Simon Brendle : Singularity formation in geometric flows
- Geometry and Topology ( 293 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.
Mark Stern : Geometry of stable Yang-Mills connections
- Geometry and Topology ( 156 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.
Dan Rutherford : Augmentations and immersed Lagrangian fillings
- Geometry and Topology ( 134 Views )This is joint work with Y. Pan that applies previous joint work with M. Sullivan. Let $\Lambda \subset \mathbb{R}^{3}$ be a Legendrian knot with respect to the standard contact structure. The Legendrian contact homology (LCH) DG-algebra, $\mathcal{A}(\Lambda)$, of $\Lambda$ is functorial for exact Lagrangian cobordisms in the symplectization of $\mathbb{R}^3$, i.e. a cobordism $L \subset \mathit{Symp}(\mathbb{R}^3)$ from $\Lambda_-$ to $\Lambda_+$ induces a DG-algebra map, $f_L:\mathcal{A}(\Lambda_+) \rightarrow \mathcal{A}(\Lambda_-).$ In particular, if $L$ is an exact Lagrangian filling ($\Lambda_-= \emptyset$) the induced map is an augmentation $\epsilon_L: \mathcal{A}(\Lambda_+) \rightarrow \mathbb{Z}/2.$ In this talk, I will discuss an extension of this construction to the case of immersed, exact Lagrangian cobordisms based on considering the Legendrian lift $\Sigma$ of $L$. When $L$ is an immersed, exact Lagrangian filling a choice of augmentation $\alpha$ for $\Sigma$ produces an induced augmentation $\epsilon_{(L, \alpha)}$ for $\Lambda_+$. Using the cellular formulation of LCH, we are able to show that any augmentation of $\Lambda$ may be induced by such a filling.
Thomas Mark : 3-manifolds not obtained by surgery on a knot
- Geometry and Topology ( 116 Views )A well-known theorem of Lickorish and Wallace states that any closed orientable 3-manifold can be obtained by surgery on a link in the 3-sphere. For a given 3-manifold one can ask how "simple" a link can be used to obtain it, e.g., whether a manifold satisfying certain obvious necessary conditions on its fundamental group always arises by surgery on a knot. This question turns out to be rather subtle, and progress has been limited, but in general the answer is known to be "no." Here Ill summarize some recent results including joint work with Matt Hedden, Min Hoon Kim, and Kyungbae Park that give the first examples of 3-manifolds with the homology of S^1 x S^2 and having fundamental group of weight 1 that do not arise by surgery on a knot in the 3-sphere.
Luca Di Cerbo : Finite volume complex hyperbolic surfaces and their compactifications
- Geometry and Topology ( 137 Views )In this talk, I will discuss the geometry of finite volume complex hyperbolic surfaces and their compactifications. Finally, applications at the common edge between Riemannian and complex algebraic geometry are given.
Larry Guth : Area-contracting maps between rectangles
- Geometry and Topology ( 137 Views )The k-dilation of a map measures how much the map stretches k-dimensional volumes. The 1-dilation is the usual Lipschitz constant. We consider the problem of finding the smallest k-dilation among all degree 1 maps from one rectangle to another rectangle. (These are n-dimensional rectangles.) In general the linear map is far from optimal.
Vladimir Matveev : Geodesically equivalent metrics in the large: Beltrami and Schouten problems.
- Geometry and Topology ( 143 Views )Two metrics are geodesically equivalent if they have the same (unparameterized) geodesics. During my talk I describe geodesically equivalent metrics on closed manifolds (which is an answer to Beltrami's question) and explain the proof of Lichnerowicz-Obata conjecture (which is an answer on the infinitesimal version of the Beltrami question known as Schouten problem).
Carla Cederbaum : From Newton to Einstein: a guided tour through space and time
- Geometry and Topology ( 135 Views )The cosmos and its laws have fascinated people since the ancient times. Many scientists and philosophers have tried to describe and explain what they saw in the sky. And almost all of them have used mathematics to formulate their ideas and compute predictions for the future. Today, we have made huge progress in understanding and predicting how planets, stars, and galaxies behave. But still, the mysteries of our universe are formulated and resolved in mathematical language and always with new mathematical methods and ideas. In this lecture, you will hear about two of the most famous physicists of all times, Isaac Newton (1643-1727) and Albert Einstein (1879-1955), and about their theories of the universe. You will learn about common features and central differences in their viewpoints and in the mathematics they used to formulate their theories. In passing, you will also encounter the famous mathematician Carl Friedrich Gauß (1777-1855) and his beautiful ideas about curvature.
Yu Wang : Quantitative stratification of stationary Yang-Mills and recent progress on global gauge problem
- Geometry and Topology ( 140 Views )Given a stationary Yang-Mills connection A, we are interested in studying its singular structure. In this talk we introduce a quantitative way to stratify the singular sets. Our main results include a Minkowski Volume estimate and the rectifiability of this quantitative stratification, which leads to the rectifiability of the classical stratifications S^k(A) for all integer k. To be precise, we first recall certain background preliminaries needed for this talk. After giving the statements of the main results, I will briefly describe the machinary used in the proof, and explain the new points and the major difficulty that we have faced. The main results in the talk are based on a work by myself last year. If time allows I will further discuss some open problems regarding global gauge in this field, and recent progress in those directions made jointly with Aaron Naber.
Yao Xiao : Equivariant Lagrangian Floer theory on compact toric manifolds
- Geometry and Topology ( 78 Views )We define an equivariant Lagrangian Floer theory on compact symplectic toric manifolds for the subtorus actions. We prove that the set of Lagrangian torus fibers (with weak bounding cochain data) with non-vanishing equivariant Lagrangian Floer cohomology forms a rigid analytic space. We can apply tropical geometry to locate such Lagrangian torus fibers in the moment map. We show that these Lagrangian submanifolds are nondisplaceable by equivariant Hamiltonian diffeomorphisms.
Oguz Savk : Bridging the gaps between homology planes and Mazur manifolds.
- Geometry and Topology,Uploaded Videos ( 265 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.
Michael Lipnowski : The Seiberg-Witten equations and the length spectrum of hyperbolic three-manifolds
- Geometry and Topology ( 108 Views )We exhibit examples of hyperbolic three-manifolds for which the Seiberg-Witten equations do not admit any irreducible solution. Our approach relies hyperbolic geometry in an essential way; it combines an explicit upper bound for the first eigenvalue on coexact 1-forms \lambda_1^* on rational homology spheres which admit irreducible solutions together with a version of the Selberg trace formula relating the spectrum of the Laplacian on coexact 1-forms with the volume and complex length spectrum of a hyperbolic three-manifold. Using these relationships, we also provide precise certified numerical bounds on \lambda_1^* for several hyperbolic rational homology spheres.
Michael Willis : The Khovanov homology of infinite braids
- Geometry and Topology ( 107 Views )In this talk, I will show that the limiting Khovanov chain complex of any infinite positive braid categorifies the Jones-Wenzl projector, extending Lev Rozansky's work with infinite torus braids. I will also show a similar result for the limiting Lipshitz-Sarkar-Khovanov homotopy types of the closures of such braids. Extensions to more general infinite braids will also be considered. This is joint work with Gabriel Islambouli.
Daniel Stern : Min-Max Methods for Ginzburg-Landau Functionals and Connections to Geometric Measure Theory
- Geometry and Topology ( 129 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.
Thomas Walpuski : G2instantons over twisted connected sums
- Geometry and Topology ( 167 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.
Adam Levine : Non-orientable surfaces in homology cobordisms
- Geometry and Topology ( 105 Views )We study the minimal genus problem for embeddings of closed, non-orientable surfaces in a homology cobordism between rational homology spheres, using obstructions derived from Heegaard Floer homology and from the Atiyah-Singer index theorem. For instance, we show that if a non-orientable surface embeds essentially in the product of a lens space with an interval, its genus and normal Euler number are the same as those of a stabilization of a non-orientable surface embedded in the lens space itself. This is joint work with Danny Ruberman and Saso Strle.
Bulent Tosun : Legendrian and transverse knots in cabled knot types
- Geometry and Topology ( 111 Views )In this talk we will exhibit many new phenomena in the structure of Legendrian and transverse knots by giving a complete classification of all cables of the positive torus knots. We will also provide two structural theorems to ensure when cable of a Legendrian simple knot type is also Legendrian simple. Part of the results are joint work with John Etnyre and Douglas LaFountain
Fédéric Rochon : On the uniqueness of certain families of holomorphic disks
- Geometry and Topology ( 135 Views )A Zoll metric is a metric whose geodesics are all circles of equal length. In this talk, we will first review the definition of the twistor correspondence of LeBrun and Mason for Zoll metrics on the sphere $S^{2}$. It associates to a Zoll metric on $S^{2}$ a family of holomorphic disks in $CP_{2}$ with boundary in a totally real submanifold $P\subset CP_{2}$. For a fixed $P\subset CP_{2}$, we will indicate how one can show that such a family is unique whenever it exists, implying that the twistor correspondence of LeBrun and Mason is in some sense injective. One of the key ingredients in the proof will be the blow-up and blow-down constructions in the sense of Melrose.
Marcus Khuri : On the Penrose Inequality
- Geometry and Topology ( 165 Views )The cosmic censorship conjecture roughly states that singularities in the evolution of spacetime are always hidden from the outside world by event horizons. As a test for this conjecture Penrose proposed the inequality M >= (A/16pi)^1/2, relating the total ADM mass M of a spacetime to the area A of an event horizon. For time symmetric initial data sets of Einstein's equations this inequality has been confirmed, independently by Huisken and Ilmanen (for one black hole) and by Bray (for multiple black holes). The purpose of this talk is to show how the time symmetric proofs can be generalized to apply to general initial data, assuming existence for a canonical degenerate elliptic system of equations. This is joint work with Hubert Bray.
Lev Rozansky : A categorification of the stable Witten-Reshetikhin-Turaev invariant of links in S2 x S1
- Geometry and Topology ( 163 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.
Robert Lipshitz : Planar grid diagrams and bordered Floer homology
- Geometry and Topology ( 97 Views )Heegaard Floer homology, a kind of (3+1)-dimensional field theory, associates chain complexes to 3-manifolds and chain maps to 4-manifolds with boundary. These complexes and maps are defined by counting holomorphic curves, and are hard to compute. Bordered Floer homology extends Heegaard Floer theory one dimension lower, assigning algebras to surfaces and differential modules to 3-manifolds with (parameterized) boundary. After introducing the bordered Floer framework, we will illustrate its construction in a toy case where it is explicit and combinatorial: planar grid diagrams. This is joint work with Peter Ozsvath and Dylan Thurston.
Lilian Hsiao : Colloids with tunable geometry and their effects on viscoelastic materials and suspensions
- Geometry and Topology ( 112 Views )A central challenge in soft matter and materials science is the microscopic engineering of functional materials. Incorporating anisotropy here is of general interest, for example in actin networks, clay platelets, and polymer nanocomposites where geometry, ordering, and kinetics all play important roles in determining their properties. Nevertheless, forming a general connection between microstructure and macroscopic properties is not trivial. Here, I focus on the self-assembly and mechanics of colloidal materials with an emphasis on how shape anisotropy and interaction potential can be used to guide their design. I will first discuss the relevance of the physical interactions that give rise to a general class of colloidal gels, followed by how shape anisotropy can introduce metastable gelled states. I will also show that the slowed rotational dynamics caused by surface roughness and friction can lead to enhanced shear thickening that is not seen with smooth colloids. These results collectively show that particle-level interactions provide a powerful means to design soft materials at multiple length scales.