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**Michel van Garrel (University of Birmingham)**

[ Talk 1 ]

**Title: **Introduction to tropical geometry

**Abstract:** Tropical geometry is sometimes described as the shadow of algebraic geometry. It transforms algebraic geometry into integral affine geometry by turning algebraic varieties into piecewise affine integral manifolds. Perhaps surprisingly, a lot of information is preserved this way and problems related to algebraic varieties can often be solved at the level of their tropicalisations. I will describe how this works and explain the special case of toric varieties.

[ Talk 2 ]

**Title:** Polarised cluster varieties

**Abstract:** Cluster varieties are a natural generalisation of toric varieties, providing a more flexible construction while keeping combinatorial control of their geometry. Then, the existence of a polarisation is a fairly restrictive condition which dictates the possible mutations of cluster charts. They play an important role in mirror symmetry as the conjectured mirrors of Fano varieties.

[ Talk 3 ]

**Title:** Fano varieties and their mirror polarised cluster varieties

**Abstract:** Fano varieties are the algebraic varieties of positive curvature. A conjecture by Corti states that each Fano variety is mirror-dual to a polarised cluster variety. Using tropical geometry, I will explain the geometric origin of this conjecture and describe some examples where it is known to hold.

**You Qi (University of Virginia)**

[ Talk 1 ]

**Title: **I. Introduction to Hopfological algebra.

**Abstract: **Hopfological algebra, introduced by Khovanov, is a generalization of homological algebra which can be associated to any finite dimensional Hopf algebra. Traditional homological algebra (chain complexes, homotopies, dg-algebras, the derived category, etcetera) is a special case, and all of these concepts can be generalized. Another special case is the category of p-complexes, which categorifies the cyclotomic ring at a prime root of unity. In this talk we discuss the motivation behind categorification at a root of unity, and the basics of Hopfological algebra.

[ Talk 2 ]

**Title: **II. Categorified quantum sl(2) at a root of unity.

**Abstract: **To categorify a module over the cyclotomic ring, one needs to produce a p-dg category, which is the analogue of a dg-algebra for this kind of Hopfological algebra. When there is already a graded category which categorifies an algebra at generic parameter q (such as the Khovanov-Lauda-Rouquier category for the positive half of the quantum group, or Lauda's categorification of all of quantum sl(2)), one might hope to equip that category with a p-dg structure. However, determining the p-dg Grothendieck group is still difficult, and requires understanding a new kind of direct sum decomposition. This talk focuses on this issue in the context of categorified quantum sl(2) at a root of unity.

[ Talk 3 ]

**Title: **III. Towards tensor product representations.

**Abstract: **This lecture will focus on categorified tensor product representations of quantum sl(2) at a prime root of unity. In particular, we introduce some new tools which help to transfer categorical actions from one setting to another via a Soergel-like functor.

**Chul-hee Lee (June E Huh Center for Mathematical Challenges, Korea Institute for Advanced Study)**

**Title: **Verlinde rings and cluster algebras

**Abstract:** The Kirillov-Reshetikhin modules form a special class of finite-dimensional representations of quantum affine algebras. In an attempt to calculate the central charges of certain conformal field theories using the dilogarithm function based on the Thermodynamic Bethe Ansatz, Kirillov and Kuniba-Nakanish-Suzuki have proposed some conjectures regarding their quantum dimensions, such as their positivity. I will review how this problem can be reformulated in terms of the Verlinde ring arising from the category of integrable representations of an affine Lie algebra. Then I will discuss the relationship between this topic and the Hernande-Leclerc category, which has been introduced in the context of the categorification of cluster algebras.

**Kyungyong Lee (The University of Alabama)**

**Title:** Broken lines and compatible pairs for rank 2 quantum cluster algebras

**Abstract:** This is a joint work with Amanda Burcroff. Around 2015, Gross, Hacking, Keel, and Kontsevich proved the positivity property for cluster algebras by establishing a novel connection between cluster algebras and scattering diagrams, which arose earlier in the study of mirror symmetry. More precisely, they introduced the theta basis, where each basis element is expressed as a sum over broken lines on a scattering diagram. We want to give a combinatorial interpretation for scattering diagrams and broken lines. We present some progress in this direction.

**Wille Liu (Institute of Mathematics, Academia Sinica)**

**Title: **Character sheaves on graded Lie algebras

**Abstract:** I will present some aspects of the theory of character sheaves on a graded reductive Lie algebra. It is first studied by Lusztig and Yun in 2016 and generalises the theory of character sheaves on reductive Lie algebras, mostly due to Lusztig. It is expected to be a geometric model for the harmonic analysis on p-adic groups. I will report on a forthcoming joint work with Tsai, Vilonen and Xue on the construction of cuspidal sheaves on graded Lie algebras.

**Seokbong Seol (Korea Institute for Advanced Study)**

**Title:** Formal exponential maps and the Atiyah class of dg manifolds** **

**Abstract:** Exponential maps arise naturally in the contexts of Lie theory and smooth manifolds. The infinite jets of these classical exponential maps are related to Poincaré --Birkhoff--Witt isomorphism and the complete symbols of differential operators.

We will investigate the question on how to extend these maps to dg manifolds. As an application, we will show there is an L-infinity structure on the space of vector fields in connection with the Atiyah class of a dg manifold.

In particular, for the dg manifold arising from a foliation, we induce an L-infinity structure on the deRham complex associated with the foliation. As a special case, it is related to Kapranov’s L-infinity structure on the Dolbeault complex of a Kähler manifold.

This is a joint work with Mathieu Stiénon and Ping Xu.

**Philsang Yoo (Seoul National University)**

**Title: **Quantum Field Theory and Factorization Algebras

**Abstract: **In this expository talk, I will outline general features of quantum field theory and introduce a mathematical structure known as a factorization algebra, which encodes essential information of quantum field theory.