Please note that, during this term, the seminar will be Covid-19 free via Microsoft Teams.

The seminar will be accessible here. If you do not have an account from Imperial College, you might need an invitation. Please send me an email and I will add you.

*Pieter Belmans (University of Bonn). ***Graph potentials as mirrors to moduli of vector bundles on curves.** Friday 9th October, 1:30-2:30pm.

**Abstract:** In a joint work with Sergey Galkin and Swarnava Mukhopadhyay we have introduced graph potentials, a class of Laurent polynomials associated to decorated trivalent graphs. These Laurent polynomials satisfy interesting symmetry and compatibility properties. Under mirror symmetry they are related to moduli spaces of rank 2 bundles (with fixed determinant of odd degree) on a curve of genus $g\geq 2$, which is a class of Fano varieties of dimension $3g-3$. I will discuss what we can say about the (quantum) period, and if time permits how the geometry of graph potentials gives further evidence for conjectural semiorthogonal decompositions of the derived category.

*Noah Arbesfeld (Imperial College). ***K-theoretic invariants of Quot schemes.** Friday 16th October, 1:30-2:30pm.

**Abstract:** I will present results on virtual invariants of Quot schemes parametrizing zero-dimensional quotients on surfaces and threefolds. I will also explain some geometric context into which these results fit. The results for surfaces yield information about certain integrals over moduli spaces of instantons. The formulas for threefolds provide evidence for a conjectural framework of Nekrasov and Okounkov that relates the Donaldson-Thomas theory of threefolds to the geometry of Calabi-Yau fivefolds. The talk is based on joint work with Yakov Kononov and joint work with Drew Johnson, Woonam Lim, Dragos Oprea and Rahul Pandharipande.

Please note the unusual time:

*Andrew Lobb (Durham University). ***The rectangular peg problem.** Friday 23rd October, 2:30-3:30pm.

**Abstract:** For any smooth Jordan curve and rectangle in the plane, we show that there exist four points on the Jordan curve forming the vertices of a rectangle similar to the given one. Joint work with Josh Greene.

*Carlos Améndola (Technische Universität München). ***Maximum Likelihood Estimation of Toric Fano Varieties.** Friday 30th October, 1:30-2:30pm.

**Abstract:** One of the success stories in algebraic statistics is the use of algebraic techniques to study and compute parameter estimates for statistical models. Conversely, statistical inference problems give rise to interesting geometric problems. In this talk we study the maximum likelihood estimation problem for several classes of toric Fano models. We start by exploring the maximum likelihood degree for all 2-dimensional Gorenstein toric Fano varieties. We show that the ML degree is equal to the degree of the surface in every case except for the quintic del Pezzo surface with two singular points of type A1 and provide explicit expressions that allow to compute the maximum likelihood estimate in closed form whenever the ML degree is less than 5. We then explore the reasons for the ML degree drop using A-discriminants and intersection theory. Based on joint work with Dimitra Kosta and Kaie Kubjas.

* Florian Naef (University of Dublin). ***The string coproduct.** Friday 6th November, 1:30-2:30pm.

**Abstract:** Given an oriented manifold M, Chas and Sullivan define operations on the homology of free loop space H(LM), obtained by a combination of geometrically intersecting cycles in M, and concatenating paths. There is a rough hierarchy of these operations. The lowest level includes an associative product on H(LM), the string coproduct. As shown by Cohen-Klein-Sullivan this operation merely uses the underlying homotopy type of the manifold. The simplest operation at the next level is the string coproduct. I will discuss several aspects of this operation: an interpretation in terms of a topological field theory, a connection with the geometry of the configuration space of points, a conjectural connection with Reidemeister torsion of the manifold and an algebraic description over the rationals.

This is based on joint works with T. Willwacher and P. Safronov.

*Steve Oudot (École polytechnique).*** Local characterizations of poset representations: the case of the plan.** Friday 13th November, 1:30-2:30pm.

**Abstract:** Thanks to recent work by Botnan and Crawley-Boevey, we know that all pointwise finite-dimensional representations of the plane R^2 (viewed as a poset equipped with the product order) decompose into direct sums of indecomposables. A question that arises immediately is to classify these indecomposables. As it is a difficult question in full generality, we want to tackle it from different perspectives. One of them, inspired by applications in topological data analysis, is to try to work out “local” or easily-verifiable conditions under which certain types of summands will (or will not) appear in the decomposition. In this talk I will present two results in this vein:

– first, a condition called “middle exactness”, which guarantees that the indecomposables are indicator modules supported on blocks (i.e. upper-right or lower-left quadrants, horizontal or vertical bands);

– second, a somewhat weakened condition which guarantees that the indecomposables are indicator modules supported on axis-aligned rectangles.

I will also present some applications of these results, together with some counter-examples showing the limits of this approach.

*Anthea Monod (Imperial College). ***Tropical Geometry of Phylogenetic Tree Spaces.** Friday 20th November, 1:30-2:30pm.

**Abstract:** BHV space is a well-studied moduli space of phylogenetic trees that appears in many scientific disciplines, including computational biology, computer vision, combinatorics, and category theory. Speyer and Sturmfels identify a homeomorphism between BHV space and a version of the Grassmannian using tropical geometry, endowing the space of phylogenetic trees with a tropical structure, which turns out to be advantageous for computational studies. In this talk, I will present the coincidence between BHV space and the tropical Grassmannian. I will then give an overview of some recent work I have done that studies the tropical Grassmannian as a metric space and the practical implications of these results on probabilistic and statistical studies on sets of phylogenetic trees.

*Boris L Tsygan (Northwestern University). ***Notes on noncommutative forms and Hochschild chains.** Friday 27th November, 1:30-2:30pm.

**Abstract:** I will start from the very beginning and explain how one arrives at a generalisation of calculus from a ring of functions on a manifold to a noncommutative ring, getting the Ginzburg-Schedler construction in terms of noncommutative forms. I will then relate this to other situations, including crystalline cohomology and it’s noncommutative generalisations in terms of Hochschild and cyclic complexes. I will discuss the common algebraic structure in all these situations, namely the category in cocategories.

*David Fernandez (Bielefeld University).*** Noncommutative Poisson geometry and pre-Calabi-Yau algebras.** Friday 4th December, 1:30-2:30pm.

**Abstract:** A long-standing problem in Poisson geometry has been to define appropriate “noncommutative Poisson structures”. To solve it, M. Van den Bergh introduced double Poisson algebras and double quasi-Poisson algebras that can be regarded as noncommutative analogues of usual Poisson manifolds and quasi-Poisson manifolds, respectively. Recently, N. Iyudu and M. Kontsevich found an insightful correspondence between double Poisson algebras and pre-Calabi-Yau algebras; certain cyclic A_infty-algebras that can be seen as noncommutative versions of shifted Poisson manifolds. In this talk, I will present an extension of Iyudu-Kontsevich’s correspondence to the differential graded setting. Moreover, I will explain how double quasi-Poisson algebras give rise to pre-Calabi-Yau algebras. Interestingly, they involve an infinite number of nonvanishing higher multiplications weighted by the Bernoulli numbers. This is a joint work with E. Herscovich (Grenoble).

*Spencer Bloch (University of Chicago).*** The world’s simplest degeneration of algebraic varieties.** Friday 11th December, 1:30-2:30pm.

**Abstract:** We consider a one parameter degeneration of smooth varieties of odd dimension 2n-1. Assume the central fibre has an isolated ordinary double point and no other singularities. (Example: neighborhood of a singular fibre in a Lefschetz pencil). Pretty simple, no? Not from the viewpoint of arithmetic algebraic geometry! There are two biextensions of Hodge structure involved. I will explain biextensions and talk about the problem of calculating the heights of these objects. Of particular interest is the case of a hypergeometric family, where the heights must be functions of the hypergeometric parameters. From an arithmetic viewpoint, the Beilinson conjectures relate one of the heights to the vanishing at the center of the critical strip of the L-function associated to H^{2n-1} of the resolved central fibre.

*Jeremy Lane (McMaster University).*** Stratified gradient-Hamiltonian vector fields and applications in symplectic geometry.** Friday 18th December, 1:30-2:30pm.

**Abstract:** Gradient-Hamiltonian vector fields have emerged in recent years as an important bridge between algebraic geometry and symplectic geometry. As shown in the work of Nishinou-Nohara-Ueda and Harada-Kaveh, gradient-Hamiltonian vector fields allow one to construct integrable systems on smooth projective varieties from toric degenerations.

In this talk I will describe how this construction can be extended to degenerations of quasi-projective varieties which are not necessarily smooth. Such degenerations are common and arise in many important contexts, such as toric degenerations of the base affine space G//N of a reductive algebraic group G associated Lusztig’s to dual canonical basis. As a consequence, we are able to construct integrable systems with nice properties such as convexity on arbitrary Hamiltonian K-manifolds, for K a compact connected Lie group.

This based on joint work with Benjamin Hoffman.