## Mark Stern : Nahm transforms and ALF Spaces

- Uploaded by schrett ( 10 Views )In this talk we consider the moduli space of Yang-Mills instantons on the family of hyperkahler 4 manifolds known as multi-center TaubNUT spaces. We describe the Nahm transform for flat manifolds. Then we sketch its extension to the above hyperkahler family, where it defines an isometry between the moduli space of instantons on the multi-center TaubNUT and the moduli space of solutions of a rococo system of ordinary differential equations. This is joint work with Sergey Cherkis and Andres Larrain Hubach

## Luca Di Cerbo : Extended Graph Manifolds, Dehn Fillings, and Einstein Metrics

- Uploaded by schrett ( 102 Views )In this talk, I will present some new topological obstructions for solving the Einstein equations (in Riemannian signature) on a large class of closed four-manifolds. Finally, I will show that complex-hyperbolic Einstein Dehn filling compactification cannot possibly performed in dimension four. This is in striking contrast with the real-hyperbolic case, and it answers (negatively) a fifteen years old question of Michael Anderson. If time permits, I will conclude with some tantalizing open problems both in dimension four and in higher dimensions. Part of this work is joint with M. Golla (Universit\’e de Nantes).

## Daniel Stern : Spectral shape optimization and new behaviors for free boundary minimal surfaces

- Uploaded by schrett ( 110 Views )Though the study of isoperimetric problems for Laplacian eigenvalues dates back to the 19th century, the subject has undergone a renaissance in recent decades, due in part to the discovery of connections with harmonic maps and minimal surfaces. By the combined work of several authors, we now know that unit-area metrics maximizing the first nonzero Laplace eigenvalue exist on any closed surface, and are realized by minimal surfaces in spheres. At the same time, work of Fraser-Schoen, Matthiesen-Petrides and others yields analogous results for the first eigenvalue of the Dirichlet-to-Neumann map on surfaces with boundary, with maximizing metrics induced by free boundary minimal immersions into Euclidean balls. In this talk, I'll describe a series of recent results characterizing the (perhaps surprising) asymptotic behavior of these free boundary minimal immersions (and associated Steklov-maximizing metrics) as the number of boundary components becomes large. (Based on joint work with Mikhail Karpukhin.)

## Viktor Burghardt : The Dual Motivic Witt Cohomology Steenrod Algebra

- Uploaded by schrett ( 144 Views )Over a field k, the zeroth homotopy group of the motivic sphere spectrum is given by the Grothendieck-Witt ring of symmetric bilinear forms GW(k). The Grothendieck-Witt ring GW(k) modulo the hyperbolic plane is isomorphic to the Witt ring of symmetric bilinear forms W(k) which further surjectively maps to Z/2. We may take motivic Eilenberg-Maclane spectra of Z/2, W(k) and GW(k). Voevodsky has computed the motivic Steenrod algebra of HZ/2 and solved the Bloch-Kato conjecture with its help. We move one step up in the above picture; we study the motivic Eilenberg-Maclane spectrum corresponding to the Witt ring and compute its dual Steenrod algebra.

## Isaac Sundberg : The Khovanov homology of slice disks

- Uploaded by schrett ( 100 Views )To a cobordism between links, Khovanov homology assigns a linear map that is invariant under boundary-preserving isotopy of the cobordism. In this talk, we study those maps arising from surfaces in the 4-ball and apply our findings to existence and uniqueness questions regarding slice disks bounding a given knot. This reflects joint works with Jonah Swann and Kyle Hayden.

## Demetre Kazaras : If Ricci is bounded below, then mass is in control!

- Uploaded by schrett ( 211 Views )The ADM mass of an isolated gravitational system is a geometric invariant measuring the total mass due to matter and other fields. In a previous work, we showed how to compute this invariant (in 3 spatial dimensions) by studying harmonic functions. Now I'll use this formula to consider the question: How flat is an asymptotically flat manifold with very little total mass? In the presence of a lower bound on Ricci curvature, we make progress on this question and confirm special cases of conjectures made by Ilmanen and Sormani.

## Calvin McPhail-Snyder : Making the Jones polynomial more geometric

- Uploaded by root ( 257 Views )The colored Jones polynomials are conjectured to detect geometric information about knot complements, such as hyperbolic volume. These relationships ("volume conjectures") are known in a number of special cases but are in general quite mysterious. In this talk I will discuss a program to better understand them by constructing holonomy invariants, which depend on both a knot K and a representation of its knot group into SL_2(C). By defining a version of the Jones polynomial that knows about geometric data, we hope to better understand why the ordinary Jones polynomial does too. Along the way we can obtain more powerful quantum invariants of knots and other topological objects.

## Mark Stern : Introduction to nonlinear harmonic forms.

- Uploaded by root ( 102 Views )We motivate and introduce nonlinear harmonic forms. These are de Rham representatives $z$ of cohomology classes which minimize the energy $\|z\|_{L_2}^2$ subject to a nonlinear constraint. We give basic existence results for quadratic constraints, discuss the rich Euler Lagrange equations, and ask many regularity questions.

## Ana-Maria Brecan : On the intersection pairing between cycles in SU(p,q)-flag domains and maximally real Schubert varieties

- Uploaded by root ( 92 Views )An SU(p, q)-flag domain is an open orbit of the real Lie group SU(p, q) acting on the complex flag manifold associated to its complexification SL(p + q, C). Any such flag domain contains certain compact complex submanifolds, called cycles, which encode much of the topological, complex geometric and repre- sentation theoretical properties of the flag domain. This talk is concerned with the description of these cycles in homology using a specific type of Schubert varieties. They are defined by the condition that the fixed point of the Borel group in question is in the closed SU(p,q)-orbit in the ambient manifold. We consider the Schubert varieties of this type which are of com- plementary dimension to the cycles. It is known that if such a variety has non-empty intersection with a certain base cycle, then it does so transversally (in finitely many points). With the goal of understanding this duality, we describe these points of intersection in terms of flags as well as in terms of fixed points of a given maximal torus. The relevant Schubert varieties are described in terms of Weyl group elements.

## Kyle Hayden : Complex curves through a contact lens

- Uploaded by root ( 79 Views )Every four-dimensional Stein domain has a height function whose regular level sets are contact three-manifolds. This allows us to study complex curves in the Stein domain via their intersection with these contact level sets, where we can comfortably apply three-dimensional tools. We use this perspective to characterize the links in Stein-fillable contact manifolds that bound complex curves in their Stein fillings. (Some of this is joint work with Baykur, Etnyre, Hedden, Kawamuro, and Van Horn-Morris.)

## Richard Hain : Fundamental groups of branched coverings of certain Kaehler manifolds

- Uploaded by root ( 80 Views )In this talk I will discuss a result that describes (under certain conditions) the fundamental group of certain branched coverings of quasi-projective varieties that admits a complete Kaehler metric with non-positive curvature. When applied to the period map for genus 3 curves, it implies that the Torelli group in genus 3 is finitely generated. Combined with recent work of Putman and Hatcher-Margalit, it gives a new proof of Dennis Johnson's result that the Torelli group is genus g is finitely generated for all g > 2.

## Lev Rozansky : A categorification of the stable Witten-Reshetikhin-Turaev invariant of links in S2 x S1

- Uploaded by root ( 116 Views )This work was done in close collaboration with M. Khovanov. The Witten-Reshetikhin-Turaev invariant Z(M,L;r) of a link L in a 3-manifold M is a seemingly random function of an integer r. However, for a small class of 3-manifolds constructed by identical gluing of two handlebodies (e.g., for S3 and for S2 x S1) and for sufficiently large values of r the ratio Z(M,L;r)/Z(M;r) is equal to a rational function J(M,L;q) of q evaluated at the first 2r-th root of unity. If M = S3, then J is the Jones polynomial. Khovanov categorified J(S3,L), that is, to a link L in S3 he assigned a homology H(L) with an extra Z-grading such that its graded Euler characteristic equals J(S3,L). We extend Khovanov's construction to links in S2 x S1 thus categorifying J(S2xS1,L). In his work on categorification of the Jones polynomial, Khovanov introduced special algebras H_n and assigned a H_m x H_n module to every (2m,2n)-tangle. We show that if a link L in S2 x S1 is presented as a closure of a (2n,2n)-tangle, then the Hochschild homology of its H_n bimodule is determined by the link itself and serves as a categorificaiton of J(S2xS1,L). Moreover, we show that this Hochschild homology can be approximated by Khovanov homology of the circular closure of the tangle within S3 by a high twist torus braid, thus providing a practical method of its computation.

## Thomas Walpuski : G2instantons over twisted connected sums

- Uploaded by root ( 90 Views )In joint work with H. Sá Earp we introduced a method to construct G2instantons over compact G2manifolds arising as the twisted connected sum of a matching pair of building blocks. I will recall some of the background (including the twisted connected sum construction and a short discussion as to why one should care about G2instantons), discuss our main result and explain how to interpret it in terms of certain Lagrangian subspaces of a moduli space of stable bundles on a K3 surface. If time permits, I will discuss an idea to construct the input required by our gluing theorem.

## Daniel Stern : Min-Max Methods for Ginzburg-Landau Functionals and Connections to Geometric Measure Theory

- Uploaded by root ( 75 Views )I will present a natural method for producing unstable critical points of the Ginzburg-Landau functionals on an arbitrary manifold, and describe results showing that a nontrivial portion of the energy must concentrate on a generalized minimal submanifold of codimension two.

## Mark Stern : Geometry of stable Yang-Mills connections

- Uploaded by root ( 100 Views )On a compact 4-manifold, every self-dual connection and every anti-self-dual connection minimizes the Yang-Mills energy. In this talk, I will answer the converse question for compact homogeneous 4-manifolds. I will also survey related stability results in other dimensions.