## 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.

## Brian Krummel : Higher codimension relative isoperimetric inequality outside a convex set

- Geometry and Topology ( 223 Views )We consider an isoperimetric inequality for area minimizing submanifolds $R$ lying outside a convex body $K$ in $\mathbb{R}^{n+1}$. Here $R$ is an $(m+1)$-dimensional submanifold whose boundary consists of a submanifold $T$ in $\mathbb{R}^{n+1} \setminus K$ and a free boundary (possibly not rectifiable) along $\partial K$. An isoperimetric inequality outside a convex body was previously proven by Choe, Ghomi, and Ritore in the codimension one setting where $m = n$. We extend their result to higher codimension. A key aspect of the proof are estimates on the concentration of mass of $T$ and $R$ near $\partial K$.

## 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.

## Thomas Walpuski : G2Â?instantons over twisted connected sums

- Geometry and Topology ( 178 Views )In joint work with H. SÃ¡ Earp we introduced a method to construct G2Â?instantons over compact G2Â?manifolds 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 G2Â?instantons), 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.

## Vestislav Apostolov : Old and new trends in Bihermitian geometry

- Geometry and Topology ( 171 Views )A bihermitian structure is a Riemannian metric compatible with two distinct orthogonal complex structures. In the mathematical literature this notion appeared in 90's in the study of the curvature of conformal 4-manifolds. However, bihermitian metrics were already studied in the physics literature in the 80's, as a building bloc of what Gates, Hull and Rocek call `the target space for a (2,2) super-symmetric sigma model'. There has been a great deal of interest in bihermitian geometry more recently, motivated by its link with the notion of generalized Kaehler geometry, introduced by Gualtieri and Hitchin. In this talk I will explain some main features of 4-dimensional bihermitian manifolds, as developed in the 90's, and report on recent classification results that I obtained with M. Gualtieri and G. Dloussky.

## Bulent Tosun : Fillability of contact surgeries and Lagrangian discs

- Geometry and Topology ( 165 Views )It is well known that all contact 3-manifolds can be obtained from the standard contact structure on the 3-sphere by contact surgery on a Legendrian link. Hence, an interesting and much studied question asks what properties of a contact structure are preserved under various types of contact surgeries. The case for the negative contact surgeries is fairly well understood. In this talk, we will discuss some new results about positive contact surgeries and in particular completely characterize when contact (r) surgery is symplectically/Stein fillable for r in (0,1]. This is joint work with James Conway and John Etnyre.

## Zhou Zhang : Volume Form and Scalar Curvature for K\ahler-Ricci Flow over General Type Manifold

- Geometry and Topology ( 164 Views )It is an interesting project guided by Tian's conjecture to use K\"ahler-Ricci flow with changing cohomology class in the study of general type manifold. The locally smooth convergence leaves quite some freedom for the global geometry. Meanwhile, volume form and scalar curvature have shown different behavior in infinite and finite time cases.

## Simon Brendle : Minimal Lagrangian diffeomorphisms between domains in the hyperbolic plane

- Geometry and Topology ( 163 Views )Let $\Omega$ and $\tilde{\Omega}$ be domains in the hyperbolic plane with smooth boundary. Assume that both domains are uniformly convex, and have the same area. We show that there exists an area-preserving, orientation-preserving diffeomorphism $f: \Omega \to \tilde{\Omega}$ such that the graph of $f$ is a minimal surface in $\mathbb{H}^2 \times \mathbb{H}^2$. Moreover, we show that the set of all such diffeomorphisms is parametrized by the circle.

## Faramarz Vafaee : Floer homology and Dehn surgery

- Geometry and Topology ( 157 Views )The past thirty years have witnessed the birth of a beautiful array of approaches to the field of low dimensional topology, drawing on diverse tools from algebra, analysis, and combinatorics. One particular tool that has made a dramatic impact on the field is the Heegaard Floer theory of Ozsvath and Szabo. Defined 17 years ago, this theory has produced an encompassing package of invariants, which have significantly impacted the study of many areas of low dimensional topology, including Dehn surgery. In this talk, we will focus on two questions: a) which 3-manifolds do arise by Dehn surgery along a knot in the 3-sphere? b) what are all ways to obtain a fixed 3-manifold by Dehn surgery along a knot in the 3-sphere?

## Chindu Mohanakumar : Coherent orientations of DGA maps associated to exact Lagrangian cobordisms

- Geometry and Topology ( 152 Views )We discuss the DGA map induced by an exact Lagrangian cobordism, and an analytic strategy to lift the map to integer coefficients, introduced by Fukaya, Oh, Ohta and Ono and further adapted by Ekholm, Etnyre, and Sullivan and Karlsson respectively. We then explain how this strategy can be applied to find a concrete combinatorial formula for a mini-dipped pinch move, thereby completely determining the integral DGA maps for all decomposable, orientable Lagrangian cobordisms. If time permits, we will show how to obtain this formula in a model case. We will also go into future potential work, including applications to Heegaard Floer Homology and nonorientable cobordisms.

## Dan Rutherford : Generating families and invariants of Legendrian knots

- Geometry and Topology ( 151 Views )Legendrian knots in standard contact R3 have in addition to their topological knot type two classical invariants known as the Thurston-Bennequin and rotation numbers. Over the past decade several invariants have been developed which are capable of distinguishing between knots with identical classical invariants. The purpose of this talk is to describe interesting relationships between some of these new invariants. Major players in this talk are the Chekanov-Eliashberg DGA (Legendrian contact homology) and related objects, as well as combinatorial structures on front diagrams and homological invariants arising from the theory of generating families (due to Chekanov-Pushkar, Fuchs, and Traynor). The main new result (joint with Fuchs) is that, when a Legendrian knot is defined by a generating family, homology groups obtained by linearizing the Chekanov-Eliashberg DGA are isomorphic to the homology of a pair of spaces associated with the generating family.

## Ahmad Issa : Embedding Seifert fibered spaces in the 4-sphere

- Geometry and Topology ( 141 Views )Which 3-manifolds smoothly embed in the 4-sphere? This seemingly simple question turns out to be rather subtle. Using Donaldson's theorem, we derive strong restrictions to embedding a Seifert fibered space over an orientable base surface, which in particular gives a complete classification when e > k/2, where k is the number of exceptional fibers and e is the normalized central weight. Our results point towards an interesting conjecture which I'll discuss. This is joint work with Duncan McCoy.

## Lan-Hsuan Huang : Constant mean curvature foliations for isolated systems in general relativity

- Geometry and Topology ( 138 Views )We will discuss the existence and uniqueness of the foliation by stable spheres with constant mean curvature for asymptotically flat manifolds satisfying the Regge-Teitelboim condition at infinity. This work generalizes the earlier results of Huisken/Yau, Ye, and Metzger. We will also discuss the concept of the center of mass in general relativity.

## Fernando Marques : The space of positive scalar curvature metrics on the three-sphere

- Geometry and Topology ( 135 Views )In this talk we will discuss a proof of the path-connectedness of the space of positive scalar curvature metrics on the three-sphere. The proof uses the Ricci flow with surgery and the connected sum construction of Gromov and Lawson. The work of Perelman on Hamilton's Ricci flow is fundamental. If time permits we will also discuss an application to general relativity.

## Goncalo Oliveira : Monopoles in Higher Dimensions

- Geometry and Topology ( 130 Views )The Monopole (Bogomolnyi) equations are Geometric PDEs in 3 dimensions. In this talk I shall introduce a generalization of the monopole equations to both Calabi Yau and G_2 manifolds. I will motivate the possible relations of conjectural enumerative theories arising from "counting" monopoles and calibrated cycles of codimension 3. Then, I plan to state the existence of solutions and sketch how these examples are constructed.

## Eylem Zeliha Yildiz : Braids in planar open books and fillable surgeries.

- Geometry and Topology ( 129 Views )We'll give a useful description of braids in $\underset{n}{\#}(S^1\times S^2)$ using surgery diagrams, which will allow us to address families of knots in lens spaces that admit fillable positive contact surgery. We also demonstrate that smooth $16$ surgery to the knot $P(-2,3,7)$ bounds a rational ball, which admits a Stein handlebody. This answers a question left open by Thomas Mark and BÃ¼lent Tosun.

## Jeremy Van Horn-Morris : Fiber genus and the topology of symplectic fillings

- Geometry and Topology ( 127 Views )Work of Donaldson, Giroux, and many others shows how to associate a singular surface fibration to a symplectic 4-manifold, either closed or with boundary, as well as to a contact 3-manifold. These are Lefschetz pencils, fibrations and open books, resp. It was asked by Stipsicz, Ozbagci, Korkmaz and others, whether the genus (or genus and self intersection) of the fiber of these fibrations gave an a priori bound on the topological complexity of the symplectic manifold. This question is equivalent to asking for a bound on the length of a factorization of a mapping class element of the fiber into right handed Dehn twists. We will discuss some of the known conditions which can produce such a bound, as well as present examples where such a bound does not exist. This is joint work with I. Baykur.

## Ralph Howard : Tangent cones and regularity of real hypersurfaces

- Geometry and Topology ( 125 Views )We characterize $C^1$ embedded hypersurfaces of $R^n$ as the only locally closed sets with continuously varying flat tangent cones whose measure-theoretic-multiplicity is at most $m < 3/2$. It follows any (topological) hypersurface which has flat tangent cones and is supported everywhere by balls of uniform radius is $C^1$. In the real analytic case the same conclusion holds under the weakened hypothesis that each tangent cone be a hypersurface. In particular, any convex real analytic hypersurface $X$ of $R^n$ is $C^1$. Furthermore, if $X$ is real algebraic, strictly convex, and unbounded then its projective closure is a $C^1$ hypersurface as well, which shows that $X$ is the graph of a function defined over an entire hyperplane. This is joint work with Mohammad Ghomi.

## Richard Hain : On a problem of Eliashberg

- Geometry and Topology ( 123 Views )Suppose that (d_1, ..., d_n) is an n-tuple of integers satisfying sum_j d_j = 0. Eliashberg posed the problem of computing the class of the locus in the moduli space of n-pointed, genus g curves [C;x_1,...,x_n] where sum d_j x_j = 0 in the jacobian of C. In this talk I will give the solution and sketch the proof, which uses known facts about the structure of mapping class groups.

## Steven Sivek : A contact invariant in sutured monopole homology

- Geometry and Topology ( 123 Views )Kronheimer and Mrowka recently used monopole Floer homology to define an invariant of sutured manifolds, following work of JuhÃ¡sz in Heegaard Floer homology. Contact 3-manifolds with boundary are natural examples of such manifolds. In this talk, I will construct an invariant of a contact structure as an element of the associated sutured monopole homology group. I will discuss several interesting properties of this invariant, including gluing maps which are analogous to the Heegaard Floer sutured gluing maps of Honda, Kazez, and Matić and applications to Legendrian knots. This is joint work with John Baldwin.

## Jonathan Hanselman : Bordered Heegaard Floer homology and graph manifolds

- Geometry and Topology ( 119 Views )Heegaard Floer homology is a powerful 3-manifold invariant developed by Oszvath and Szabo. Bordered Heegaard Floer homology is an extension of the Heegaard Floer theory to 3-manifolds with boundary, which lets us compute the "hat" version of Heegaard Floer for complicated manifolds by cutting them into simpler pieces. Graph manifolds are an important class of 3-manifolds which decompose in a particularly nice way; all the components of their JSJ decomposition are Seifert fibered. The majority of the talk will be devoted to introducing the terms above, starting with a brief overview of Heegaard Floer homology. At the end we see how to use bordered Heegaard Floer to compute HF-hat for any graph manifold.

## Natasa Sesum : On the extension of the mean curvature flow and the Ricci flow

- Geometry and Topology ( 116 Views )In the talk we will discuss curvature conditions under which we can guarantee the existence of a smooth solution to the Ricci flow and the mean curvature flow equation. These are improvements of Hamilton's and Husiken's results on extending the Ricci flow and the mean curvature flow, under conditions that the norm of Riemannian curvature and the norm of the second fundamental form are uniformly bounded along the flow in finite time, respectively.

## Daniel Scofield : Patterns in Khovanov link and chromatic graph homology

- Geometry and Topology ( 108 Views )Khovanov homology of a link and chromatic graph homology are known to be isomorphic in a range of homological gradings that depend on the girth of a graph. In this talk, we discuss patterns shared by these two homology theories. In particular, we improve the bounds for the homological span of chromatic homology by Helme-Guizon, Przytycki and Rong. An explicit formula for the rank of the third chromatic homology group on the main diagonal is given and used to compute the corresponding Khovanov homology group and the fourth coefficient of the Jones polynomial for links with certain diagrams.

## Kai Xu : pi_2-systolic inequalities for 3-manifolds with positive scalar curvature

- Geometry and Topology ( 100 Views )We discuss the following recent result of the speaker. Suppose a closed 3-manifold M has scalar curvature at least 1, and has nontrivial second homotopy group, and is not covered by the cylinder (S^2)*R. Then the pi_2-systole of M (i.e. the minimal area in the second homotopy group) is bounded by a constant that is approximately 5.44pi. If we include quotients of cylinder into consideration, then the best upper bound is weakened to 8_pi. This shows a topological gap in the pi_2-systolic inequality. We will discuss the ideas behind this theorem, as well as the proof using Huisken and Ilmanenâ??s weak inverse mean curvature flow.

## Mohammed Abouzaid : Bordism of derived orbifolds

- Geometry and Topology ( 0 Views )Among the first significant results of algebraic topology is the computation, by Thom, Milnor, Novikov, and Wall among others, of the bordism groups of stably complex and oriented manifolds. After reviewing these results, I will discuss the notion of derived orbifolds, and briefly indicate how the bordism groups of these objects appear as universal recipients of invariants arising in Gromov-Witten theory and symplectic topology. Finally, I will state what is known about them, as well as some conjectures about the structure of these groups.

## Laura Wakelin : Finding characterising slopes for all knots

- Geometry and Topology ( 0 Views )A slope p/q is characterising for a knot K if the oriented homeomorphism type of the 3-manifold obtained by performing Dehn surgery of slope p/q on K uniquely determines the knot K. For any knot K, there exists a bound C(K) such that any slope p/q with |q|â?¥C(K) is characterising for K. This bound has previously been constructed for certain classes of knots, including torus knots, hyperbolic knots and composite knots. In this talk, I will give an overview of joint work with Patricia Sorya in which we complete this realisation problem for all remaining knots.