## Yang Li : On the Donaldson-Scaduto conjecture

- Geometry and Topology ( 708 Views )Motivated by G2-manifolds with coassociative fibrations in the adiabatic limit, Donaldson and Scaduto conjectured the existence of associative submanifolds homeomorphic to a three-holed 3-sphere with three asymptotically cylindrical ends in X \times R^3, where X is an A2-type ALE hyperkähler manifold. We prove this conjecture by solving a real Monge-Ampère equation with singular right hand side. The method produces many other asymptotically cylindrical U(1)-invariant special Lagrangians in X \times R^2, where X arises from the Gibbons-Hawking construction. This is joint work in progress with Saman Habibi Esfahani.

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

- Geometry and Topology ( 553 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.

## Renato Bettiol : Positive biorthogonal curvature in dimension 4

- Geometry and Topology ( 384 Views )A 4-manifold is said to have positive biorthogonal curvature if the average of sectional curvatures of any pair of orthogonal planes is positive. In this talk, I will describe a construction of metrics with positive biorthogonal curvature on the product of spheres, and then combine it with recent surgery stability results of Hoelzel to classify (up to homeomorphism) the closed simply-connected 4-manifolds that admit a metric with positive biorthogonal curvature.

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

- Geometry and Topology ( 370 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.

## Oguz Savk : Bridging the gaps between homology planes and Mazur manifolds.

- Geometry and Topology,Uploaded Videos ( 315 Views )We call a non-trivial homology 3-sphere a Kirby-Ramanujam sphere if it bounds a homology plane, an algebraic complex smooth surface with the same homology groups of the complex plane. In this talk, we present several infinite families of Kirby-Ramanujam spheres bounding Mazur type 4-manifolds, compact contractible smooth 4-manifolds built with only 0-, 1-, and 2-handles. Such an interplay between complex surfaces and 4-manifolds was first observed by Ramanujam and Kirby around nineteen-eighties. This is upcoming joint work with Rodolfo Aguilar Aguilar.

## Simon Brendle : Singularity formation in geometric flows

- Geometry and Topology ( 309 Views )Geometric evolution equations like the Ricci flow and the mean curvature flow play a central role in differential geometry. The main problem is to understand singularity formation. In this talk, I will discuss recent results which give a complete picture of all the possible limit flows in 2D mean curvature flow with positive mean curvature, and in 3D Ricci flow.

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

- Geometry and Topology ( 301 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).

## Robert Bryant : The affine Bonnet problem

- Geometry and Topology ( 296 Views )The classical Euclidean problem studied by Bonnet in the 19th century was to determine whether, and in how many ways, a Riemannian surface can be isometrically embedded into Euclidean 3-space so that its mean curvature is a prescribed function. He found that, generically, specifying a metric and mean curvature admitted no solution but that there are special cases in which, not only are there solutions, but there are even 1-parameter families of distinct (i.e., mutually noncongruent) solutions. Much later, these Bonnet surfaces were found to be intimately connected with integrable systems and Lax pairs. In this talk, I will consider the analogous problem in affine geometry: To determine whether, and in how many ways, a surface endowed with a Riemannian metric g and a function H can be immersed into affine 3-space in such a way that the induced Blaschke metric is g and the induced affine mean curvature is H. This affine problem is, in many ways, richer and more interesting than the corresponding Euclidean problem. I will classify the pairs (g,H) that display the greatest flexibility in their solution space and explain what is known about the (suspected) links with integrable systems and Lax pairs.

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

- Geometry and Topology ( 279 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.

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

- Geometry and Topology ( 259 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.)

## Siqi He : Classification of Nahm Pole Solutions to the KW Equations on $S^1\times\Sigma\times R^+$

- Geometry and Topology ( 251 Views )We will discuss Witten’s gauge theory approach to Jones polynomial by counting solutions to the Kapustin-Witten (KW) equations with singular boundary conditions over 4-manifolds. We will give a classification of solutions to the KW equations over $S^1\times\Sigma\times R^+$. We prove that all solutions to the KW equations over $S^1\times\Sigma\times R^+$ are $S^1$ direction invariant and we give a classification of the KW monopole over $\Sigma\times R^+$ based on the Hermitian-Yang-Mills type structure of KW monopole equation. This is based on joint works with Rafe Mazzeo.

## Subhankar Dey : Cable knots are not thin

- Geometry and Topology ( 246 Views )Thurston's geometrization conjecture and its subsequent proof for Haken manifolds distinguish knots in S^3 by the geometries in the complement of the knots. While the definition of alternating knots make use of nice knot diagrams, Knot Floer homology, a knot invariant toolbox, defined by Ozsvath-Szabo and Rasumussen, generalizes the definition of alternating knots in the context of knot Floer homology and defines family of quasi-alternating knots which contains all alternating knots. Using Lipshitz-Ozsvath-Thurston's bordered Floer homology, we prove a partial affirmation of a folklore conjecture in knot Floer theory, which bridges these two viewpoints of looking at knots.

## Tye Lidman : Homology cobordisms with no 3-handles

- Geometry and Topology ( 243 Views )Homology cobordisms are a special type of manifold which are relevant to a variety of areas in geometric topology, including knot theory and triangulability. We study the behavior of a variety of invariants under a particular family of four-dimensional homology cobordisms which naturally arise from Stein manifolds. This is joint work with Ali Daemi, Jen Hom, Shea Vela-Vick, and Mike Wong.

## Daniel Stern : Scalar curvature and circle-valued harmonic maps

- Geometry and Topology ( 242 Views )We introduce a new tool for relating the scalar curvature of a Riemannian manifold to its global geometry and topology, based on the study of level sets of harmonic functions and harmonic maps to the circle. We will explain how these ideas lead to simple new proofs and improvements upon some well-known results in three-manifold geometry and general relativity, previously studied primarily via minimal surface and Dirac operator methods.

## Nathan Dowlin : A spectral sequence from Khovanov homology to knot Floer homology

- Geometry and Topology ( 238 Views )Khovanov homology and knot Floer homology are two knot invariants which are defined using very different techniques, with Khovanov homology having its roots in representation theory and knot Floer homology in symplectic geometry. However, they seem to contain a lot of the same topological data about knots. Rasmussen conjectured that this similarity stems from a spectral sequence from Khovanov homology to knot Floer homology. In this talk I will give a construction of this spectral sequence. The construction utilizes a recently defined knot homology theory HFK_2 which provides a framework in which the two theories can be related.

## Isaac Sundberg : The Khovanov homology of slice disks

- Geometry and Topology ( 236 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.

## Alex Waldron : Yang-Mills flow on special holonomy manifolds

- Geometry and Topology ( 234 Views )I will describe an upcoming paper with Goncalo Oliveira investigating the properties of Yang-Mills flow on base manifolds with restricted holonomy, generalizing known results from the 4-d and Kahler cases. We show that finite-time blowup is governed by the F^7 component of the curvature in the G_2 and Spin(7) cases, and by the appropriate curvature component in the remaining cases on Berger's list. Assuming that this component remains bounded along the flow, we show that the infinite-time bubbling set is calibrated by the defining (n-4)-form.

## Yanir Rubinstein : Einstein metrics on Kahler manifolds

- Geometry and Topology ( 232 Views )The Uniformization Theorem implies that any compact Riemann surface has a constant curvature metric. Kahler-Einstein (KE) metrics are a natural generalization of such metrics, and the search for them has a long and rich history, going back to Schouten, Kahler (30's), Calabi (50's), Aubin, Yau (70's) and Tian (90's), among others. Yet, despite much progress, a complete picture is available only in complex dimension 2. In contrast to such smooth KE metrics, in the mid 90's Tian conjectured the existence of KE metrics with conical singularities along a divisor (i.e., for which the manifold is `bent' at some angle along a complex hypersurface), motivated by applications to algebraic geometry and Calabi-Yau manifolds. More recently, Donaldson suggested a program for constructing smooth KE metrics of positive curvature out of such singular ones, and put forward several influential conjectures. In this talk I will try to give an introduction to Kahler-Einstein geometry and briefly describe some recent work mostly joint with R. Mazzeo that resolves some of these conjectures. One key ingredient is a new C^{2,\alpha} a priori estimate and continuity method for the complex Monge-Ampere equation. It follows that many algebraic varieties that may not admit smooth KE metrics (e.g., Fano or minimal varieties) nevertheless admit KE metrics bent along a simple normal crossing divisor.

## Jimmy Petean : On the Yamabe invariant of Riemannian products

- Geometry and Topology ( 219 Views )The Yamabe invariant of a closed manifold appears naturally when studying the total scalar curvature functional on the space of Riemannian metrics on the manifold. Computations are difficult, in particular in the positive case (when the manifold admits metrics of positive scalar curvarture, and there is no unicity of metrics of constant scalar curvature on a conformal class). In this talk I will review a little of what is known about the computation of the invariant and discuss some recent joint work with K. Akutagawa and L. Florit on the Yamabe constants of Riemannian products.

## Richard Hain : The Lie Algebra of the Mapping Class Group, Part 2

- Geometry and Topology ( 218 Views )In this talk I will review the construction of the Lie algebra associated to the mapping class group of a (possibly decorated) surface and explain how this generalizes the Lie algebra associated to the pure braid group. I will also explain the analogue of the KZ-equation in the mapping class group case. In the second talk I will discuss filtrations of this Lie algebra associated to curve systems on the surface and their relation to handlebody groups.

## Curtis Porter : Straightening out degeneracy in CR geometry: When can it be done?

- Geometry and Topology ( 216 Views )CR geometry studies boundaries of domains in C^n and their generalizations. A central role is played by the Levi form L of a CR manifold M, which measures the failure of the CR bundle to be integrable, so that when L has a nontrivial kernel of constant rank, M is foliated by complex manifolds. If the local transverse structure to this foliation still determines a CR manifold N, then we say M is CR-straightenable, and the Tanaka-Chern-Moser classification of CR hypersurfaces with nondegenerate Levi form can be applied to N. It remains to classify those M for which L is degenerate and no such straightening exists. This was accomplished in dimension 5 by Ebenfelt, Isaev-Zaitzev, and Medori-Spiro. I will discuss their results as well as my recent progress on the problem in dimension 7 (http://arxiv.org/abs/1511.04019).

## Luca Di Cerbo : Seiberg-Witten equations on manifolds with cusps and geometric applications.

- Geometry and Topology ( 214 Views )In this talk, I will discuss the Seiberg-Witten equations on finite volume Riemannian manifolds which are diffeomorphic to the product of two hyperbolic Riemann surfaces of finite topological type. Finally, using a Seiberg-Witten scalar curvature estimate I will present several results concerning the Riemannian geometry of these spaces.

## Shubham Dwivedi : Geometric flows of $G_2$ structures

- Geometry and Topology ( 213 Views )We will start by discussing a flow of isometric $G_2$ structures. We consider the negative gradient flow of the energy functional restricted to the class of $G_2$ structures inducing a given Riemannian metric. We will discuss various analytic aspects of the flow including global and local derivative estimates, a compactness theorem and a monotonicity formula for the solutions. After defining an entropy functional we will prove that low entropy initial data lead to solutions that exist for all time and converge smoothly to a $G_2$ structure with divergence free torsion. We will also discuss finite time singularities and the singular set of the solutions. Finally, we will discuss the isometric flow "coupled” with the Ricci flow of the underlying metric, which again is a flow of $G_2$ structures, and discuss some of its properties. This is a based on two separate joint works with Panagiotis Gianniotis (University of Athens) and Spiro Karigiannis (University of Waterloo).

## Gonçalo Oliveira : Gauge theory on Aloff-Wallach spaces

- Geometry and Topology ( 204 Views )I will describe joint work with Gavin Ball where we classify certain G2-Instantons on Aloff-Wallach spaces. This classification can be used to test ideas and explicitly observe various interesting phenomena. For instance, we can: (1) Vary the underlying structure and find out what happens to the G2-instantons along the way; (2) Distinguish certain G2-structures (called nearly parallel) using G2-Instantons; (3) Find G2-Instantons, with respect to these structures, which are not absolute minima of the Yang-Mills functional.

## Richard Hain : Hodge theory and the Goldman-Turaev Lie bialgebra

- Geometry and Topology ( 203 Views )In the 1980s, Bill Goldman used intersection theory to define
a Lie algebra structure on the free **Z** module *L(X)* generated by the
closed geodesics on a hyperbolic surface *X*. This bracket is related to
a formula for the Poisson bracket of functions on the variety of flat
*G*-bundles over *X*. In related work (1970s and 1990s), Vladimir Turaev
(with contributions by Kawazumi and Kuno in the 2000s) constructed a
cobracket on *L(X)* that depends on the choice of a framing. In this
talk, I will review the definition of the Goldman-Turaev Lie bialgebra
of a framed surface and discuss its relevance to questions in other
areas of mathematics. I'll discuss how Hodge theory can be applied to
these questions. I may also discuss some related questions, such as
the classification of mapping class group orbits of framings of a
punctured surface.

## Yu Pan : Exact Lagrangian cobordisms and the augmentation category

- Geometry and Topology ( 196 Views )To a Legendrian knot, one can associate an $A_{\infty}$ category, the augmentation category. An exact Lagrangian cobordism between two Legendrian knots gives a functor of the augmentation categories of the two knots. We study the functor and establish a long exact sequence relating the corresponding cohomology of morphisms of the two ends. As applications, we prove that the functor between augmentation categories is injective on the level of equivalence classes of objects and find new obstructions to the existence of exact Lagrangian cobordisms in terms of linearized contact homology and ruling polynomials.

## John Berman : Measuring Ramification with Topological Hochschild Homology

- Geometry and Topology ( 196 Views )Topological Hochschild homology (THH) has recently been popular as an approximation to algebraic K-theory, but it is also a measure of ramification in the sense of number theory. I will survey the interaction between THH and number theory, along with some surprising connections to classical algebraic topology. This will culminate in a new computation of THH of any ring of integers R, suggesting the philosophy: Spec(R) -> Spec(Z) is one point away from being etale.

## Jim Isenberg : Construcing solutions of the Einstein constraint equations

- Geometry and Topology ( 195 Views )The first step in finding a spacetime solution to the Einstein gravitational field equations via the inital value formulation is to construct initial data which satisfy the Einstein constraint equations. There are three ways of carrying out this construction which have been found to be useful: the conformal and conformal thin sandwich methods, the gluing techniques, and the quasi-spherical approaches. We describe each of these, we discuss their advantages and disadvantages, we outline some of their recent successful applications, and we present some of the outstanding questions remaining to be solved from each of these perspectives.

## Jer-Chin Chuang : Subdivisions and Transgressive Chains

- Geometry and Topology ( 192 Views )Combinatorial transgressions are secondary invariants of a space admitting triangulations. They arise from subdivisions and are analogous to transgressive forms such as those in Chern-Weil theory. In this talk, I characterize transgressions that are path-independent of subdivision sequence. The result is obtained by using a cohomology on posets that is shown to be equivalent to higher derived functors of the inverse (or projective) limit over the opposite poset.

## Lee Deville : Synchrony vs. Asynchrony due to Large Deviations in Stochastic Neuronal Networks

- Geometry and Topology ( 191 Views )We consider idealized stochastic models for a network of pulse-coupled oscillators where there is randomness both in input and in network architecture. We describe the various types of dynamics which arise in this system, analyze scalings which arise in the infinite-network limit, and study the various "finite-size" effects as perturbations of these limits. Most notably, the networks we consider can simultaneously support both synchronous and asynchronous modes of behavior and will switch stochastically between these modes due to "rare events". We also relate the analysis of certain scaling limits of this network to classical graph-theoretical results involving the size of components in the Erdos-Renyi random graph. This work is joint with Charles Peskin and Joel Spencer.