Title: Gauge theory on Eguchi-Hanson fibrations

Abstract: This talk is motivated by the old problem of constructing Donaldson-Thomas invariants for symplectic six-manifolds. The basic idea is to count connections satisfying a 6d instanton equation. Alas, such a count is not an invariant and has to be corrected by contributions from pseudo-holomorphic curves. Making this proposal rigorous is a fascinating yet challenging programme plagued with analytic difficulties. I will discuss a way of defining correction terms, following Haydys-Walpuski, and then focus on the case of particularly simple six-manifolds: fibrations over Riemann surfaces with fibers diffeomorphic to the resolution of C^2 /Z_2. Based on an ongoing project with Greg Parker and Thomas Walpuski.

Title: Solutions to the IIA and the IIB systems

Abstract: The IIA and IIB systems in theoretical physics can be formulated as coupled systems of PDEs for an SU(3) structure on a 6-manifold endowed with additional structure (an abelian monopole for the IIA system, a dilaton and a torsion 3-form for the IIB system). The IIA system turns out to be equivalent to the dimensional reduction of the equations for a torsion-free G2 structure with a Killing field, while the IIB system is closely related to the Hull–Strominger system of Heterotic String Theory and the existence of special Hermitian metrics on non-Kähler Calabi–Yau 3-folds. In the talk I will discuss existence results for complete solutions (necessarily non-compact) of the two systems. The results on the IIA system are based on joint work with M. Haskins and J. Nordström, while the results on the IIB system are based on upcoming joint work with M. Garcia-Fernandez.

Title: Distinguishing exotic R^4's with Heegaard Floer homology

Abstract: Attaching a Casson handle to a slice disk complement yields a smooth 4-manifold that is homeomorphic to R^4. We show that if two slice knots have sufficiently different knot Floer homology, then the resulting exotic R^4’s made using the simplest positive Casson handle are not diffeomorphic, giving us a countably infinite family of pairwise nondiffeomorphic exotic R^4’s. Our main tool is Gadgil’s end Floer homology. This is joint work with Sean Eli and Tye Lidman.

Title: Partition Functions of 5d SCFTs and K-Theoretic Four-Manifold Invariants

Abstract: I will discuss partition functions of five-dimensional supersymmetric gauge theories on circle bundles over closed smooth four-manifolds M. When the spacetime is a product, M\times S^1, these partition functions can be identified with indices of Dirac operators on moduli spaces of instantons, which provide special cases of K-theoretic Donaldson invariants. I will discuss the derivation of these partition functions for b_2^+(M)>0 from the effective field theory description on the Coulomb branch of the four-dimensional low-energy theory, and briefly discuss their generalization to nontrivial circle bundles.

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Title: The family Seiberg-Witten invariants and diffeomorphisms on 4-manifolds

Abstract: This lecture series explores the interplay between Seiberg-Witten theory and the diffeomorphism groups of 4-manifolds. I will first provides a foundational introduction to the Seiberg-Witten equations. Then I will extend this framework to the family Seiberg-Witten invariants and examines their implications for mapping class groups of 4-manifolds. Last, I will discuss connections with symplectic geometry and singularity theory.

Title: The heterotic SU(3) moduli space

Abstract: Arising from theoretical physis, the heterotic SU(3) system is a complicated system of geometric PDEs connecting gauge theory to complex 3-dimensional geometry.  Geometrically, this system has been studied in connection with conifold transitions and hoped-for links to Reid's fantasy.  I will describe recent work and work in progress on geometry associated with the moduli space of solutions to the heterotic SU(3) system, with the goal of searching for invariants.

Title: Froyshov's invariant and slice surfaces

Abstract: I will discuss some applications of Froyshov's invariant q3 to Dehn surgery problems, the existence of slice surfaces in definite 4-manifolds, and Z/2-sliceness. Time permitting, I will discuss some conjectures on the generalization to SU(p) instanton theory over the field Z/p. 

Title: Limiting configurations of $G$-Higgs bundles

Abstract: Let $X$ be a compact Riemann surface and $G$ a semisimple complex Lie group. A $G$-Higgs bundle on $X$ is a holomorphic principal $G$-bundle $P_G$ equipped with a holomorphic section $\theta$ of the adjoint bundle twisted by the canonical bundle. Under certain assumptions, we obtain a harmonic metric $h_t$ on the $G$-Higgs bundle $(P_G,t\theta)$ for each $t>0$. If $G$ is a special linear group and the associated spectral curve is smooth, the family $h_t$ converges exponentially to a singular decoupled harmonic metric $h_{\infty}$ as $t\to\infty$. Such a limiting metric $h_{\infty}$ is called the limiting configuration.

In this talk, we discuss a generalization to cases where $G$ is a general semisimple complex Lie group.

Title: Bow varieties and 3d mirror symmetry

Abstract: Bow varieties are hyper-Kaehler manifolds, introduced by Cherkis, as moduli spaces of singular BPS monopoles on $\mathbb R^3$ and instantons on multi-Taub NUT spaces. They have quiver description, found in my joint work with Takayama, useful to analyze singularities of bow varieties. They are closed under 3d mirror symmetry, i.e., 3d mirror of a bow variety is another bow variety. Therefore, they have been used to check conjectures in 3d mirror as examples. In this minicourse, I will give a survey on bow varieties.

Title: 6d theories and 4-manifolds

Abstract: We will discuss some possible applications of quantum field theories in six dimensions to the study of 4-manifolds.

Title: An exact triangle in monopole Floer spectra.

Abstract:  We describe an exact triangle relating the monopole Floer spectra of surgeries on a knot in an integer homology sphere (including the unfolded monopole Floer spectrum defined by Khandhawit-Lin-Sasahira). This generalizes a special case of the exact triangle in monopole Floer homology proved by Kronheimer-Mrowka-Ozsvath-Szabo.  In this talk, we give some of the set-up of the exact triangle and also indicate some applications to calculating monopole Floer spectra of Seifert spaces. This is joint work with Irving Dai and Hirofumi Sasahira.

Title: Analysis for the Vafa-Witten equations

Abstract: The two lectures will describe some of the fundamental analytical results regarding the moduli spaces of solutions to the Vafa-Witten equations on compact, Riemannian 4-manifolds.  The focus will be on the compactness question for these moduli spaces; why they need not be compact, and what can be said a priori about non-converging sequences in the non-compact moduli spaces.  

Title: Z/2Z harmonic spinors: progress and open problems, I and II. 

Abstract: More than a decade ago, Taubes and his disciples began to observe that non-compactness phenomena in a variety of “new” low-dimensional gauge theories lead to Z/2Z harmonic spinors, roughly speaking, harmonic spinors with codimension two defects. These have

since also appeared in a related questions in gauge theory and calibrated geometry.

In light of the above it is pressing to understand: the nature of the defects in Z/2Z harmonic spinors, when Z/2Z harmonic spinors, and how they deform. The purpose of this two talk mini-course is to survey some of the recent progress and point out a number of remaining open problems.