Winter 2025
The Seminar runs on Tuesdays at STRAND BLDG S4.29. This term, some talks will run from 15:00 to 16:00, and others will be in two parts – an introductory talk from 15:00 to 15:40 and a research talk from 15:45 to 16:30.
21 January, 15:00-16:00, STRAND BLDG S4.29
Speaker: Konstanze Rietsch (King’s College London)
Title: Generalising Euler’s Tonnetz
Abstract: Euler in 1739 wrote about a way to visualise harmonic relationships in music thus creating what is now called `Euler’s Tonnetz’. This talk is about Euler’s tonnetz from a modern point of view, and how to generalise it. Our ‘Tonnetze’ will take place on triangulated surfaces. We will, in particular, consider a set of examples that live on triangulations of tori and are related to crystallographic reflection groups, and a diatonic example related to a famous finite geometry.
28 January, Part I: 15:00-15:40, Part II: 15:45-16:30, STRAND BLDG S4.29
Speaker: Jeehoon Park (Seoul National Univeristy)
Title: Flat F-manifolds for Calabi-Yau smooth projective complete intersection varieties
Abstract: In this talk, I will explain how to construct formal flat F-manifold structures on the primitive cohomology H of a Calabi-Yau smooth projective complete intersection variety X. My strategy is to convert an analysis on H to study on non-isolated hypersurface singularities using the Cayley trick; we associate a dGBV (differential Gerstenhaber-Batalin-Vilkovisky) algebra A to H, and find a special solution to Maurer-Cartan equation in order to construct formal flat F-manifolds.
4 February, Part I: 15:00-15:40, Part II: 15:45-16:30, STRAND BLDG S4.29
Speaker: Yanki Lekili (Imperial College London)
Title: Deformations of cyclic quotient surface singularities via mirror symmetry
Abstract: Let $X_0$ be a rational surface with a cyclic quotient singularity $(1,a)/r$. Kawamata constructed a remarkable vector bundle $K_0$ on $X_0$ such that the finite-dimensional algebra $End(K_0)$, called the Kalck-Karmazyn algebra, “absorbs’’ the singularity of $X_0$ in a categorical sense. If we deform over an irreducible component of the versal deformation space of $X_0$ (as described by Kollár and Shepherd-Barron), the bundle $K_0$ also deforms to a vector bundle $K$. These results were established using abstract methods of birational geometry, making the explicit computation of the family of algebras challenging. We will utilise homological mirror symmetry to compute $End(K)$ explicitly. In the case of a Q-Gorenstein smoothing, this algebra $End(K)$ is a matrix order deforming the Kalck-Karmazyn algebra, and “absorbs” the singularity of the corresponding terminal 3-fold singularity. I will also discuss a conjecture describing irreducible components of the deformation space of $X_0$ in terms of the finite-dimensional algebra $End(K_0)$.
This is based on our joint work (on arXiv) with Jenia Tevelev.
11 February, Part I: 15:00-15:40, Part II: 15:45-16:30, STRAND BLDG S4.29
Speaker: Tyler Kelly (Queen Mary University London)
Title: Applying Exoflops to Calabi-Yau Complete Intersections in Toric Varieties
Abstract: Landau-Ginzburg (LG) models consist of the data of a quotient stack X and a regular complex-valued function W on X. Here, geometry is encapsulated in the singularity theory of W. One can find that LG models are deformations of Calabi-Yau complete intersections in some sense. Exoflops essentially create new GIT problems of partial compactifications of X, expanding the tractable birational geometries related to a given Calabi-Yau complete intersection. We will explain this technique, provide some foundational results about this and then some new applications proven recently for Calabi-Yau orbifolds. This talk contains results from a series of joint works with D. Favero (UMinn) and A. Malter (BIMSA).
18 February, Part I: 15:00-15:40, Part II: 15:45-16:30, STRAND BLDG S4.29
Speaker: Henry Wilton (University of Cambridge)
Title: On the congruence subgroup property for mapping class groups of surfaces
Abstract: I will relate two notorious open questions in low-dimensional topology. The first asks whether every hyperbolic group is residually finite. The second, the congruence subgroup property, relates the finite-index subgroups of mapping class groups of surfaces to the topology of the underlying surface. I will explain why, if every hyperbolic group is residually finite, then mapping class groups enjoy the congruence subgroup property. If there’s time, I may give some further applications to the question of whether hyperbolic 3-manifolds are determined by the finite quotients of their fundamental groups.
25 February, Part I: 15:00-15:40, Part II: 15:45-16:30, STRAND BLDG S4.29
Speaker: Samuel Johnston (Imperial College London)
Title: Quantum periods, toric degenerations and intrinsic mirror symmetry
Abstract: One half of mirror symmetry for Fano varieties is typically stated as a relation between the symplectic geometry of a Fano variety $Y$ and the complex geometry of a Landau-Ginzburg model, realized as a pair $(X,W)$ with $X$ a quasi-projective variety and $W$ a regular function on $X$. The pair $(X,W)$ itself is expected to reflect a pair on the Fano side, namely a decomposition of $Y$ into a disjoint union of an affine log Calabi-Yau and an anticanonical divisor $D$, thought of as mirror to $W$. We will discuss recent work which shows how the intrinsic mirror construction of Gross and Siebert naturally produce potential LG models assuming milder conditions on the singularities of $D$ than typically required for the intrinsic mirror construction. In particular, we show that classical periods of this LG model recover the quantum periods of $Y$. In the setting when $Y\setminus D$ is an affine cluster variety, we will describe how these LG models naturally give rise to Laurent polynomial mirrors and encode certain toric degenerations of $Y$. As an example, we consider $Y = Gr(k,n)$, $D$ a particular choice of anticanonical divisor with affine cluster variety complement and give an explicit description of the intrinsic LG model in terms of Plücker coordinates on $Gr(n-k,n)$, recovering mirrors constructed and investigated by Marsh-Rietsch and Rietsch-Williams.
4 March, Part I: 15:00-16:00 STRAND BLDG S4.29
Speaker: Ximena Fernández (City St George’s, University of London)
Title: The Fermat principle in Riemannian geometry
Abstract: In many situations in physics, the path of light is determined not only by spatial geometry but also by an underlying local density (e.g., mass concentration in general relativity, refractive index in optics). Consider a scenario where a Riemannian manifold in Euclidean space is shaped by a density function, with only a finite sample of points available. How can we infer the original metric and determine the manifold’s topology? This talk introduces a density-based method for estimating topological features from data in high-dimensional Euclidean spaces, assuming a manifold structure. The key to our approach lies in the Fermat distance, a sample metric that robustly infers the deformed Riemannian metric. Theoretical convergence results and implications in the homology inference of the manifold will be presented. Additionally, I will show practical applications in time series analysis with examples from real-world data. This talk is based on the article: X. Fernandez, E. Borghini, G. Mindlin, and P. Groisman. “Intrinsic Persistent Homology via Density-Based Metric Learning.” Journal of Machine Learning Research 24 (2023) 1-42.
11 March, Part I: 15:00-15:40, Part II: 15:45-16:30, STRAND BLDG S4.29
Speaker: Misha Karpukhin (UCL)
Title: Eigenvalues and minimal surfaces
Abstract: Given a Riemannian surface, the study of sharp upper bounds for Laplacian eigenvalues under the area constraint is a classical problem of spectral geometry going back to J. Hersch, P. Li, S.-T. Yau and N. Nadirashvili. The particular interest in this problem stems from the remarkable fact that the optimal metrics for such bounds arise as metrics on minimal surfaces in spheres. In the talk I will survey recent results on the subject with an emphasis on the fruitful interaction between the geometry and spectral bounds. In particular, I will describe a surprisingly effective method of constructing new minimal surfaces based on the eigenvalue optimisation with a prescribed symmetry group.
18 March, Part I: 15:00-15:40, Part II: 15:45-16:30, STRAND BLDG S4.29
Speaker: Monika Kudlinska (University of Cambridge)
Title: Analogues of the Thurston norm in groups
Abstract: The Thurston norm of a 3-manifold $M$ measures the minimal topological complexity of a surface dual to a character of $M$. In this talk, we will introduce a real-valued function on the first cohomology of an arbitrary group which generalises the Thurston norm. We will propose a strategy for proving that such a function defines a seminorm using the theory of $L^2$-invariants. Finally, we will implement this strategy for some classes of right-angled Artin groups.
25 March, 15:00-16:00, STRAND BLDG S4.29
Speaker: Andras Juhasz (University of Oxford)
Title: Examples of topologically unknotted tori
Abstract: I will discuss three different constructions of smooth tori in $S^4$ whose complements have fundamental group $\mathbb{Z}$: turned 1-twist-spun tori due to Boyle, the union of a ribbon disc with a genus one Seifert surface constructed by Cochran and Davis, and certain tori with four critical points. They are all topologically unknotted, but it is not known whether they are smoothly standard, except for tori with four critical points whose middle level set is a split link. The branched double cover of $S^4$ along any of these surfaces is a potentially exotic copy of $S^2 \times S^2$, though, in the case of Boyle’s example, it cannot be distinguished from the standard $S^2 \times S^2$ using Seiberg-Witten invariants. This is joint work with Mark Powell.