Abstract: Let X be a complex manifold with a holomorphic symplectic structure, i.e., a holomorphic two-form that gives a non-degenerate skew-symmetric pairing on the tangent bundle. A Lagrangian fibration is a fibration X->B whose general fibre is a Lagrangian torus, wrt the holomorphic symplectic structure.

In this talk we consider (Tate-Shafarevich) twists of X->B. These are fibrations X'->B that locally over the base B are isomorphic to X->B. We describe the cohomology groups parametrizing such twists and exhibit some interesting examples. Our main new discovery is that these twists can produce non-Kahler spaces from projective spaces; namely, twists of generalized Kummer varieties yield Guan's examples of non-Kahler holomorphic symplectic manifolds.

Abstract: I will describe a version of global Strominger-Yau-Zaslow (SYZ) mirror symmetry and homological mirror symmetry for a theta divisor in a principally polarized abelian variety of any dimension, over the complex moduli of principally polarized and SYZ fibered abelian varieties. This is based on joint work with Haniya Azam, Catherine Cannizzo, and Heather Lee. 

Abstract: A Higgs bundle comprises a holomorphic bundle over a Riemann surface accompanied by a Higgs field, they are pivotal in gauge theories as solutions to dimensionally reduced Yang-Mills equations. Extensively studied for decades, we delve into the intriguing realm of Higgs bundles on non-compact Riemann surfaces, introducing poles on the Higgs field and augmenting the geometry. Our discussion dives into the Poisson structure analysis and the encoded integrable systems, providing an accessible overview of Higgs bundles without assuming prior knowledge. This presentation draws on collaborative work with Biswas, Martens, Peón-Nieto, and Szabo. 

Abstract: Ax-Schanuel and Ax-Lindenmann type results specify the kind of algebraic relations that are preserved by some transcendental functions. For instance, if there is some algebraic relation x and y and also between j(x) and j(y), being j the modular function, then there x and y are related by a Moebius transformation with rational coefficients and j(x) and j(y) satisfy a modular polynomial. In this talk we will review recent research on the topic and speak about the link of this kind of results and the differential Galois theory for principal connections. This is part of an ongoing project with Guy Casale, James Freitag and Joel Nagloo 

Abstract: In the talk I will outline our (joint with Irit Huq-Kuruvilla) attempt to develop the theory of Gromov-Witten invariants based on Euler characteristics rather than intersection numbers. The starting observation is that in the category of almost complex manifolds the usual Euler characteristic is bordism-invariant. At the homotopy theory level, this leads to the specialization of stably almost complex cobordisms in which the total Chern class occurs in the role of the abstract Todd class. Our goal, however, is to apply this idea in the context of Gromov-Witten theory. After sketching the underlying philosophy, the talk will zoom in on some elementary examples.  

Abstract:  Holomorphic modular forms on the Shimura variety S(G) attached to the reductive group G can be interpreted naturally as sections of automorphic vector bundles: locally free sheaves that can be defined analytically by exploiting the structure of a Shimura variety as a quotient of a symmetric space. The construction can also be made algebraic, and in this way one gets a canonical functor from the tensor category of representations of a certain Levi subgroup K of G to the tensor category of vector bundles on S(G), and thus a homomorphism from the representation ring of K to K_0(S(G)). When S(G) is compact we determine how the image of this homomorphism behaves under Chern characters to Deligne cohomology and continuous l-adic cohomology. When S(G) is non-compact and of abelian type, we use perfectoid geometry to define Chern classes in the l-adic cohomology of the minimal compactification of S(G); these are analogous to the topological cohomology classes defined by Goresky and Pardon, using differential geometry. (Joint work with Helene Esnault.) 

Abstract: We'll discuss universal construction of topological theories in dimensions 2 and 1 (the latter for theories with defects) and explain how they relate to Hankel matrices, automata, and pseudocharacters (if time allows). 

Abstract: Following a conjecture of C. Teleman, we discuss how to relate the quantum cohomology of a GIT quotient with the equivariant quantum cohomology via Fourier transformation. We use this to show the decomposition of quantum cohomology of projective bundles and blowups.

Abstract: Mirror Symmetry phenomenon originating in string theory predicts that any Calabi-Yau manifold should pair up with another Calabi-Yau manifold such that their symplectic geometry and complex geometry get swapped. In 1996, Strominger, Yau and Zaslow proposed a geometric picture that any pair of Calabi-Yau should admit Lagrangian torus fibration at the same time with a shared base manifold. Along the way, there have been several mirror constructions tailed to some special kinds of manifolds such as Calabi-Yau hypersurfaces and complete intersections by Batyrev and Batyrev-Borisov. Nowadays, the Gross-Siebert program offers an intrinsic mirror construction which turns out to be the most successful one in the sense that it can be applied in much more generality and recovers all existing mirror constructions. Gross-Siebert mirror ring out of the program at a large complex structure limit point, with their structure coefficients defined using a new kind of Gromov-Witten invariants so called punctured Gromov-Witten invariants developed by Abramovich, Chen, Gross and Siebert, should be regarded as the ring of regular functions on its mirror. In this talk, I am going to illustrate that the Gross-Siebert mirror ring can be fully determined by expressing its structure coefficients/ punctured invariants in terms of 2-pointed ordinary relative Gromov-Witten invariants in the smooth log Calabi-Yau case.  

Abstract: Homological mirror symmetry is the most commonly accepted mathematical definition of mirror symmetry. On the other hand, the Gross-Siebert program provides an explicit way to construct mirror pairs via tropical geometry and toric degenerations. It is so tempting to put these two beautiful subjects together. In this talk, I am going to introduce the notion of tropical Lagrangian multi-sections, which is the ``tropicalization" of locally free sheaves on the B-model and Lagrangian multi-sections on the A-model. Algebraically, I will illustrate how to construct locally free sheaves on the central fiber of a toric degeneration and discuss their smoothability in dimension 2. Symplectically, I will construct Lagrangian multi-sections with asymptotic conditions that are mirror to toric vector bundles. 

Abstract:  Dubrovin's conjecture, arising from physics, relates two different areas of algebraic geometry, quantum cohomology and derived categories. It states that, for any Fano variety, the derived category admits a full exceptional decomposition if and only if the quantum cohomology is generically semisimple. Test cases to show that the conjecture holds are homogeneous projective spaces: for some of them, among which (co)adjoint varieties, the conjecture has been shown to be true. In this talk we will focus on the cohomology side of the conjecture and we will study hyperplane sections of (co)adjoint varieties. These varieties, even though they are not homogeneous, admit an action of a torus with a finite number of fixed points. The aim of the talk is to show how to use this action to compute the cohomology and the quantum cohomology of such varieties; if time permits, we will summarize what is known with respect to Dubrovin’s conjecture in this context. This is a joint work with N. Perrin. 

Abstract: Commutative Rings are not appropriate for Arithmetic Geometry (and also for Physics !) because addition is “singular at the real prime”. We generalise them by considering commutative Props (all the matrices over a given ring), or commutative Bios (all the raws and columns vectors over the ring). We give the compactified spec(Z) (and similarly for any number field) as a pro object of the category of Prop or Bio Scheme. Considering the associated (rational) Witt lambda ring of this compactified spec(Z) we give a natural algebraic home for the Ramanujan’s sums. 

Abstract：In recent years, our understanding of symmetries has greatly evolved in terms of topological defect surfaces in QFTs of various dimensions. In this talk, I will discuss the features of a class of topological defect lines (TDLs) which correspond to (non-invertible) 0-form or 1-form symmetries in 2d para-fermionic CFTs or 3d anyon theories respectively. The talk is three-folded. First, TDLs can be studied in a pure 2d CFT perspective via a bottom-up approach; On the other hand, we can lift the TDLs into 3d anyon theories and understand them from the viewpoint of para-fermionic condensation in a top-down way; In addition, I will also explain the mathematical structures behind the TDLs in para-fermion/anyon theories, in terms of a generalized fusion category, Zn para-fusion categories.

Abstract: In this talk I will outline the history of instanton sheaves on 3 dimensional varieties, from the Atiyah--Ward correspondence in the 1970s until the most recent advances involving Bridgeland stability conditions. This is based on joint work with G. Comaschi, C. Martinez, and D. Mu. 

Abstract: Generalized complex geometry was introduced by N.Hitchin partly motivated by Mirror Symmetry. It is an interpolation between complex geometry and symplectic geometry. I will describe a symplectic approach to generalized complex geometry. This approach utilizes shifted symplectic geometry and a new theory of homotopy complex structures on derived differential stacks. The theory of homotopy complex structure is built on a strict model of differential forms on formal differential stacks based on polydifferential operators. I will also explain the corresponding results for generalized complex branes which are geared towards a construction of a category of generalized complex branes. 

Abstract: The small quantum connection on a Fano variety is one of the simplest objects in enumerative geometry. Nevertheless, it is the subject of far-reaching conjectures known as the Dubrovin/Gamma conjectures. Traditionally, these conjectures are made for manifolds with semi-simple quantum cohomology or more generally for Fano manifolds whose quantum connection is of unramified exponential type at q=\infty. 

 I will explain a program, joint with Paul Seidel, to show that this unramified exponential type property holds for all Fano manifolds M carrying a smooth anticanonical divisor D. The basic idea of our argument is to view these structures through the lens of a noncommutative Landau-Ginzburg model intrinsically attached to (M,D).

Abstract: I'll explain a new approach to some cohomology theories for algebraic varieties over p-adic fields. It is based on the motivic theory of rigid-analytic varieties, and gives rise, in a unified way, to monodromy operators in Hodge theory, ell-adic and p-adic cohomology. As an application, we answer two questions in the literature (of Fontaine and Flach-Morin, respectively). (Joint with Federico Binda and Alberto Vezzani.)

Abstract: Gauge Linear Sigma Models (GLSM) are Gromov-Witten type curve counting theories. Conceptually, GLSM count maps to a critical locus of a function on a GIT quotient of a vector space. Recently, GLSM were defined mathematically by several groups of people. In the  talk I will explain how to construct and compute generating functions of GLSM invariants assigned to matrix factorizations in the case where the GIT quotient is a toric orbifold. Integral representations for these generating functions lead to various mathematical statements such as Higgs-Coulomb correspondence, wall-crossing and mirror symmetry.

Abstract: The talk is devoted to several real and tropical enumerative problems.

We suggest new invariants of the projective plane (and, more generally, of certain toric surfaces) that arise from appropriate enumeration of real elliptic curves.These invariants admit a refinement (according to the quantum index) similar to the one introduced by Grigory Mikhalkin in the rational case. We discuss the combinatorics of tropical counterparts of the elliptic invariants under consideration and establish a tropical algorithm allowing one to compute them.

This is a joint work with Eugenii Shustin.

Abstract: Quantum K-theory studies a K-theoretic analogue of Gromov-Witten invariants defined as holomorphic Euler characteristics over the moduli spaces of stable maps. The invariants so defined satisfy similar axioms as their cohomological counterparts do, except the divisor equation. Inspired by Verlinde/Grassmannian correspondence, level structures are introduced into quantum K-theory as determinant-type twistings. Such structures admit various symmetries, reveal surprising connections of quantum K-theory to mock theta functions, and appear in the so-called quantum Serre duality as well.

Abstract : We will start with a general overview of the interactions of piecewise-linear geometry with non-archimedean geometry, originating in the classical work of Bieri-Groves. We will then  present a recent finiteness result in tropical geometry obtained in joint work with A. Ducros, E. Hrushovski and J. Ye

Abstract: I will describe the framework of Exponential Networks for computing the BPS spectrum of M-theory on a local Calabi-Yau threefold times R^4 x S^1.  Exponential networks define the counting of special Lagrangians in the mirror Calabi-Yau threefold.  Therefore, from the geometric data of mirror curves, we propose how to compute the related DT invariants. I will briefly sketch how this is achieved for some concrete examples and discuss some relations with BPS quivers. 

Abstract:"Holomorphic Floer Theory" (HFT for short) is the name of the program initiated by Maxim Kontsevich and myself about 2014. The key technical tool for HFT is the notion of wall-crossing structure which we introduced in 2013 and recently revisited in 2020. In this talk I am going to discuss the wall-crossing structure which appears in the study of exponential integrals in finite and infinite dimensions. If time permits I will explain a conjectural wall-crossing structure related to the Chern-Simons theory as well as its application to resurgence of perturbative series arising in the Chern Simons theory.