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**♦ ****Amina Abdurrahman (Princeton University)**

Title: Overview of results related to finding square roots of symplectic L-functions

Abstract: We aim to touch on several different results in number theory and

topology related to the question of finding square roots of symplectic

L-functions in order to provide motivation for the next talks.

This includes Meyer's signature formula in topology and Deligne's

result about epsilon factors attached to an orthogonal representation

in number theory among other things.

Title: A global cohomological formula for Reidemeister torsion

Abstract: We give a global cohomological formula for Reidemeister torsion of a

3-manifold together with a symplectic local system.

This generalizes a formula of Meyer discussed in the previous talk and

can be considered as the topological analogue of a formula for the

central value of a symplectic L-function, which we will discuss in the

next talk.

This is joint work with Akshay Venkatesh.

Title: Square roots of symplectic L-functions

Abstract: In the 70s Deligne gave a topological formula for the local epsilon

factors attached to an orthogonal representation. We consider the case

of a symplectic representation and present a conjecture giving a

topological formula for a finer invariant, the square class of its

central value.

This is joint work with Akshay Venkatesh.

Title: Kazhdan-Lusztig polynomials of matroids

Abstract: I will introduce Kazhdan-Lusztig polynomials of matroids and survey combinatorial and geometric theories built around them. The focus will be on the conjecture of Gedeon, Proudfoot, and Young that all zeros of the Kazhdan-Lusztig polynomial of a matroidlie on the negative real axis.

**♦ ****Clark Barwick (University of Edinburgh)**

Title: The stratified homotopy type of number rings.

Abstract: (Note: My goal with these talks is to spark interest in using the new subject of stratified homotopy theory and exodromy. So these talks will be of an introductory and foundational nature, and I am hoping to provoke lots of discussion. It's less important that I arrive at a particular set of theorems than that I describe these ideas intelligibly.)

Talk 1: Stratified homotopy theory. I'll begin by describing homotopy types of topological spaces that is different from the definitions we saw as students. This will involve some higher category theory to say properly; I'll explain why. Then I'll discuss stratified spaces, and I'll explain how our attitude toward homotopy types leads us to a description of the homotopy type of a stratified space. By the end, we will have three ways of thinking about stratified homotopy types, all of which will be relevant.

Talk 2: Homotopy types in algebraic geometry. I'll begin by repeating Grothendieck's definition of the étale fundamental group of a scheme or variety, and I'll give some examples. Using our attitude toward homotopy types from the last lecture, I'll define the étale homotopy type of a scheme. I will go further, however, and define a stratified étale homotopy type of a scheme. If there is time, I'll explain how the pyknotic/condensed formalism helps us manage "topological" information in homotopical contexts.

Talk 3: Studying number rings. I'll describe the usual statement of global duality for number rings, which is at the root of the analogy between arithmetic geometry and low-dimensional topology. I will mention some fun facts about this analogy, but I will also highlight some problems with the analogy. I will then describe how stratified homotopy theory can be used to repair some of these problems. If there is time, I'll explain some recent work of Tomer Schlank and Ariel Davis that takes this vision seriously to generate conjectures in analytic number theory.

**♦ ****Ben Davison (University of Edinburgh)**

Title : The decomposition theorem and nonabelian Hodge theory for stacks

Abstract: The nonabelian Hodge correspondence provides a diffeomorphism between certain coarse moduli spaces of semistable Higgs bundles on a smooth projective curve C (the Dolbeault side) and coarse moduli spaces of representations of the fundamental group of C (the Betti side). In the case of coprime rank and degree, these spaces are smooth, and the famous P=W conjecture states that the isomorphism in cohomology provided by the above diffeomorphism takes the weight filtration on the Betti side to the perverse filtration on the Dolbeault side. The purpose of these talks is to use recent advances in cohomological Donaldson-Thomas theory to extend this story to moduli stacks.

For coprime rank and degree, two key features in the study of classical nonabelian Hodge theory are the perverse filtration with respect to the Hitchin base, and the purity of the cohomology of the Dolbeault moduli space. I will present an extension of the BBDG decomposition theorem to moduli stacks of objects in 2CY categories, which enables us to reproduce both of the above features for stacks in nonabelian Hodge theory.

These results, along with cohomological Hall algebras, allow us to connect the intersection cohomology of coarse moduli spaces with the Borel-Moore homology of the above stacks, providing the connection between three versions of the P=W conjecture: the original conjecture for smooth moduli spaces, the version for intersection cohomology of singular moduli spaces, and a new version for stacks.

**♦ ****Tudor Dan Dimofte (University of Edinburgh)**

Title: 3d Mirror Symmetry and Link Homology

Abstract: These lectures will center around some work from the past few years (with N.Garner, J. Hilburn, A. Oblomkov, and L. Rozansky) constructing "triply graded" HOMFLY link homology in topological twists of 3d N=4 gauge theories.

I will begin by recalling various historical (and less historical) realizations of link invariants and link homology in physics, all ultimately descending from conifold constructions of Witten, Hori-Vafa, and Gukov-Schwarz-Vafa. To set the stage for the remaining talks, I'll zoom in on one of these constructions and reduce it to a 3d N=4 gauge theory living on certain branes.

In the second talk, I will summarize work of Oblomkov and Rozansky that realizes HOMFLY homology in 3d B twists, using categories of matrix factorizations, relating the Oblomkov-Rozansky construction to the brane configurations of the first talk. I will take the opportunity to review the general categorical structures that arise in 3d B-twists, and how they apply to the link-homology computation.

In the final lecture, I will then describe a "3d mirror" A-twisted setup (which forms the bulk of our new work), and in the process connect with affine Springer fibers and the Braverman-Finkelberg-Nakajima construction of Coulomb branches, and highlight some known and some yet mysterious aspects of higher categorical structures in the 3d A twist.

Lecture 1: Conifolds and Branes

Lecture 2: HOMFLY Homology in 3d B-Models

Lecture 3: A-Model Mirrors and Affine Springer Fibers

**♦ ****Lotte Hollands (Heriot-Watt University)**

Title: Quantum field theory in the 1/2 Omega-background

Abstract: In this talk I'll discuss various aspects of supersymmetric QFT's in the 1/2 Omega-background, such as a relation to the Hitchin integrable system, the brane of opers and cluster coordinates on the moduli space of flat connections. Central to this story is the partition function of the QFT with respect to certain 1/2 BPS boundary conditions. We'll argue that this partition function may be obtained from an exact WKB analysis and defines a section of a distinguished line bundle over the moduli space of flat connections.

**♦ ****Pavel Safronov (University of Edinburgh)**

Title: Reidemeister torsion in topology, mathematical physics and geometry

Abstract: In this series of talks I will outline many applications of the theory of Reidemeister torsion in various fields of mathematics. This invariant is related to a "higher" analog of the Euler characteristic. I will review its appearance in topological quantum field theories (BF theory), knot theory (Alexander polynomial) and relate it to the theory of epsilon-factors of Deligne, Beilinson, Patel, ... Finally, I will explain a work in progress providing an interpretation of the Reidemeister torsion as a volume form in derived algebraic geometry.

**♦ Bernd Johannes Schroers (Heriot-Watt University)**

Title: Particles and Fusion Rules in Quantum Field Theory

Abstract: Particles can be modelled in two basic ways in QFT: as quantum excitations like electrons or photons, or as soliton solutions of classical field equations like vortices or non-abelian magnetic monopoles. It is also possible to combine the two concepts to solitons carrying quantum excitations, for example as flux-charge composites or as dyons in Yang-Mills-Higgs theory. One of the most basic challenges in analysing a QFT is to understand the rules according to which particles fuse. This is well-understood for quantum excitations, partly understood for solitons, but barely understood for mixed excitations like dyons. In my talk I will explain this problem and discuss a few examples where there is a partial understanding.

**♦ Alexander Shapiro (University of Edinburgh)**

Title: Cluster structure on character varieties and Coulomb branches.

Abstract: Given (a mutation equivalence class of) a quiver one can define a cluster variety — certain affine Poisson variety with rich combinatorial structure. The two main sources of cluster varieties arising in mathematical physics are the decorated character varieties and the K-theoretic Coulomb branches of 4d N=2 quiver gauge theories compactified on a circle. I will discuss these constructions with a focus on the following two examples which lie in the intersection of the two worlds: the quantum group and the spherical double affine Hecke algebra, which can be simultaneously thought as quantum character varieties and as quantized K-theoretic Coulomb branches.

**♦ Brian R. Williams (Boston University)**

Title: Twisted holography and holomorphic fivebranes

Abstract: In physics, the holographic principle gives a powerful correspondence between gravity and gauge theory. Recently, Costello and Li have proposed a program for a `twisted’ version of holography in terms of a mathematical phenomena called Koszul duality. In this series of lectures we introduce this program and highlight some important examples worked out by Costello, Gaiotto, and Paquette in the context of topological string theory. In the last lecture, we will turn to recent work with Raghavendran which uses the twisted holographic principle to gain insight into the holomorphic twist of the ubiquitous six-dimensional superconformal field theory and discuss applications to the AGT correspondence.