*Yuhan Sun (Imperial College London). ***Symplectic cohomology of nodal Lagrangian torus fibrations.** Friday 12th Jan, 1:30-2:30pm. Huxley 341.

**Abstract: **We will start with a gentle introduction of symplectic cohomology, reviewing its importance in dynamics, symplectic topology and mirror symmetry. Then we discuss a covering formula to compute it when our symplectic manifold admits a Lagrangian torus fibration. The main examples are in real dimension 4 with nodal fibers, called symplectic cluster manifolds. This is joint with U.Varolgunes.

*Anand Deopurkar (Australian National University). ***How to count using equivariant cohomology.** Friday 19th Jan, 1:30-2:30pm. Huxley 341.

**Abstract: **I will discuss a natural and fascinating enumerative question in algebraic geometry. Consider a homogeneous form in n variables, and its orbit under the action of GL(n) by linear change of coordinates. What is is the degree of this orbit? The case of binary forms was settled in the 1890s by Enriques and Fano. The case of ternary forms was settled in the 1990s by Aluffi and Faber. I will recast these questions using equivariant cohomology and use this reformulation to settle the case of a quaternary cubic form. The degree of its orbit turns out to be 96120. This is joint work with Anand Patel and Dennis Tseng.

*Yue Ren (Durham University). *** Tropical geometry of generic root counts.** Friday 26th Jan, 1:30-2:30pm. Huxley 341.

**Abstract: **In this talk, we will discuss the problem of determining the generic root counts of parametrised polynomial systems over an algebraically closed field.

We will briefly touch upon the motivation from polynomial system solving, and describe how tropical geometry can help in this task. We will give a brief glimpse on the algebraic geometry behind the tropical intersection product, and discuss various strategies how the latter can be computed. We conclude the talk by highlighting applications to chemical reaction networks, coupled oscillators, and graph rigidity.

If time permits, we will tease an exciting new development in applied tropical geometry in collaboration with Anthea Monod (Imperial), Paul Lezeau (Imperial), and Yiannis Fam (Imperial): non-Archimedean optimisation.

*Raf Bocklandt (University of Amsterdam). ***Dimer Duality in Different Dimensions.** Friday 2nd Feb, 1:30-2:30pm. Huxley 341.

**Abstract: **A dimer is a bipartite graph on a Riemann surface. Dimer duality is a combinatorial involution on the set of dimers that often changes the genus of the surface.

This duality offers a combinatorial approach to model certain instances of mirror symmetry. Depending on what dimension you wish to work in you get different types of mirror pairs.

In this talk we will discuss how one can construct combinatorial Fukaya-type categories to model this and what their algebraic counterparts look like.

*Artem Pulemotov (University of Queensland). ***Palais–Smale sequences for the prescribed Ricci curvature functional.** Friday 9th Feb, 1:30-2:30pm. Huxley 341.

**Abstract: **On homogeneous spaces, solutions to the prescribed Ricci curvature equation coincide with the critical points of the scalar curvature functional subject to a constraint. We provide a complete description of Palais–Smale sequences for this functional. As an application, we obtain new existence results for the prescribed Ricci curvature equation, which enables us to observe previously unseen phenomena. Joint work with Wolfgang Ziller (University of Pennsylvania).

*Nikita Nikolaev (University of Birmingham). ***From WKB to BPS and DT.** Friday 16th Feb, 1:30-2:30pm. Huxley 341.

**Abstract: **I will give an overview of the exact WKB method and abelianisation of meromorphic connections on curves. Remarkably, these methods access a wealth of geometric structures on the moduli spaces of connections. This includes symplectic cluster coordinates which solve Riemann-Hilbert problems associated with BPS structures, as well as complex hyperkähler structures that conjecturally encode DT invariants.

*Navid Nabijou (Queen Mary University of London). ***Logarithms, roots, and negative tangencies.** Friday 23rd Feb, 1:30-2:30pm. Huxley 341.

**Abstract: **Logarithmic and orbifold structures provide two independent ways to model curves in a variety with tangency along a normal crossings divisor. The associated systems of Gromov-Witten invariants benefit from complementary techniques; this has motivated extensive interest in comparing the two approaches.

I will report on recent work in which we establish a complete comparison which, crucially, incorporates negative tangency orders. Negative tangency orders appear naturally in the boundary splitting formalisms of both theories. As such, our comparison opens the way for the wholesale importation of techniques from one side to the other. Contemporaneous work of Sam Johnston uses our comparison to give a new proof of the associativity of the Gross-Siebert intrinsic mirror ring.

Along the way, I will discuss the pathological geometry of negative tangency mapping spaces, and how this can be described and controlled tropically. A crucial step in our work is the discovery of a “refined virtual class” on the logarithmic moduli space, which gives rise to a distinguished sector of the Gromov-Witten theory.

This is joint work with Luca Battistella and Dhruv Ranganathan.

*Dario Beraldo (UCL). ***On the geometric Langlands conjecture.** Friday 1st Mar, 1:30-2:30pm. Huxley 341.

**Abstract: **I will outline the recent proof of the (global, unramified) geometric Langlands conjecture, obtained in collaboration with Arinkin, Chen, Gaitsgory, Faergeman, Lin, Raskin and Rozenblyum.

*Steven Sivek (Imperial College London).*

**Rational homology 3-spheres and SL(2,C) representations.**Friday 8th Mar, 1:30-2:30pm. Huxley 341.

**Abstract: **Building on non-vanishing theorems of Kronheimer and Mrowka in instanton Floer homology, Zentner proved that if Y is a homology 3-sphere other than S^3, then its fundamental group admits a homomorphism to SL(2,C) with non-abelian image. In this talk, I’ll explain how to generalize this to any Y whose first homology is 2-torsion or 3-torsion, other than #^n RP^3 for any n or lens spaces of order 3. This is joint work with Sudipta Ghosh and Raphael Zentner.

*Pierre Descombes (Imperial College London). ***Scattering diagrams for noncommutative resolutions.** Friday 15th Mar, 1:30-2:30pm. Huxley 341.

**Abstract: **The space of Bridgeland stability conditions is a complex manifold that connect slope stability and King stability of various derived equivalent commutative and noncommutative varieties. The moduli spaces of stable objects are locally constant but can change on walls in the space of stability conditions; a phenomenon called wall crossing, which has deep links with birational geometry and the minimal model program. Van den Bergh has introduced the notion of noncommutative crepant resolutions and has shown how to build from them derived equivalent commutative crepant resolutions.

In the CY3 case, wall crossing gives that the DT invariants counting stable object form some combinatorial structure called a scattering diagram, which allows to compute them recursively from some (hopefully simple) initial data. After an introduction to these ideas, we will present recent work aiming to describe initial data for scattering diagrams associated to noncommutative crepant resolutions of CY3 singularities.

*Christopher Mahadeo (University of Illinois at Chicago). ***Topological recursion and twisted Higgs bundles.** Friday 22nd Mar, 1:30-2:30pm. Huxley 341.

**Abstract: **Prior works relating meromorphic Higgs bundles to topological recursion have considered non-singular models that allow the recursion to be carried out on a smooth Riemann surface. I will discuss some recent work where we define a “twisted topological recursion” on the spectral curve of a twisted Higgs bundle, and show that the g=0 components of the recursion compute the Taylor expansion of the period matrix of the spectral curve, mirroring a result of for ordinary Higgs bundles and topological recursion.