
*Transcendence of 1-periods*

Abstract: (joint work with G. Wüstholz) 1-periods are complex numbers obtained as path intervals of algebraic 1-forms on algebraic varieties over the field of algebraic numbers. The set contains famous numbers like 2pi i or values of log in algebraic numbers. They are a long-standing object of transcendence theory. We will explain a sharp transcendence criterion and describe more generally all linear relations between 1-periods. The proof uses the theory of 1-motives in an essential way, allowing us to reduce the question to the seminal Analytic Subgroup Theorem of Wüstholz. In the second half of the talk, we will discuss 1-motives in more detail. This leads to quantitive versions of the theorem, i.e., formulas for the dimension of the space of periods of a given

1-motive.

- Title:

**Stefan Kebekus**()- Title:

Abstract:

- Title:

**Ilaria Viglino ()**- Title:
*Almost sure convergence of least common multiple of ideals for polynomials over a number fields*

Abstract: For f an irreducible polynomial with integer coefficients of degree n, Cilleruelo’s

conjecture states that log(lcm(f(1),…,f (M))) is asymptotic to (n – 1)M log M, as M tends to infinity. The Prime Number Theorem for arithmetic progressions can be exploited to obtain an asymptotic estimate when f is a linear polynomial. Cilleruelo extended this result to quadratic polynomials. The asymptotic remains unknown for irreducible polynomials of higher degree. Recently the conjecture was shown on average for a large family of polynomials of any degree by Rudnick and Zehavi. We investigate the case of S_n-polynomials with coefficients in the ring of algebraic integers of a fixed number field extension K/Q by considering the least common multiple of ideals in the ring of algebraic integers.

- Title:

**Riccardo Ontani****()**- Title: Jeffrey-Kirwan localisation for quotients of linear spaces

Abstract: Many interesting varieties can be built as GIT quotients of linear spaces by actions of reductive groups, a big family of examples being given by quiver varieties. In this talk we will describe a localisation formula (due to Jeffrey and Kirwan) for computing integrals over this kind of quotients. If the quotient variety is not proper but it admits a circle action with proper fixed locus, we can still define the integrals by localisation and obtain a formula for them.

- Title: Jeffrey-Kirwan localisation for quotients of linear spaces

**Matthew Satriano****(University of Waterloo)**- Title: Beyond twisted maps: crepant resolutions of log terminal singularities and a motivic McKay correspondence

Abstract: Crepant resolutions have inspired connections between birational geometry, derived categories, representation theory, and motivic integration. In this talk, we prove that every variety with log-terminal singularities admits a crepant resolution by a smooth stack. We additionally prove a motivic McKay correspondence for stack-theoretic resolutions. Finally, we show how our work naturally leads to a generalization of twisted mapping spaces. This is joint work with Jeremy Usatine.

- Title: Beyond twisted maps: crepant resolutions of log terminal singularities and a motivic McKay correspondence

**Alexei Skorobogatov (Imperial)**- Title: On p-primary torsion of the Brauer group in characteristic p

Abstract: Let k be a finitely generated field. Relation between the Tate conjecture for divisors and finiteness properties of the Brauer groups of varieties over k is well known, at least for torsion coprime to char(k). Much less is known about p-primary torsion in characteristic p. In a recent paper, D’Addezio clarified the situation for abelian varieties over fields of positive characteristic. Using similar ideas, I will show that for varieties X and Y satisfying some mild conditions, the cokernel of the map from Br(X) \oplus Br(Y) to Br(X\times Y) is a direct sum of a finite group and a p-group of finite exponent (which can be infinite). This implies, for example, that the transcendental Brauer group of surfaces dominated by products of curves has finite exponent.

- Title: On p-primary torsion of the Brauer group in characteristic p

**Gergely Bérczi (Aarhus University)**- Title:
*The geometry of Hilbert scheme of points on manifolds*

Abstract: While the Hilbert scheme of points on surfaces is extensively studied, the Hilbert scheme of points on higher-dimensional manifolds proves to be notoriously complex, exhibiting wild and pathological properties. Our understanding of their topology and geometry is very limited. In this survey I will present recent developments in various aspects of their geometry. I will discuss the distribution of torus fixed points within the components of the Hilbert scheme (work with J. Svendsen), a new formula for tautological integrals over geometric components (work with A. Szenes), and p-adic integration on Hilbert schemes leading to new invariants of hypersurface singularities (work with I. Rossinelli).

- Title:

**Tommaso Botta (ETZH)**- Title:
*Bow varieties, stable envelopes, and their 3d mirror symmetry*

Abstract: Mirror symmetry for 3d N=4SUSY QFTs has recently received much attention in geometry and representation theory. Theories within this class give rise to interesting moduli spaces of vacua, whose most relevant components are called the Higgs and Coulomb branches. Nakajima initiated the mathematical study of Higgs branches in the 90s; since then, their geometry has been pivotal in diverse areas of enumerative geometry and geometric representation theory. On the other hand, a mathematically precise definition of the Coulomb branch has only recently been proposed, and its study has started. Physically, 3d mirror symmetry is understood as a duality for pairs of theories whose Higgs and Coulomb branches are interchanged. Mathematically, it descends to a number of statements relating invariants attached to the dual sides. One of its key predictions is the identification of dual pairs of elliptic stable envelopes, which are certain topological classes intimately related to elliptic quantum groups.

- Title:

**Stevan Gajovic (Charles University Prague)**- Title:
*Rational points on curves and the method of Chabauty and Coleman*

Abstract: In this talk, we will discuss the method of Chabauty and Coleman and its variations. The original method is powerful for determining rational points on curves satisfying a certain rank condition. Depending on this rank condition, we also discuss the other variations that we can use to determine rational (or more general) points. We pay special attention to the case when the rank condition is violated – in some cases, we can use p-adic heights as an alternative solution. We will also briefly explain how to compute p-adic heights on hyperelliptic curves and their application to the variant called quadratic Chabauty.

- Title:

**Alessandro Giacchetto (ETZH)**- Title:
*Resurgent large genus asymptotics of intersection numbers*

Abstract: Factorials can be recursively computed through their definition, but the computation gets difficult quite quickly when the number of interest gets larger and larger. A workaround is given by Stirling’s approximation: a closed, asymptotic formula for factorials. A similar (but much more complicated) situation occurs when trying to compute Witten’s intersection numbers. Virasoroconstraints recursively compute these numbers, but the computation gets difficult when the genus gets larger and larger. An approximation has been recently proved by Aggarwal by studying the structure of the associated Virasoro constraints. I will present an alternative proof of Aggarwal’s result based on quantum curves and resurgence. The advantage of this strategy is that it easily generalises to several problems (such as r-spin intersection numbers, Norbury’s intersection numbers, etc) and gives higher-order corrections. Based on the joint work with B. Eynard, E. Garcia-Failde, P. Gregori, and D. Lewański.

- Title:

**Xiaoyu Zhang (Duisburg-Essen)**- Title:
*Superspecial reductions of CM abelian varieties*

Abstract: The CM points in a Shimura variety contain very important arithmetic information of the whole variety. The famous conjecture of André-Oort says that the Zariski closure of a family of CM points in a Shimura variety is in fact (a translation of) a subvariety of Hodge type. In this talk, we would like to say something about a variant of this conjecture mod a prime: take a p-power isogeny orbit C of a fixed CM abelian variety A, if A has superspecial reduction modulo a prime q, then a certain simultaneous reduction of elements in C (by action of a torus) is in fact surjective onto the set of all superspecial mod q abelian varieties. I will present these results and indicate the ideas of the proofs.

- Title:

**Domenico Valloni**(EPFL)- Title:
*Noether problem in mixed characteristic*

Abstract: Let k be any field and let V be a linear and faithful representation of a finite group G. The Noether problem asks whether V/G is a (stably) rational variety over k. It is known that if the characteristic p of k is bigger than 0 and G is a p-group, then V/G is always rational.

On the other hand, Saltman and later Bogomolov constructed many examples of p-groups such that V/G is not stably rational over the complex numbers. The aim of the talk is to study what happens over a DVR of mixed characteristic (0,p). We show for instance that for all the examples found by Saltaman and Bogomolov, there cannot exist a smooth projective scheme over R whose special resp. generic fibre are stably birational to V/G (and, hence, that P^n_R/G never admits a relative resolution of singularities over R). The proof combines integral p-adic Hodge theory and the study of differential forms in positive characteristic.

- Title:

**Lucien Hennecart**(**Edinburgh**)- Title:
*Nonabelian Hodge isomorphisms for cohomological Hall algebras*

Abstract: Nonabelian Hodge theory relates three different kinds of objects on smooth, projective, algebraic varieties: Higgs bundles, local systems, and connections. The corresponding moduli spaces are diffeomorphic but not isomorphic. Nevertheless, this suffices to compare their cohomologies. When considering the moduli stacks, the situation is less favourable: it is not known how to compare the topologies of the Dolbeault and de Rham moduli stacks. I will explain how cohomological Hall algebras give us the right tool to study these stacks for a smooth projective curve. In addition, it is even possible to compare the cohomological Hall algebra structures through the nonabelian Hodge isomorphisms we obtain.

- Title:

**Alapan Mukhopadhyay**(EPFL)- Title:
*Generators of bounded derived categories in prime characteristics*

Abstract: Since Bondal- van den Bergh’s work on the representability of functors, proving existence of strong generators of the bounded derived category of a scheme has been a central problem. While for a quasi-excellent, separated scheme the existence of strong generators is established, explicit examples of such generators are not common. In this talk, we show that explicit generators can be produced in prime characteristics using the Frobenius pushforward functor. As a consequence, we will see that for a domain R with finite Frobeniusendomorphism, R1/pn – for large enough n- generates the bounded derived category of R-modules. This recovers Kunz’s characterization of regularity in terms of flatness of Frobenius. Part of the talk is based on a joint work with Matthew Ballard, Srikanth Iyengar, Patrick Lank and Josh Pollitz.

- Title:

**Asbjorn Nordentoft (Paris)**- Title:
*Non-vanishing of finite order twists of L-values via horizontal p-adic L-functions*

Abstract: Goldfeld’s Conjecture predicts that exactly 50% of quadratic twists of a fixed elliptic curve have L-function non-vanishing at the central point. Correspondingly, when considering twists by higher order Dirichlet characters, it has been predicted by David-Fearnly-Kisilevsky that 100% should be non-vanishing. Very little was previously known beyond the quadratic case as the problem lies beyond the current technology of e.g. analytic number theory. In this talk I will present a p-adic approach relying on the construction of ‘horizontal p-adic L-functions’. This yields strong quantitative non-vanishing results for general order twists. In particular, we obtain the best result towards Goldfeld’s Conjecture for one hundred percent of elliptic curves (improving on a result of Ono). I will also present applications to simultaneous non-vanishing and Diophantine stability. This talk is based on joint work with Daniel Kriz. In the introductory talk, I will give an introduction to the concept in arithmetic statistics known as ‘Diophantine stability’ and the connections to the Birch-Swinnerton-Dyer Conjecture. In particular, I will discuss Goldfeld’s Conjecture, as well as the case of the cyclotomic p-tower studied in Iwasawa theory.

- Title:

**Philip Engel (University of Bonn)**- Title:
*The non-abelian Hodge locus*

Abstract: Given a family of smooth projective varieties, one can consider the relative de Rham space of flat vector bundles of rank n on the fibers. The flat vector bundles which underlie a polarized Z-variation of Hodge structure form the “non-abelian Hodge locus”. Simpson proved that this locus is analytic, and he conjectured it is algebraic. This would imply a conjecture of Deligne that only finitely many representations of the fundamental group of a fiber appear. I will discuss a proof of Deligne’s and Simpson’s conjectures, under the additional hypothesis that the Z-zariski closure of monodromy is a cocompact arithmetic group. This is joint work with Salim Tayou.

- Title:

**Dinakar Ramakrishnan (Caltech)**- Title:
*An analog of a theorem of Manin for the p-primary torsion of certain Abelian 3-folds*

Abstract: Given any elliptic curve E over a number field k, Manin showed (about 50 years ago) that for any prime p, the p-primary torsion of E(k) is bounded uniformly for all E/k. He also showed the uniform irreducibility of the associated p-adic Galois representation modulo p^r for r uniform in E if E does not have CM. In this lecture, we will discuss an analog proven with Mladen Dimitrov for abelian 3-folds with multiplication by an imaginary quadratic field.

- Title:

**Marta Benozzo (UCL)**- Title:
*On the canonical bundle formula in positive characteristic*

Abstract: An important problem in birational geometry is trying to relate in a meaningful way the canonical bundles of the source and the base of a fibration. The first instance of such a formula is Kodaira’s canonical bundle formula for surfaces which admit a fibration with elliptic fibres. It describes the relation between the canonical bundles in terms of the singularities of the fibres and their j-invariants. In higher dimension, we do not have an equivalent of the j-invariant, but we can still define a moduli part. Over fields of characteristic 0, positivity properties of the moduli part have been studied using variations of Hodge structures. Recently, the problem has been approached with techniques from the minimal model program. These methods can be used to prove a canonical bundle formula result in positive characteristic.

- Title:

**Yalong Cao****(Riken)**- Title:
*From curve counting on Calabi-Yau 4-folds to quasimaps for quivers with potentials*

Abstract: I will start by reviewing an old joint work with Davesh Maulik and Yukinobu Toda on relating Gromov-Witten, Gopakumar-Vafa (in the sense of Klemm-Pandharipande) and stable pair invariants on compact Calabi-Yau 4-folds. For non-compact CY4 like local curves, similar invariants can be studied via the perspective of quasimaps to quivers with potentials. In a recent joint work with Gufang Zhao, we define a virtual count for such quasimaps and prove a gluing formula. Computations of examples will also be discussed.

- Title:

**Eric Chen****(EPFL)**- Title:
*Duality of automorphic periods*

Abstract: The study of automorphic integrals has a long history which began with Hecke’s Mellin transform of a modular form, and more recently it has provided an indispensable language with which to describe functoriality phenomena in the Langlands program. In the first half of this presentation, I will explain these developments via emblematic examples while rephrasing them in terms of the recent framework of relative Langlands duality proposed by Ben-Zvi–Sakellaridis–Venkatesh. In the second half, I will present joint work with Venkatesh in which we establish relative duality in certain “singular” examples with the aid of new numerical invariants of Galois representations that we call “nonabelian L-functions”.

- Title:

**Dimitar Jetchev****(EPFL)**- Title:
*The Birch and Swinnerton-Dyer formula in analytic ranks zero and one for modular forms of higher weight*

Abstract: In this talk, I will report on recent results on the computation of the p-part of the leading term of the L-function of a modular form of arbitrary weight at the central point in the cases when the order of vanishing is at most one. Unlike the classical case of weight 2 modular forms, qualitatively different arguments are needed in the higher-weight case. After explaining the difference, I will indicate how one can use level-raising and (non-ordinary) p-adic deformations together with some of the arguments in weight 2 to obtain results in the case of general weights. This is joint work with Chris Skinner and Xin Wan.

- Title:

**Paolo Ghiggini****(Institut Fourier, Grenoble)**- Title:
*Knot invariants from Reeb orbits*

Abstract: Vincent Colin, Ko Honda, Michael Hutchings and I defined embedded contact homology groups for knots in a three-manifold 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 Ozsváth, Szabó and Rasmussen’s knot 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 knot Floer homology will be necessary beyond the fact that it exists and is interesting. On the other hand the definition of embedded contact homology, both for knots and closed three-manifolds, will be sketched. This is a joint work in progress with Vincent Colin and Ko Honda.

- Title:

**Giovanni Inchiostro (University of Washington)**- Title:
*Moduli of boundary polarized Calabi-Yau pairs*

Abstract: I will discuss a new approach to build a moduli space of pairs (X,D) where X is a Q-Fano variety and D is a Q-divisor such that K_X + D is Q-rationally equivalent to 0. In the case of X=P^2, our approach gives a projective moduli space, which interpolates between KSBA-stability and K-stability. This is joint work with K. Ascher, D. Bejleri, H. Blum, K. DeVleming, Y. Liu, X. Wang.

- Title:

**Joaquin Moraga (UCLA)**- Title:
*Fundamental groups of log Calabi-Yau surfaces*

Abstract: Let (X,B) be a log Calabi-Yau surface. We show that the fundamental group

pi_1(X,B) admits a normal metabelian subgroup of rank at most 4 and index at most 7200.

We show that this statement is optimal in the following sense: the bounds 4 and 7200 are sharp and there are examples in which pi_1(X,B) is not a virtually abelian group. This is joint work in progress with Cecile Gachet and Zhining Liu. During the pre-talk I will introduce smooth Fano varieties, smooth Calabi-Yau varieties, and compute the fundamental groups of some Fano surfaces and Calabi-Yau surfaces with canonical singularities, i.e., with Du Val singularities.

- Title:

**Sam Payne (University of Texas at Austin)**- Title:
*Motivic structures in the cohomology of moduli spaces of curves*

Abstract: Cohomology groups of moduli spaces of algebraic curves consist of characteristic classes for surface bundles and appear in many disparate areas of mathematics from low-dimensional topology to algebraic geometry, mathematical physics, and stable homotopy theory. Algebraic geometry endows these cohomology groups with additional motivic structures, i.e. (mixed) Hodge structures and l-adic Galois representations. I will recall the motivation and important prior results on these cohomology groups before discussing new predictions arising from work of Chenevier and Lannes on L-functions attached to Galois representations with everywhere good reduction and recent results that unconditionally confirm several of these predictions.

- Title:

**Egor Yasinsky (École Polytechnique)**- Title:
*Equivariant geometry of Fano varieties*

Abstract: I will discuss some questions related to equivariant birational geometry of del Pezzo surfaces and, possibly, of higher-dimensional Fano varieties, focusing on the notion of birational rigidity.

- Title:

**Shenyuang Zhao (Université de Toulouse)**- Title:
*Anosov group action on flag varieties*

Abstract: Given a self-map of a projective variety, we may ask how subvarieties of the projective variety behave under iterations of the self-map. Brolin and Lyubich discovered an equidistribution phenomenon for self-maps of the projective line. People expect the same for higher dimensional varieties, i.e. different subvarieties converge in the same way under iterations. We can also ask the same question for iterations under a group action instead of a single self-map. In dimension one Patterson-Sullivan theory can be thought of as an anologue of Brolin-Lyubich’s work in the group action setting. In this ongoing joint work with Bac Dang, Michael Kapovich and Mikhail Lyubich, we try to generalize some aspects of the one dimensional theory to higher dimension.

- Title:

**Nikolaos Tsakanikas (EPFL)**- Title:
*Applications of Zariski decompositions*

Abstract: The Minimal Model Program (MMP) plays a central role in the classification theory of projective varieties. In this talk I will discuss recent progress towards the existence of minimal models, which is one of the main open problems in the MMP. In particular, I will explain the close relationship between the existence of minimal models and the existence of so-called weak Zariski decompositions, focusing mainly on applications of the latter. This talk is based on joint works with Xiaowei Jiang, Vladimir Lazić and Lingyao Xie.

- Title:

**Marion Jeannin (Université d’Uppsala)**- Title:
*Semistability of G-torsors and parabolic subgroups in positive characteristic*

Abstract: Let k be a field and X be a k-curve. Let also G be a reductive group scheme over X. Semistability for G-torsors can be defined by several ways that depend on assumptions on k and G. These approaches are both well defined and equivalent when k is of characteristic zero. In this talk I will explain in which generalities it is possible to extend some of these approaches to the positive characteristic framework and compare them. This requires to investigate whether some well known results in representation theory in characteristic zero still hold true in characteristic p > 0. More specifically, an analogous statement of a theorem of Morozov (which classifies, in characteristic 0, parabolic subalgebras of the Lie algebra of a reductive group by means of their nilradical) is a cornerstone of all this unification attempt. In the first part of the talk, I will provide an overview of the geometric content and emphasize the role of parabolic subgroups in all this theory of semistability. The second part of the talk will be dedicated to the extension of Morozov’s theorem to positive characteristics, and the way it allows one to get a more uniform vision of the different historical definitions of semistability of G-torsors.

- Title:

**Prahlad Sharma (Renyi Institute)**- Title:
*Bilinear sums with $GL(2)$ coefficients and the exponent of distribution of $d_3$*

Abstract: We obtain the exponent of distribution $1/2+1/30$ for the ternary divisor function $d_3$ to square-free and prime power moduli, improving the previous results of Fouvry–Kowalski–Michel, Heath-Brown, and Friedlander–Iwaniec. The key input is certain estimates on bilinear sums with $GL(2)$ coefficients obtained using the delta symbol approach.

- Title:

**Sebastián Herrero (PUCV Chile & ETH Zürich)**- Title:
*Special values of modular functions and S-units*

Abstract: The values of modular functions at complex multiplication points are classically used to construct explicit abelian extensions of quadratic imaginary fields. The factorization of (the algebraic norm of) these numbers is also a problem that has been extensively studied, at least since the beginnings of the 1900s. More recently, particular attention has been given to the question of when these values are algebraic units, and more generally how often these special values are S-units when S is a fixed finite set of prime numbers. In this talk I will present a brief overview of known results and open related problems, and I will explain joint work with Ricardo Menares (PUC Chile) and Juan Rivera-Letelier (U. of Rochester) on the finiteness of special values of modular functions that are S-units.

- Title:

**Marc Abboud (University of Rennes 1)**- Title:
*Dynamical degrees of endomorphisms of affine surfaces*

Abstract: Let $f: \mathbf C^2 \rightarrow \mathbf C^2$ be a polynomial transformation. The dynamical degree of $f$ is defined as $\lim_n (\text{deg} f^n)^{1/n}$, where $\text{deg} f^n$ is the degree of the $n$-th iterate of $f$. In 2007, Favre and Jonsson showed that the dynamical degree of any polynomial endomorphism of $\mathbf C^2$ is an algebraic integer of degree $\leq 2$. For any affine surface, there is a definition of the dynamical degree that generalizes the one on the affine plane. We show that the result still holds in this case: the dynamical degree of an endomorphism of any complex affine surface is an algebraic integer of degree $\leq 2$. In this talk, I will give an overview of the recent results obtained on dynamical degrees on algebraic varieties and explain the key tools of the proof.

- Title:

**Gebhard Martin (University of Bonn)**- Title:
*On the non-degeneracy of Enriques surfaces*

Abstract: I will report on joint work with G. Mezzedimi and D. Veniani on Enriques surfaces with small non-degeneracy. The non-degeneracy of an Enriques surface X is the maximal length of an isotropic sequence of effective curves on X. Roughly speaking, the higher the non-degeneracy of an Enriques surface is, the more well-behaved are its projective models. For example, Enriques surfaces of maximal non-degeneracy 10 are isomorphic to surfaces of degree 10 in P^5. We prove that, in characteristic different from 2, every Enriques surface has non-degeneracy at least 4, which implies that all of them arise via Enriques’ classical construction as a minimal resolution of a sextic in P^3 which is non-normal along the edges of a tetrahedron and all Enriques surfaces are birational to normal quintics in P^3.

- Title:

**Loïs Faisant (Université Grenoble Alpes)**- Title:
*Motivic distribution of rational curves*

Abstract: In diophantine geometry, the Batyrev-Manin-Peyre conjecture originally concerns rational points on Fano varieties. It describes the asymptotic behaviour of the number of rational points of bounded height, when the bound becomes arbitrary large. A geometric analogue of this conjecture deals with the asymptotic behaviour of the moduli space of rational curves on a complex Fano variety, when the « degree » of the curves « goes to infinity ». Various examples support the claim that, after renormalisation in a relevant ring of motivic integration, the class of this moduli space may converge to a constant which has an interpretation as a motivic Euler product. In this talk, we will state this motivic version of the Batyrev-Manin-Peyre conjecture and give some examples for which it is known to hold : projective space, more generally toric varieties, and equivariant compactifications of vector spaces. In a second part we will introduce the notion of equidistribution of curves and show that it opens a path to new types of results.

- Title:

**Michel Van Garrel (University of Birmingham)**- Title:
*Geometry of Enumerative Mirror Symmetry*

Abstract: For the pair (Y,D) of a smooth Fano variety and smooth anticanonical divisor, mirror symmetry computes in a complicated fashion the counts of rational curves in Y that meet D in one point only (rather, the corresponding log Gromov-Witten invariants). In this talk, I will show how these computations are in fact the consequence of a simple geometric duality between (Y,D) and its Gross-Siebert intrinsic mirror family. I will focus on the case of Y the projective plane and D an elliptic curve. Then the mirror family is the moduli space of elliptic curves with certain level structure. This is joint work with Helge Ruddat and Bernd Siebert and is part of a long term project to de-mystify mirror symmetry.

- Title:

**Stéphane Lamy (Université Paul Sabatier, Toulouse)**- Title:
*Tits alternative for the Cremona group*

Abstract: A group G satisfies the Tits alternative if any subgroup contains either a free group over 2 generators, of a solvable group of finite index. The classical result by Jacques Tits is that the linear groups satisfy this alternative, up to restricting to finitely generated subgroups in the case of a ground field of positive characteristic. In this talk I will explain the strategy of proof to show that the plane Cremona group also satisfies the Tits alternative. It will essentially be a survey of previous results by Cantat and Urech, with a few contributions by myself mainly for the case of positive characteristic, and also for the classification of torsion subgroups.

- Title:

**Emmanuel Letellier****(IMJ-PRG)**- Title:
*Fourier transform from the symmetric square representation of PGL(2) and SL(2)*

Abstract: In this talk we will first review Braverman-Kazhdan’s approach of Langlands functoriality via Fourier transform (in the finite field case). We will then explain in the case of symmetric square representation of PGL(2) and SL(2) how to extend Braverman-Kazhdan’s Fourier operator (which is not involutive) to an involutive Fourier transform. This is a joint work with Gérard Laumon.

- Title:

**Ben Briggs (University of Copenhagen**)- Title:
*Syzygies of the cotangent complex*

Abstract: The cotangent complex is an important but difficult to understand object associated to a map of commutative rings (or schemes). It is connected with some easier to compute invariants: the module of differential forms, the conormal module, and Koszul homology can all be seen as syzygies inside the cotangent complex. Quillen conjectured that, for maps of finite flat dimension, the cotangent complex can only be bounded for locally complete intersection homomorphisms. This was proven by Avramov in 1999. I will explain how to get a new proof by paying attention to these syzygies, and how to simultaneously prove a conjecture of Vasconcelos on the conormal module.

- Title:

)**Navid Nabijou (Queen Mary University of London**- Title:
*Roots and logs in the enumerative forest*

Abstract: Logarithmic and orbifold structures provide two paths to the enumeration of curves with fixed tangencies to a normal crossings divisor. Simple examples demonstrate that the resulting systems of invariants differ, but a more structural explanation of this defect has remained elusive. I will discuss joint work with Luca Battistella and Dhruv Ranganathan, in which we identify birational invariance as the key property distinguishing the two theories. The logarithmic theory is stable under strata blowups of the target, while the orbifold theory is not. By identifying a suitable system of “slope-sensitive” blowups, we define a limit orbifold theory and prove that it coincides with the logarithmic theory. Our proof hinges on a technique – rank reduction – for reducing questions about normal crossings divisors to questions about smooth divisors, where the situation is much-better understood.

- Title:

**Sam Molcho (ETHZ)**- Title:
*Jacobians, Stable Maps, and Cycles on the Moduli Space of Curves*

Abstract: The cohomology ring of the moduli space of curves contains two families of classes with different geometric origins: one comes from the universal Jacobian and Brill-Noether theory, and the other from the moduli space of stable maps and Gromov-Witten theory. After reviewing their construction, I will explain how new ideas from logarithmic geometry connect these classes, and discuss how the connection allows us to explicitly calculate them in the tautological ring, in terms of its standard generators. Time permitting, I will indicate the implications these calculations have for the study of relations in the tautological ring. The talk will draw from several joint works and some work in progress, carried out with Holmes-Pandharipande-Pixton-Schmitt, Abreu-Pagani, Ranganthan, and Wise.

- Title:

**Patricio Almirón (IMAG)**- Title:
*The underlying topological nature of the Poincaré series of a plane curve*

Abstract: In 2003, Campillo, Delgado and Gusein-Zade show the equality between the

Poincaré series of a reducible plane curve singularity $C$ and the Alexander polynomial $\Delta_L$ of the corresponding link $L$. However, their proof lacks of a conceptual explanation for this coincidence. In this talk I will show some new theorems of factorization of the Poincaré series $P_C$ depending on some key values of the semigroup of values of

$C$ with purely algebraic methods. As a consequence of these theorems, we will show that our procedure supplies a new proof of the theorem of Campillo, Delgado and Gusein-Zade. More concretely, we will focus on the translation of our algebraic construction to the iterated toric

structure of the link $L$. This is a joint work with Julio José Moyano-Fernández.

- Title:

**Richard Griffon (Université Clermont-Auvergne)**- Title:
*Isogenies of Elliptic Curves over Function Fields*

Abstract: This talk is based on a joint work with Fabien Pazuki, in which we study elliptic curves over function fields and the isogenies between them. Specifically, we prove analogues in the function field setting of two famous theorems about isogenous elliptic curves over number fields. The function field versions of these theorems, though having a similar flavour to their number field counterparts, display some striking differences. The first of these results completely describes the variation of the Weil height of the $j$-invariant of elliptic curves within an isogeny class. In particular, we show that the modular height remains constant under an isogeny of degree prime to the characteristic. Our second main theorem is an “isogeny estimate” in the spirit of theorems by Masser–Wüstholz and by Gaudron–Rémond. Unavoidable inseparability issues aside, we prove a uniform isogeny bound in this setting. After stating our results and giving sketches of their proof I will, time permitting, mention a few Diophantine applications.

- Title:

**Pierre Descombes (Jussieu)**- Title:
*Toric localization for cohomological DT invariants*

Abstract: In this talk, we will first review the construction of cohomological DT invariants, a kind of virtual cohomology for moduli spaces of sheaves on Calabi-Yau threefolds. We will explain their construction by Brav, Bussi, Dupont, Joyce, and Szendroï, using perverse sheaves of vanishing cycles. We will then present a toric localization formula for these invariants,

which can be seen as a virtual version of the Bialinicky-Birula decomposition. The proof of this formula uses in a crucial way the formalism of hyperbolic localization and its interaction with perverse sheaves.

- Title:

**Alessio Cela, ETH Zürich**- Title:
*Fixed-domain curve counts for blow-ups of projective space*

Abstract: In this talk I will explain some new results about the problem of counting pointed curves of fixed complex structure in blow-ups of projective space at general points. The geometric and the virtual Gromov-Witten counts in genus 0 and in higher genus for large anti-canonical degree curve classes agree in the Fano (and some $(-K)$-nef) examples, but not in general. For toric blow-ups, geometric counts can be expressed in terms of integrals on products of Jacobians and symmetric products of the domain curves, and evaluated explicitly in genus 0 and in the case of Bl_q(P^r), obtaining very simple closed formulas.

- Title:

**Giacomo Mezzedimi (Universität Bonn)**- Title:
*Entropy on K3 surfaces*

Abstract: To any diffeomorphism of a manifold we can associate a real number, called the entropy, which measures how fast its iterates spread out the points of the manifold. In this talk we will investigate the entropy of automorphisms of complex algebraic surfaces, and more

specifically of K3 surfaces. I will explain how we can obtain a classification of K3 surfaces of zero entropy, which exhibit the simplest dynamics among all K3 surfaces. I will then discuss some applications concerning the distribution of rational points on K3 surfaces over number fields. This is joint work with Simon Brandhorst.

- Title:

**Jean Fasel (Université Grenoble Alpes)**- Title:
*Vectors bundles on threefolds*

Abstract: In this talk, I will survey classification results for vector bundles on threefolds. I will start with classical results in the affine case, and then show how to complete the classification in that case. Then, I will pass to quasi-projective threefolds, focusing on the case of complex varieties.

- Title:

**Iacopo Brivio (NCTS, Taiwan)**- Title:
*Invariance of plurigenera and KSBA moduli in positive and mixed characteristi*c

Abstract: A famous theorem by Siu states that plurigenera are invariant under smooth deformations for complex projective manifolds, a result which is a cornerstone of higher dimensional moduli theory. In this talk we will explore some examples showing that Siu’s theorem fails in positive and mixed characteristic, then discuss the implications at the level of moduli theory, as well as some related questions.

- Title:

**Zhixin Xie (Saarland University, Saarbrücken)**- Title:
*Classification problem of rationally connected threefolds with nef anticanonical bundle*

Abstract: Complex projective varieties with nef anticanonical bundle appear as natural generalisations of Fano varieties. Fano manifolds are classified up to dimension three and many of their properties are well studied. However, the classification of manifolds with nef anticanonical divisor is more complicated as new phenomena arise and many results for

Fano varieties no longer hold for this class of varieties. In this talk, we will focus on the delicate case when the anticanonical bundle is nef but not semiample. I will explain a partial classification in this case and discuss some related problems on this class of varieties in comparison with Fano varieties.

- Title:

**Stefan Kebekus (University of Freiburg)**- Title:
*An Albanese construction for Campana’s C-pairs*

Abstract: Almost twenty years ago, Campana introduced C-pairs to complex geometry. Interpolating between compact and non-compact geometry, C-pairs capture the notion of “fractional positivity”in the “fractional logarithmic tangent bundle”. Today, they are an indispensible tool in the study of hyperbolicity, higher-dimensional birational geometry and several branches of arithmetic geometry. This talk reports on joint work with Erwan Rousseau. Aiming to construct a “C-Albanese variety”, we clarify the notion of a “morphism of C-pairs” and discuss the beginnings of a Nevanlinnatheory for “orbifold entire curves”.

- Title:

**Tobias Rossmann (NUI Galway)**- Title:
*Zeta functions in asymptotic algebra*

Abstract: Over the past decades, the study of zeta functions arising from algebraic counting problems has evolved into a distinct branch of asymptotic algebra. An appealing feature of this area is that it constitutes a meeting ground for several different mathematical subjects such as algebra, combinatorics, geometry, and logic. The first part of my talk will be a (biased) introduction to this area, in particular to the study of zeta functions enumerating subobjects (e.g. subgroups). I will then turn to recent developments surrounding the enumeration of (linear) orbits of unipotent groups. A key theme will be the development of tools for proving the absence or presence of geometric features in (at first glance) unexpected places.

- Title:

**Sasha Minets (University of Edinburgh)**- Title:
*A proof of P=W conjecture*

Abstract: Let C be a smooth projective curve. The non-abelian Hodge theory (NAHT) of Simpson is a diffeomorphism between the character variety M_B of C and the moduli of (semi)stable Higgs bundles M_D on C. Since this diffeomorphism is not algebraic, it induces an isomorphism of cohomology rings, but does not preserve finer information, such as the weight filtration. Based on computations in small rank, de Cataldo-Hausel-Migliorini conjectured that the weight filtration on H^*(M_B) gets sent to the perverse filtration on H^*(M_D) under NAHT. In this talk, I will explain a recent proof of this conjecture, which crucially uses the action of Hecke correspondences on H^*(M_D). This is a joint work with T. Hausel, A. Mellit, O. Schiffmann.

- Title:

**Kaisa Matomaki (University of Turku)**- Title:
*Products of primes in arithmetic progressions*

Abstract: Erdös conjectured that, when $q$ is a sufficiently large prime, every residue class $\pmod{q}$ can be represented as a product of two primes $p_1p_2$ with $p_1, p_2 \leq q$. This can be seen as a multiplicative analogue of the Goldbach conjecture claiming that every even integer greater than two can be written as a sum of two primes. I will discuss my on-going work with Joni Teräväinen establishing among other things a ternary variant of Erdös’ conjecture that, for every sufficiently large cube-free $q$, every reduced residue class $\pmod{q}$ can be represented as a product of three primes $p_1 p_2 p_3$ with $p_1, p_2, p_3 \leq q$. This improves on very recent works of Szabo and Zhao showing that one has such presentations with products of six primes. In the first part of the talk I will give a general overview of the topic as well as discuss some very fundamental ideas in the proof, in particular why the problem is more difficult than the ternary Goldbach problem and how we overcome this difficulty. In the second part of the talk I will give a more detailled description of the proof ideas.

- Title:

**Raymond Cheng (Hannover)**- Title:
*q-bic Hypersurfaces*

Abstract: Let’s count: 1, 2, q+1. The eponymous objects are special projective hypersurfaces of degree q+1, where q is a power of the positive ground field characteristic. This talk will sketch an analogy between the geometry of q-bic hypersurfaces and that of quadric and cubic hypersurfaces. For instance, the moduli spaces of linear spaces in q-bics are smooth and themselves have rich geometry. In the case of q-bic threefolds, I will describe an analogue of result of Clemens and Griffiths, which relates the intermediate Jacobian of the q-bic with the Albanese of its surface of lines.

- Title:

**Eva Bayer (EPFL)**- Title:
*Automorphisms of K3 surfaces, knots and isometries of lattices*

Abstract: The starting point of this research is a question of Curt McMullen : which Salem numbers are realized as dynamical degrees of automorphisms of K3 surfaces? We show that this is the case for Salem numbers of degree 4, 6, 8, 12, 14 and 16. The proofs use results on isometries of lattices, and these have other geometric applications : for instance, to characterize the signatures of knots with a given Alexander polynomial.

- Title:

**Federico Caucci (Università degli Studi di Milano)**- Title:
*Derived categories and motivic classes of irregular varieties*

Abstract: I will show that smooth projective varieties with equivalent bounded derived categories have, at least under certain natural hypothesis, the same class in a suitable variant of the Grothendieck ring of varieties. This has some applications to the derived invariance of Hodge numbers. It is a joint work with Luigi Lombardi and Giuseppe Pareschi.

- Title:

**Woonam Lim (ETHZ)**- Title: V
*irasoro constraints in sheaf theory and vertex algebras*

Abstract: In enumerative geometry, Virasoro constraints were first conjectured in Gromov-Witten theory with many new recent developments in the sheaf theoretic context. In joint work with A. Bojko and M. Moreira, we rephrase the sheaf-theoretic Virasoro constraints in terms of primary states coming from a natural conformal vector in Joyce’s vertex algebra. This shows that Virasoro constraints are preserved under wall-crossing. As an application, we prove the conjectural Virasoro constraints for moduli spaces of torsion-free sheaves on any curve and on surfaces with only (p,p) cohomology classes.

- Title: V

**Tanguy Vernet (EPFL)**- Title:
*Rational singularities for moment maps of totally negative quivers*

Abstract: Moment maps of quivers are key to several geometric realizations of quantum groups and counting their points over finite fields has proven to be an efficient technique to compute graded dimensions of those. More recently, Wyss counted jets of quiver moment maps over finite fields in a particular case and related their asymptotic behaviour to Igusa zeta functions. In the first half of the talk, I will introduce these countings of jets and relate their asymptotic behaviour to rational singularities of quiver moment maps. In the second part, I will prove that a large class of quiver moment maps have rational singularities, namely moment maps of totally negative quivers. If time allows, I will discuss some applications to singularities of other moduli spaces.

- Title:

**Yajnaseni Dutta (University of Bonn)**- Title:
*Birational self-maps of hyperkähler manifolds of K3[n]-type*

Abstract: Birational self-maps that are not biregular are hard to find on hyperkähler manifolds. For instance, there aren’t any on K3 surfaces. In this joint work with D. Mattei and Y. Prieto we showed that a general projective Hyperkähler manifold that is deformation equivalent to the Hilbert scheme of n-points on a K3 surface (i.e., of K3[n]-type) cannot admit certain birational self-maps of finite order. This prompted us to investigate birational self-maps moduli of sheaves on K3 surfaces which are of K3[n]-type. Using Markman’s theory of hyperkähler lattices and Bayer–Macri’s study of Bridgeland stability on K3 surfaces, we impose explicit numerical constraints on the topological invariants of the sheaves so that certain birational involutions exist on their moduli space.

- Title:

**Josh Lam (Humboldt University, Berlin)**- Title:
*Local systems on curves over finite fields and boundedness of trace fields*

Abstract: For a local system on a curve over a finite field, the work of Lafforgue shows that there is a well defined number field, referred to as the trace field, generated by the traces of Frobenius elements at the closed points. A basic question is how do such trace fields vary as the curve and the local system vary. Based on computations of Kontsevich, as well as Maeda’s conjecture in the number field setting, one expects that generically such fields are as large as they are allowed to be. I will show that, in the case of rank two local systems, as we vary over all pointed curves of type (g,n) over all finite fields, the set of trace fields of fixed degree is finite. This can be viewed as a uniform (across the moduli of curves) version of a finiteness result of Deligne’s in positive characteristic.

- Title:

**Petru Constantinescu (EPFL)**- Title:
*Dissipation of correlations of automorphic forms*

Abstract: Mass equidistribution of eigenfunctions is a central topic in quantum chaos and number theory. In this talk we highlight a generalisation of the Quantum Unique Ergodicity for holomorphic cusp forms in the weight aspect. We show that correlations of masses coming from off-diagonal terms dissipate as the weight tends to infinity. This corresponds to classifying the possible quantum limits along any sequence of Hecke eigenforms of increasing weight. The first half of the talk will be a gentle introduction to the area of quantum ergodicity in number theory, aimed for a general graduate audience. In the second half I will highlight some ingredients in the proof.

- Title:

**Daniele Turchetti (University of Warwick)**- Title:
*Models of curves via Berkovich geometry*

Abstract: The theory of models of varieties is an important tool for topics such as deformation theory, moduli spaces, and degenerations. In the 1960s, Deligne and Mumford proved that any smooth projective curve C over a discretely valued field K has a semi-stable model after base-change to a finite Galois extension L|K. The question of determining such extension has been investigated ever since but has been settled only in the case where L|K is tamely ramified. In this talk, I will present two results on the behaviour of models of curves under finite base-change. The first (joint with Lorenzo Fantini) exploits the geometry of the Berkovich analytification of C to describe the extension L|K in terms of regular models; the second (joint with Andrew Obus) investigates more in detail the case of potentially multiplicative reduction yielding new results in the case where L|K is wildly ramified.

- Title:

**Maxim Mornev (EPFL)**- Title:
*A secret proof of Dirichlet’s unit theorem*

Abstract: Common textbooks derive this result from Minkowski’s convex body theorem, making use of cumbersome logarithms in the process. However experts know that there is an arguably more conceptual proof which involves exponentials, adeles and a bit of algebra. I will explain this proof. I will also highlight a curious connection to Birch and Swinnerton-Dyer conjecture which has not made it to the literature yet. No knowledge of adeles will be assumed. If time permits I will alsodiscuss the relation with the class number formula.

- Title:

**Oscar Kivinen (EPFL)**- Title:
*Old and new geometry for Shalika germs*

Abstract: In recent work with Tsai, we were able to give a completely combinatorial formula for so called Shalika germs of tamely ramified regular semisimple elements in GL_n over a non-archimedean local field F. This was done using methods from harmonic analysis, representation theory, and rather surprisingly, knot theory. Shalika germs are a family of motivic functions on Lie(GL_n(F)) indexed by nilpotent orbits(=partitions), which refine e.g. regular semisimple orbital integrals, point-counts of compactified Jacobians of plane curve singularities, and supercuspidal character values of the p-adic group (none of which have been well understood in the last 40-50 years). I will give an introduction to Shalika germs and explain their relation to point-counting on compactified Jacobians. I will then give some examples and outline our method of computation, which reveals an intimate connection to the Hilbert scheme of points on C^2.

- Title:

**Julia Schneider (EPFL)**- Title:
*Birational maps of Severi-Brauer surfaces, with applications to Cremona groups of higher rank*

Abstract: The Cremona group of rank N over a field K is the group of birational transformations of the projective N-space that are defined over K. In this talk, however, we will first focus on birational transformations of (non-trivial) Severi-Brauersurfaces, that is, surfaces that become isomorphic to the projective plane over the algebraic closure of K. In particular, we will prove that if such a surface contains a point of degree 6, then its group of birational transformations is not generated by elementsof finite order as it admits a surjective group homomorphism to the integers. As an application, we use this result to study Mori fiber spaces over the field of complex numbers, for which the generic fiber is a non-trivial Severi-Brauer surface. Weprove that any group of cardinality at most the one of the complex numbers is a quotient of the Cremona group of rank 4 (and higher). This is joint work in progress with Jérémy Blanc and Egor Yasinsky.

- Title:

**Ramon Nunes (MPIM, Bonn/Université Fédérale du Ceará)**- Title:
*Integral representations of L-functions and spectral identities*

Abstract: In recent years a lot of attention has been given to the study of spectral identities between moments of automorphic L-functions. Besides their intrinsic beauty, these formulas are also very powerful as one is able to deduce interesting information about moments without performing a delicate study of the geometric side of a trace formula. In this talk I will show some instances of spectral identities in the literature and show a recent result obtained jointly with Subhajit Jana on a higher rank spectral identity which relates mixed moments of certain Rankin–Selberg L-functions on $GL(n)$.

- Title:

**Ruadhai Dervan (University of Cambridge)**- Title:
*Stability conditions for varieties*

Abstract: Enormous progress in algebraic geometry has been achieved through linking with differential geometry and geometric analysis. A modern example of this is the “Yau-Tian-Donaldson conjecture”, which relates the algebro-geometric notion of K-stability of a projective variety to the existence of solutions to a special PDE on the variety. In this setting, the PDE is the “constant scalar curvature equation”, which can be thought of as giving the variety a canonical choice of metric. I will describe a general framework associating geometric PDEs on projective varieties to notions of algegbro-geometric stability, and will sketch a proof showing that existence of solutions is equivalent to stability in a model case. The framework can be seen as a loose analogue in the setting of varieties of Bridgeland’s stability conditions.

- Title:

**Nguyen-Bac Dang (Institut de Mathématiques d’Orsay)**- Title:
*Degree growth of iterates of rational maps: a functional analytic approach*

Abstract: In this talk, based on a joint work with Charles Favre, I will explain one problem that arise in algebraic dynamics, concerning the growth under iteration of the algebraic degree of a given rational map. We will see how this question can be related to the study of the spectral properties of certain pullback operators acting on a suitable Banach space of algebraic cycles.

- Title:

**Ben Davison (The University of Edinburgh)**- Title:
*Nonabelian Hodge isomorphism for moduli stacks*

Abstract: If C is a smooth projective complex curve, then by classical nonabelian Hodge theory there is a diffeomorphism between the coarse moduli space of representations of the fundamental group of C, and the coarse moduli space of degree zero semistable Higgs bundles on C. In particular, the Borel-Moore homology of these two moduli spaces is isomorphic. In this talk I will construct an isomorphism between the Borel-Moore homologies of the full stack of representations of the fundamental group and the full stack of degree zero semistable Higgs bundles. In the absence of any kind of isomorphism between these two stacks, the isomorphism in BM homology has to take a roundabout route, via an isomorphism of the “BPS cohomology” of the two moduli problems. This in turn is provided by the classical nonabelian Hodge theory, along with a freeness result regarding the BPS Lie algebra on both sides of nonabelian Hodge theory. This is joint work in progress with Lucien Hennecart and Sebastain Schlegel-Mejia.In the first part of the talk I will introduce some of the objects mentioned above, including the BPS Lie algebra, which plays a leading role in the story.

- Title:

**José Simental Rodriguez (Max Planck Institute for Mathematics, Bonn)**- Title:
*Cluster structures on braid varieties*

Abstract: Given a simply-laced complex simple Lie group and an element $\beta$ of its positive braid monoid, we construct an algebraic variety $X(\beta)$ called the braid variety. These are smooth, affine varieties that generalize many well-known varieties in Lie theory, including open Richardson varieties. In joint work with Roger Casals, Eugene Gorsky, Mikhail Gorsky, Ian Le and Linhui Shen, we give an explicit cluster structure to the coordinate ring of $X(\beta)$ using the combinatorics of algebraic weaves. In particular, this shows that open Richardson varieties are cluster varieties.

- Title:

**Lorenzo Fantini (Ecole Polytechnique Paris)**- Title:
*Lipschitz geometry of complex surfaces*

Abstract: Lipschitz geometry is a branch of singularity theory that studies a complex analytic germ (X,0) in (C^n,0) by equipping it with either one of two metrics: its outer metric, induced by the euclidean metric of the ambient space, and its inner metric, given by measuring the length of arcs on (X,0). Whenever those two metrics are equivalent up to a bi-Lipschitz homeomorphism, the germ is said to be Lipschitz normally embedded (LNE).

I will give an overview of several results obtained together with André Belotto, András Némethi, Walter Neumann, Helge Pedersen, Anne Pichon, and Bernd Schober on the Lipschitz geometry of surfaces, and more precisely on their inner metric structure, properties of LNE surfaces, criteria to prove that a germ is LNE, and the so-called problem of polar exploration, which is the quest of determining the generic polar curves of a complex surface from its topology.

- Title:

**Piotr Przytycki (McGill University)**- Title:
*Tits alterative for the 3-dimensional tame automorphism group*

Abstract: This is joint work with Stephane Lamy. Let k be a field of characteristic zero. The tame automorphism group Tame(k^3) is generated by the affine automorphisms of k^3, and the automorphisms of the form (x,y,z)->(x,y,z+P(x,y)), where P is a polynomial in k[x,y]. We prove that every subgroup of Tame(k^3) is virtually solvable or contains a non-abelian free group.

- Title:

**Sergey Arkhipov (Aarhus University)**- Title:
*Logarithmic differential forms on Bott-Samelson varieties and braid relations*

Abstract: Braid relations in the coherent version of the affine Hecke category were established via a delicate study of Steinberg variety, by Bezrukavnikov and Riche. The talk is devoted to an alternative approach to establishing braid relations in a version of affine Hecke category developed in the thesis of my student Sebastian Orsted. We begin with proposing a realization of affine Hecke category Koszul dual to the one of Bezrukavnikov-Riche: we define a category of equivariant modules over the ring of differential forms on a reductive algebraic group G, equipped with convolution monoidal structure. Then we introduce the candidates for the braid group generators given by DG-modules of logarithmic differential forms on the minimal parabolic subgroups. The proof of braid relations goes via the study of logarithmic differential forms on large Bott-Samelson varieties and is based on the following observation: “Let X be a desingularization of the variety Y such that the preimage of a divisor containing singularities of Y in X is a divisor with normal crossings. Then the direct image of the sheaf of logarithmic differential forms on X to Y depends on Y, not on the resolution of singularities X”. We outline the calculation leading to the proof of this statement.

- Title:

**Andrea Ricolfi (Università di Bologna)**- Title:
*A motivic DT/PT correspondence*

Abstract: Donaldson-Thomas invariants of a Calabi-Yau 3-fold Y are related to Pandharipande-Thomas invariants via a wall-crossing formula known as the DT/PT correspondence, proved by Bridgeland and Toda. The same relation holds for the “local invariants”, those encoding the contribution of a fixed smooth curve in Y. We show how to lift the local DT/PT correspondence to the motivic level, proving an explicit formula for the local motivic DT invariants; we do so by exploiting the critical structure on certain Quot schemes acting as our local models. Joint work with Ben Davison.

- Title:

**Sergej Monavari (Universiteit Utrecht)**- Title:
*McKay correspondence in DT theory of Calabi-Yau 4-folds*

Abstract: The (generalized) McKay correspondence relates the representation theory of a finite subgroup G<SL(n,C) with the geometry of a crepant resolution of C^n/G. Lately, this correspondence has been studied at the level of derived categories and in the enumerative geometry of Calabi-Yau three-folds (Gromov-Witten/Donaldson-Thomas theories). More recently, Oh-Thomas developed an algebraic machinery to count sheaves on Calabi-Yau 4-folds. We show how the McKay correspondence naturally (and conjecturally) extends to this new setting, and how it specializes to many enumerative results already known in the literature. This is joint work (in progress!) with Y. Cao and M. Kool.

- Title:

**Cécile Gachet (Univerité Côte d’Azur, Nice)**- Title:
*Finite quotients of abelian varieties with a Calabi-Yau resolution*

Abstract: Let G be a group acting freely in codimension 1 on an abelian variety A. In terms of the Beauville-Bogomolov decomposition, the singular quotient A/G has the type of an abelian variety, whereas its terminalization (or its crepant resolution, if there is one) could be a hyperkähler or a Calabi-Yau variety: the Kummer surface is an example along these lines. In this talk, I show that however, A/G has no simply-connected crepant resolution when assuming that G acts freely in codimension 3. If G acts freely in codimension 2, there are, due to K. Oguiso, exactly two threefolds A/G with a Calabi-Yau resolution. I show that there are no such fourfolds.

- Title:

**Eleonora Anna Romano (Università di Genova)**- Title:
*Recent results on Fano varieties*

Abstract: In this talk we present some recent results on complex smooth Fano varieties. To this end, we first recall an invariant introduced by Casagrande, called Lefschetz defect. We review the literature to deduce that all Fano manifolds with Lefschetz defect greater than three are well known. Then we focus on the case in which the Lefschetz defect is equal to three, by discussing a structure theorem for such varieties. As an application, we use this result to classify all Fano 4-folds with Lefschetz defect equal to three: there are 18 families, among which 14 are toric. This is a joint work with C. Casagrande and S. Secci.

- Title:

**Silvain Rideau-Kikuchi (Université de Paris)**- Title:
*H-minimality*

Abstract: (With R. Cluckers, I. Halupczok.) The development and numerous applications of strong minimality and later o-minimality has given serious credit to the general model theoretic idea that imposing strong restrictions on the complexity of arity one sets in a structure can lead to a rich tame geometry in all dimensions. O-minimality (in an ordered field), for example, requires that subsets of the affine are finite unions of points and intervals. In this talk, I will present a new minimality notion (h-minimality), geared towards henselian valued fields of characteristic zero, generalising previously considered notions of minimality for valued fields (C,V,P …) that does not, contrary to previously defined notions, restrict the possible residue fields and value groups. By analogy with o-minimality, this notion requires that definable sets of of the affine line are controlled by a finite number of points. Contrary to o-minimality though, one has to take special care of how this finite set is defined, leading us to a whole family of notions of h-minimality. I will then describe consequences of h-minimality, among which the jacobian property that plays a central role in the development of motivic integration, but also various higher degree and arity analogs.

- Title:

**Jakub Witaszek (University of Michigan)**- Title:
*Quasi-F-splittings*

Abstract: What allowed for many developments in algebraic geometry and commutative algebra was a discovery of the notion of a Frobenius splitting, which, briefly speaking, detects how pathological positive characteristic Fano and Calabi-Yau varieties can be. Recently, Yobuko introduced a more general concept, a quasi-F-splitting, which captures much more refined arithmetic invariants. In my talk, I will discuss on-going projects in which we develop the theory of quasi-F-splittings in the context of birational geometry and derive applications, for example, to liftability of singularities. This is joint work with Tatsuro Kawakami, Hiromu Tanaka, Teppei Takamatsu, Fuetaro Yobuko, and Shou Yoshikawa.

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**Justin Lacini (University of Kansas)**- Title:
*Logarithmic bounds on Fujita’s conjecture*

Abstract: A longstanding conjecture of T. Fujita asserts that if X is a smooth complex projective variety of dimension n and if L is an ample line bundle, then K_X+mL is basepoint free for m>=n+1. The conjecture is known up to dimension five by work of Reider, Ein, Lazarsfeld, Kawamata, Ye and Zhu. In higher dimensions, breakthrough work of Angehrn, Siu, Helmke and others showed that the conjecture holds if m is larger than a quadratic function in n. We show that for n>=2 the conjecture holds for m larger than n(loglog(n)+3). This is joint work with L. Ghidelli.

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**Claire Burrin (University of Zurich)**- Title:
*Windings of closed geodesics and number theory*

Abstract: The winding of a closed geodesic around the cusp of the modular surface can be computed using a function from the theory of modular forms: the Rademacher function. In joint work with Flemming von Essen, we study how and when generalizations of the Rademacher function also encode the winding for closed geodesics around the cusps of hyperbolic surfaces. For certain families of surfaces, we use a Selberg trace formula argument to obtain precise statistical results on these winding numbers.

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**Gebhard Martin (University of Bonn)**- Title:
*Del Pezzo surfaces with global vector fields*

Abstract: The space of global vector fields on a projective variety X is the tangent space to the automorphism scheme Aut_X of X. If X is a del Pezzo surface (or a weak del Pezzo, or an RDP del Pezzo), then Aut_X is a (possibly non-reduced) affine group scheme of finite type. In particular, X has infinitely many automorphisms if and only if Aut_X is positive-dimensional and then X admits global vector fields. The last implication is an equivalence in characteristic 0, but it can fail in positive characteristic. I will explain how to classify weak and RDP (if p \neq 2) del Pezzo surfaces with global vector fields and give examples displaying interesting behaviour in small characteristics. This is joint work with Claudia Stadlmayr.

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**Giulio Codogni (University of Rome II)**- Title:
*Characterizing Jacobians via the KP equation and via flexes and degenerate trisecants to the Kummer variety: an algebra-geometric approach*

Abstract: I will present algebro-geometric proofs of a theorem by T. Shiota, and of a theorem by I. Krichever. These results characterize Jacobians of algebraic curves among all irreducible principally polarized abelian varieties. Shiota’s characterization is in terms of the KP equation. Krichever’s characterization is in terms of trisecant lines to the Kummer variety; I will discuss only the degenerate case of his result. The proofs rely on a new theorem asserting that the base locus of a complete linear system on an abelian variety is reduced. The talk is based on a joint work with E. Arbarello and G. Pareschi.

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**Luca Battistella (University of Heidelberg)**- Title:
*Logarithmic and orbifold Gromov-Witten invariants*

Abstract: Logarithmic Gromov-Witten theory can be thought of as the study of curves in open manifolds, or, in other words, curves with tangency conditions to a boundary divisor. When the divisor is smooth, several techniques have been deployed to compute the invariants, most notably twisted stable maps, and recursive schemes based on the degeneration formula. When the divisor is normal crossings, on the other hand, the logarithmic theory remains hardly accessible (with some exceptions in the surface or toric case). The strategy of rank reduction, i.e. looking at the components of the boundary one at a time, is more directly applicable to other theories than the logarithmic one (as shown in Nabijou-Ranganathan, and B.-Nabijou-Tseng-You) because of tropical obstructions. Inspired by one of the distinguishing features of the logarithmic theory – being insensitive to modifications of the boundary [Abramovich-Wise] – and further building on the work of Abramovich-Cadman-Marcus-Wise and Tseng-You, in an ongoing project with Nabijou and Ranganathan we show that genus zero tropical obstructions can be disposed of by blowing up the target sufficiently. The slogan is that the orbifold and logarithmic theories can be made to agree by imposing birational invariance on the former.

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**Johannes Flake (Aachen)**- Title:
*Interpolation tensor categories*

Abstract: Tensor categories, that is, loosely speaking, categories with two operations ⊕ and ⊗, lie at the heart of modern representation theory, various areas of algebra, and mathematical physics. A class of tensor categories of recent interest consists of so-called interpolation categories, whose study was initiated by Pierre Deligne. An interpolation category can usually be defined in three equivalent ways: representation theoretically via a family of algebraic objects, like the collection of all symmetric groups; categorically as a universal tensor category subject to specific conditions; and combinatorially via a graphical calculus involving string diagrams. In the first part of the talk, I will explain this trinity of definitions and give a gentle introduction to interpolation categories. In the second part, I will explain some of my research on the structure of interpolation categories and their monoidal centers, including joint work with N. Harman, R. Laugwitz, and S. Posur.

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**Stefan Schroer (Universität Düsseldorf)**- Title:
*Para-abelian varieties and the Albanese map*

Abstract: We show that for each scheme that is separated and of finite type over a field, and whose affinization is connected and reduced, there is a universal morphism to some para-abelian variety. The latter are schemes that acquire the structure of an abelian variety after some ground field extension. This extends a classical result of Serre. The proof relies on the corresponding result in the proper case, which was obtained before in a joint work with Bruno Laurent. The open case also relies on Macaulayfication, removal of singularities by alterations, pseudo-rational singularities, and Bockstein maps.

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**Pablo Cubides Kovacsics (Heinrich-Heine-Universität Düsseldorf)**- Title:
*Sheaf cohomology of algebraic varieties over non-archimedean fields*

Abstract: In this talk I will present a sheaf cohomology theory for algebraic varieties over non-archimedean fields based on Hrushovski-Loeser spaces. After informally framing our main results with respect to classical statements (first 30 min), I will discuss some details of our construction and the main difficulties arising in this new context. If time allows, I will further explain how our results allow us to recover results of V. Berkovich on the sheaf cohomology of the analytification of an algebraic variety over a rank 1 complete non-archimedean field. This is a joint work with Mário Edmundo and Jinhe Ye.

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**Nero Budur (KU Leuven)**- Title:
*Singularities and the monodromy conjecture*

Abstract: The monodromy conjecture claims to relate different aspects of the singularities of polynomials. The conjecture is wide open in general. We give an overview of the recent results on the singularity invariants involved, such as Bernstein-Sato ideals, contact loci of arcs, together with some recently proven particular cases of the conjecture.

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**Alex Takeda (IHES)**- Title:
*The ribbon quiver complex and operations on Hochschild invariants*

Abstract: The structure of a fully extended oriented 2d TQFT is given by a Frobenius algebra. If one wants to lift this structure to a cohomological field theory, the correct notion is of a Calabi-Yau algebra or category; the CohFT operations are then described by a certain graph complex. There are many different notions of categorical Calabi-Yau structure, all requiring some type of finiteness or dualizability. In this talk I will discuss a variation that works in non-dualizable cases as well; in this case the graphs get replaced by quivers. The resulting complex admits an algorithmic description of orientations, and calculates the homology of certain moduli spaces of open-closed surfaces. This can be used to give a fully explicit description of these operations. In the second half of the talk I will describe some of these constructions, including relative versions of Calabi-Yau structures, and some appearances of these structures in Fukaya theory and string topology. This is joint work with M. Kontsevich and Y. Vlassopoulos.

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**François Loeser (Institut Mathématique de Jussieu)**- Title:
*Tame geometry and valued fields*

Abstract: I will start by explaining how the notion of o-minimality encapsulates Grothendieck’s premonitory vision of “tame geometry”. I will then present a framework built with E. Hrushovski that allows to develop a version of tame topology over valued fields. I will focus on the study of skeleta, which are o-minimal piecewise linear subsets on which non-archimedean varieties retract.

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**Sebastian Schlegel Mejia (University of Edinburgh)**- Title:
*BPS cohomology for rank 2 degree 0 Higgs bundle*

Abstract: BPS cohomology for moduli spaces of Higgs bundles is a certain subspace of the compactly-supported cohomology of the stacks of semistable Higgs bundles that in some sense “generates” the full cohomology. It is a conjecture of Davison that the BPS Lie algebra for degree 0 Higgs bundles on a genus g>1 is freely generated as a Lie algebra by the intersection cohomology of the coarse moduli spaces. I will discuss the verification of this conjecture for the rank 2 part of the BPS Lie algebra, which follows from an analysis of the moduli stack of strictly semistable Higgs bundles that enables a comparison with the computation of the intersection cohomology of the coarse moduli space by Mauri. In the pretalk I will motivate BPS cohomology by discussing simpler examples.

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**Maxim Mornev (EPFL)**- Title:
*What is a shtuka?*and*An infinity-adic criterion of good reduction for Drinfeld modules*

Abstract: In the literature there are many seemingly incompatible definitions of a shtuka. I will explain the unifying idea which stands behind them. I will also discuss applications of shtukas to Langlands program and to the theory of Galois representations.

Abstract: J.-K. Yu discovered that a Drinfeld module has a p-adic Tate module not only for every prime p of the coefficient ring, but also for the place $p = \infty$. This contrasts with the case of abelian varieties where a Tate module can not be a vector space over the reals. I will explain how the construction of J.-K. Yu interacts with Fargues-Fontaine curve, and how this leads to a good reduction criterion of a new kind. This result supplements the classical criteria of Néron-Ogg-Shafarevich and Grothendieck-de Jong. No previous knowledge of abelian varieties, Drinfeld modules or Fargues-Fontaine curve will be necessary to understand the talk.

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**Pierrick Bousseau (ETHZ)**- Title:
*Gromov-Witten theory of complete intersections*

Abstract: I will describe an inductive algorithm computing Gromov-Witten invariants in all genera with arbitrary insertions of all smooth complete intersections in projective space. The main idea is to show that invariants with insertions of primitive cohomology classes are controlled by their monodromy and by invariants defined without primitive insertions but with imposed nodes in the domain curve. To compute these nodal Gromov-Witten invariants, we introduce the new notion of nodal relative Gromov-Witten invariants. This is joint work with Hülya Argüz, Rahul Pandharipande, and Dimitri Zvonkine (arxiv:2109.13323).

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**Andras Szenes (University of Geneva)**- Title:
*Wall-crossings and the parabolic Verlinde formula*

Abstract: I will report on joint work with Olga Trapeznikova, on a new proof of the Verlinde formula for parabolic moduli spaces. The proof only uses the basics of invariant theory and carries out a program initiated by Michael Thaddeus on following the changes of Euler characteristics of line bundles when crossing GIT walls. The necessary combinatorics is controlled by so-called diagonal bases of the relevant hyperplane arrangements.

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**Andrea Petracci (Freie U Berlin)**- Title:
*Singularities on K-moduli spaces of Fano varieties*

Abstract: Recently there has been spectacular progress, due to many scholars, on the construction of moduli stacks/spaces of K-(semi/poly)stable Fano varieties. It is a natural question to understand the geometry of these spaces. Although smooth Fano varieties have unobstructed deformations, in joint work with Kaloghiros we constructed the first examples of obstructed K-polystable Fano varieties by using toric geometry. These give singular points on K-moduli of Fanos. In this talk I will try to explain these constructions; as a corollary I will show that K-moduli of Fano of dimension at least 3 can have arbitrarily many local branches.

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**Arthur Foreyi (EPFL)**- Title:
*Equidistribution of exponential sums on abelian varieties*

Abstract: Many exponential sums over finite fields, such as Gauss or Salié-Kloosterman sums, appear as the Fourier-Melin transform of the trace function of an l-adic sheaf on a commutative algebraic group. We are interested in the equidistribution of such sums as the character varies. Generalizing work by Deligne and Katz in the cases of additive and multiplicative groups, a Tannakian formalism always controls the equidistribution. In this talk, I will illustrate this result in some concrete cases where the group is an abelian variety. This is a collaboration with Javier Fresán and Emmanuel Kowalski.

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**Manuel Luethi (EPFL)**- Title:
*Random walks on homogeneous spaces, Spectral Gaps, and Khintchine’s theorem on fractals*

Abstract: Khintchine’s theorem in Diophantine approximation gives a zero one law describing the approximability of typical points by rational points. In 1984, Mahler asked how well points on the middle third Cantor set can be approximated. His question fits into an attempt to determine conditions under which subsets of Euclidean space inherit the Diophantine properties of the ambient space. I will discuss a complete analogue of the theorem of Khintchine for certain fractal measures which was recently obtained in collaboration with Osama Khalil. Our results hold for fractals generated by rational similarities of Euclidean space that have sufficiently small Hausdorff co-dimension. The main ingredient to the proof is an effective equidistribution theorem for associated fractal measures on the space of unimodular lattices. The latter is established using a spectral gap property of a type of Markov operators associated with an S-arithmetic random walk related to the generating similarities.

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**Fabio Bernasconi (EPFL)**- Title:
*Log liftability to characteristic zero of F-split surfaces*

Abstract: Given a projective variety X over an algebraically closed field of characteristic p>0, it is an interesting question to know whether it admits a lifting to characteristic zero. This is false in general (the first examples have been constructed in the sixties), but understanding the possible obstructions or constructing new examples is still an active area of research. Given a projective variety X over an algebraically closed field of characteristic p>0, it is an interesting question to know whether it admits a lifting to characteristic zero. This is false in general (the first examples have been constructed in the sixties), but understanding the possible obstructions or constructing new examples is still an active area of research.

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**Oscar Kivinen (EPFL)**- Title:
*A coherent-constructible correspondence for affine Springer fibers*

Abstract: Affine Springer fibers are moduli spaces whose geometry plays an important role in a variety of things – for example orbital integrals on reductive groups, singularities of the Hitchin fibration, and representations of double affine Hecke algebras. The physics of 3d mirror symmetry suggests a certain equivalence of categories of constructible and coherent sheaves on a partial resolution of the commuting variety (PRCV), and following the physical heuristics it is possible to distill a particular case of this equivalence to a mathematical construction of a (quasi-)coherent sheaf on the PRCV, starting from an affine Springer fiber. In the first 30 minutes, I will give an elementary introduction to affine Springer fibers and related geometry. In the second part of the talk I will introduce BFN-Coulomb branches and the commuting variety, as well as explain the construction and some of its consequences in detail.

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