Autumn Term 2023

Giovanni Inchiostro (University of Washington). Moduli of boundary polarized Calabi-Yau pairs. Friday, Oct 6, 1:30-2:30pm. BLKT 630.

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

Ilaria Di Dedda (King’s College London). Type A symplectic higher Auslander correspondence. Friday, Oct 13, 1:30-2:30pm. BLKT 630.

Abstract: Fukaya-Seidel categories constitute a powerful and geometric derived invariant of singularities. In this talk, I will motivate and describe the study of certain isolated singularities, whose Fukaya-Seidel categories play an important role in bordered Heegaard Floer theory. Motivated by representation theory, I will relate these singularities to abstract objects associated to algebras of Dynkin type A. I will introduce “type A symplectic Auslander correspondence”, a purely geometrical construction which realises a notable result in the study of representations of these algebras.

Brent Pym (McGill University). Hodge theory for Poisson varieties and nonperturbative quantization. Friday, Oct 20, 1:30-2:30pm. BLKT 630.

Abstract: In 1997, Kontsevich solved a long-standing problem in mathematical physics by showing that every Poisson manifold admits a canonical deformation quantization, i.e. a deformation of its algebra of functions to a noncommutative ring. The formula is given by a Feynman-style series expansion, whose coefficients are periods of the moduli space of genus zero curves (multiple zeta values), making it intractable for direct calculation. Following a suggestion of Kontsevich, I will explain how K-theory and mixed Hodge structures can be used to construct natural “period coordinates” on the moduli space of algebraic Poisson varieties, in which the quantization can often be computed simply and explicitly as the exponential map for a complex torus. This talk is based on forthcoming joint work with A. Lindberg, and work in progress with T. Raedschelders and S. Sierra.

Sara Veneziale (Imperial College London). Machine Learning and the classification of Fano varieties. Friday, Oct 27, 1:30-2:30pm. BLKT 630.

Abstract: In this talk, I will describe recent work in the application of AI to explore questions in algebraic geometry, specifically in the context of the classification of Fano varieties. We ask two questions. Does the regularized quantum period know the dimension of a toric Fano variety? Is there a condition on the GIT weights that determines whether a toric Fano has at worst terminal singularities? We approach these problems using a combination of machine learning techniques and rigorous mathematical proofs. I will show how answering these questions allows us to produce very interesting sketches of the landscape of weighted projective spaces and toric Fanos of Picard rank two. This is joint work with Tom Coates and Al Kasprzyk.

Ioan Mărcuț (Radboud University Nijmegen). Linearization of Poisson structures. Friday, Nov 3, 1:30-2:30pm. BLKT 630.

Abstract: Ever since Alan Weinstein’s foundational 1983 paper, the local structure of Poisson manifolds has been a central problem in Poisson Geometry. Among other results, the paper contains the Splitting Theorem for Poisson structures, the formulation of the linearization problem and several conjectures, some of which have been solved soon thereafter by Jack Conn.

In this talk, first I will explain these classical linearization theorems and some open problems. Then I will focus on recent work of mine in this direction: the linearization problems around Poisson submanifolds (with Rui Loja Fernandes) and around singularities with isotropy sl_2(C) (with Florian Zeiser).

Soham Karwa (Imperial College London). Non-archimedean periods for log Calabi-Yau surfaces. Friday, Nov 10, 1:30-2:30pm. BLKT 630.

Abstract:  Period integrals are a fundamental concept in algebraic geometry and number theory. In this talk, we will study the notion of non-archimedean periods as introduced by Kontsevich and Soibelman.  We will give an overview of the non-archimedean SYZ program, which is a close analogue of the classical SYZ conjecture in mirror symmetry. Using the non-archimedean SYZ fibration, we will prove that non-archimedean periods recover the analytic periods for log Calabi-Yau surfaces, verifying a conjecture of Kontsevich and Soibelman. This is joint work with Jonathan Lai.

Patrick Kinnear (University of Edinburgh). Constructing the Crane-Yetter line bundle on the character stack. Friday, Nov 17, 1:30-2:30pm. BLKT 630.

Abstract: The character stack of a manifold is the moduli stack of G-local systems (principal G-bundles with flat connection). In this talk, I will describe the construction of a particular line bundle on the character stack of a 3-manifold. This is part of a bigger functorial construction called a TQFT, which assigns data for manifolds of different dimensions. For surfaces we obtain an invertible sheaf of categories on the character stack, and for 4-manifolds, a nonvanishing function. At all levels, this data has an invertibility property, which allows it to play the role of an anomaly TQFT varying over the character stack. The TQFT I construct is part of an ongoing programme to define a version of Witten-Reshetikhin-Turaev TQFT varying over this moduli stack, whose anomaly is called Crane-Yetter. Passing to global sections, we recover skein theoretic constructions from this data. The goal of this talk is to introduce the setup for thinking about the construction of a TQFT varying over the character stack,and explain why the TQFT for skein theory and WRT is invertible in the appropriate sense.

Amanda Hirschi (University of Cambridge). Global Kuranishi charts in symplectic GW theory. Friday, Nov 24, 1:30-2:30pm. BLKT 630.

Abstract: I will explain the construction of a global Kuranishi chart for moduli spaces of stable pseudoholomorphic maps to a symplectic manifold. I will show how this allows for a straightforward definition of symplectic GW invariants and sketch a proof of the Splitting axiom, the property, which implies the associativity of the quantum cup product. In the case where the symplectic manifold is semipositive, I will describe how to compare our invariants to the GW invariants constructed by Ruan and Tian. This is partially joint work with Mohan Swaminathan.

Alapan Mukhopadhyay (EPFL). Generators of bounded derived categories in prime characteristics. Friday, Dec 1, 1:30-2:30pm. BLKT 630.

Abstract:  Since the appearance of Bondal- van den Bergh’s work on the representability of functors, proving existence of strong generators of the bounded derived category of coherent sheaves on 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 prime characteristic p domain R with finite Frobenius endomorphism, $R^{1/p^n}$ – for large enough n- generates the bounded derived category of finite R-modules. This means any bounded chain complex of finite R-modules can be built from $R^{1/p^n}$ by taking summands, shifts and cones in the bounded derived category. This recovers Kunz’s characterization of regularity in terms of flatness of Frobenius. We will discuss examples indicating that in contrast to the affine situation, for a smooth projective scheme whether some Frobenius pushforward of the structure sheaf is a generator depends on the geometry of the underlying scheme. Part of the talk is based on a joint work with Matthew Ballard, Srikanth Iyengar, Patrick Lank and Josh Pollitz.

Mario B. Schulz (University of Münster). Topological control for min-max free boundary minimal surfaces. Friday, Dec 8, 1:30-2:30pm. BLKT 630.

Abstract: (joint work with Giada Franz) Free boundary minimal surfaces naturally appear in various contexts, including partitioning problems for convex bodies, capillarity problems for fluids, and extremal metrics for Steklov eigenvalues on manifolds with boundary. Constructing embedded free boundary minimal surfaces is challenging, especially in ambient manifolds like the Euclidean unit ball, which only allow unstable solutions. Min-max theory offers a promising avenue for existence results, albeit with the added complexity of requiring control over the topology of the resulting surfaces. We establish general topological lower semicontinuity results for free boundary minimal surfaces obtained through min-max methods in compact, three-dimensional ambient manifolds with mean convex boundary. We also present compelling applications, including the variational construction of a free boundary minimal trinoid in the Euclidean unit ball.

Jordi Daura Serrano (University of Barcelona). Large and iterated finite group actions on aspherical manifolds. Friday, Dec 15, 1:30-2:30pm. BLKT 630.

Abstract: The theory of finite transformation groups studies the symmetries of manifolds by means of finite group actions on them. Some fundamental questions are the following: Given a closed manifold M, which finite groups can act effectively on M? Conversely, which topological properties should M have if we know a collection of finite groups actions on M? We do not know how to answer these questions in full generality.

In this talk we will explain how we can address these questions in the context of large finite group actions on manifolds. We will discuss when the homeomorphism group of a manifold is Jordan and introduce new invariants like the discrete degree of symmetry of a manifold. We will focus on the case where the manifold is aspherical, giving new examples of closed manifolds with Jordan homeomorphism group and using the discrete degree of symmetry to obtain rigidity results. Finally, we will develop a theory of iterated finite group actions to study nilmanifolds.

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