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<h1>The topology and geometry of low-dimensional manifolds:<br>a celebration of the mathematics of Bob Gompf</h1>
<h2>University of Texas at Austin</h2>
<h2>July 12-15, 2018</h2>
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<h2>Schedule</h2>
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Talks will take place at the University of Texas at Austin. Please see below for room assignments.</p>
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<tr><th colspan="2" align=left> Thursday, July 12 - All talks in <a href="https://goo.gl/SrgFcG">ETC 2.108</a></th>
<tr><td> 9:30 </td><td> <a href="#stipsicz">András Stipsicz</a> </td></tr>
<tr><td> 11:00 </td><td> <a href="#lidman">Tye Lidman</a> </td></tr>
<tr><td> 2:00 </td><td> <a href="#miller">Allison Miller</a> </td></tr>
<tr><td> 3:45 </td><td> <a href="#li">Tian-Jun Li</a> </td></tr>
<tr><th colspan="2" align="left"><br><br>Friday, July 13 - All talks in <a href="https://goo.gl/SrgFcG">WRW 102</a></th></tr>
<tr><td> 9:30 </td><td> <a href="#ruberman">Danny Ruberman</a> </td></tr>
<tr><td> 11:00 </td><td> <a href="#baykur">0nanç Baykur</a> </td></tr>
<tr><td> 1:45 </td><td> <a href="#schwartz">Hannah Schwartz</a> </td></tr>
<tr><td> 3:00 </td><td> <a href="#yasui">Kouichi Yasui</a> </td></tr>
<tr><td> 4:15 </td><td> <a href="#mrowka">Tom Mrowka</a> </td></tr>
<tr><td> 6:30 </td><td> Banquet at <a href="https://goo.gl/maps/ZCi3P41f9ox">Clay Pit</a> </td></tr>
<tr><th colspan="2" align=left><br><br>Saturday, July 14 - All talks in <a href="https://goo.gl/SrgFcG">WRW 102</a></th></tr>
<tr><td> 9:30 </td><td> <a href="#kirby">Rob Kirby </td></tr>
<tr><td> 11:00 </td><td> <a href="#matic">Gordana Mati </td></tr>
<tr><td> 1:45 </td><td> <a href="#meier">Jeffrey Meier </td></tr>
<tr><td> 3:15 </td><td> <a href="#tosun">Bulent Tosun </td></tr>
<tr><td> 4:30 </td><td> <a href="#harvey">Shelly Harvey</a> </td></tr>
<tr><th colspan="2" align=left><br><br>Sunday, July 15 - All talks in <a href="https://goo.gl/SrgFcG">WRW 102</a></th></tr>
<tr><td> 8:50 </td><td> <a href="#eliashberg">Yasha Eliashberg</a> </td></tr>
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<h2>Titles and Abstracts</h2><br>
<a name="baykur"><b>0nanç Baykur</b>: Small symplectic and exotic 4-manifolds via positive factorizations<br><b>Abstract</b>: We will discuss new ideas and techniques for producing positive Dehn twist factorizations of surface mapping classes which yield novel constructions of interesting symplectic and smooth 4-manifolds, such as small symplectic Calabi-Yau surfaces and exotic rational surfaces, via Lefschetz fibrations and pencils.<br><br><br>
<a name="elishberg"><b>Yasha Eliashberg</b>: Simplification of singularities of mappings without pre-conditions and its applications<br><b>Abstract</b>: The h-principle type results allow to get rid of most singularities of mappings if certain homotopy conditions are met.
It turns out that one can reduce singularities to a finite geometrically understandable list without any conditions. This is a crucial step
in our joint with D. Alvarez-Gavela, D. Nadler and L. Starkston project of arborealization of Lagrangian skeleta of Weinstein manifolds.<br><br><br>
<a name="harvey"><b>Shelly Harvey</b>: A non-discrete metric on the group of topologically slice knots<br><b>Abstract</b>: Most of the 50-year history of the study of the set of smooth knot concordance classes, \(C\), has focused on its structure as an abelian group. Tim Cochran and I took a different approach, namely we studied \(C\) as a metric space (with the slice genus metric) admitting many natural geometric operators. The goal was to give evidence that the knot concordance is a fractal space. However, both of these metrics are integer valued metrics and so induce the discrete topology. Subsequently, with Mark Powell, we defined a family of real valued metrics, called the \(q\)-grope metrics, that take values in the real numbers and showed that there are sequences of knots whose \(q\)-norms get arbitrarily small for \(q>1\). However, for \(q>1\), this metric vanishes on topologically slice knots (it is really a pseudo metric). In this talk, we define a new metric (called the tower metric) based on new objects which we call positive and negative towers, using a combination of generalized handles and gropes. This is an interesting metric since it is related to the bipolar filtration, a filtration generalizing work of Gompf and Cochran. Using recent work of Cha and Kim on the non-triviality of the bipolar filtration of the group of topologically slice knots, we show that there are sequences of topologically slice knots whose \(q\)-norms get arbitrarily small but are never 0. This work is joint with Tim Cochran, Mark Powell, and Aru Ray.<br><br><br>
<a name="kirby"><b>Rob Kirby</b>: Monodromy for trisections<br><b>Abstract</b>: I will discuss joint work with Abby Thompson, first on the \(L\) invariant of a trisected 4-manifold, and then on a notion of a monodromy being a diffeomorphism of a handlebody or compression body.<br><br><br>
<a name="li"><b>Tian-Jun Li</b>: Geography of symplectic fillings in dimension 4<br><b>Abstract</b>: We introduce the Kodaira dimension of contact 3-manifolds and show that contact 3-manifolds with distinct Kodaria dimensions behave differently when it comes to the geography of various kinds of fillings. We also prove that, given any contact 3-manifold, there is a lower bound of \(2\chi+3\sigma\) for all its minimal symplectic fillings. This generalizes the similar bound of Stipsicz for Stein fillings. This talk is based on joint works with Cheuk Yu Mak, and partly with Koichi Yasui.<br><br><br>
<a name="lidman"><b>Tye Lidman</b>: Spineless four-manifolds<br><b>Abstract</b>: We construct smooth, compact four-manifolds homotopy equivalent to the two-sphere for which no homotopy equivalence is realized by a PL embedding.<br><br><br>
<a name="matic"><b>Gordana Mati</b>: Spectral order - filtering the Ozsvath-Szabo contact invariant<br><b>Abstract: </b>We define a refinement of the Ozsvath-Szabo contact invariant in Heegaard Floer homology. Using an analogue of a filtration defined on the ECH complex by Hutchings in order to capture the Latchev-Wendl algebraic torsion invariant of contact structures, we filter the Heegard Floer chain complex, and look at the order of vanishing of the contact element. This gives us an invariant that is nondecreasing under Legendrian surgery, 0 for overtwisted contact structures and infinite for the Stein fillable ones. I will talk about the filtration, some of the key properties of the invariant, and present some examples. This is joint work with Cagatay Kutluhan, Jeremy Van Horn-Morris, and Andy Wand.<br><br><br>
<a name="meier"><b>Jeffrey Meier</b>: Generalized square knots with weak property 2R<br><b>Abstract</b>: For the last forty years, potential counterexamples to the Smooth 4-Dimensional Poincaré Conjecture have been constructed, illustrated, and, subsequently, standardized. Many of these examples are geometrically simply connected, meaning they can be built without 1-handles. In this case, the 4-manifold is encoded by an \(n\)-component link with an integral Dehn surgery to the connected sum of \(n\) copies of \(S^1\times S^2\). A knot is said to have <i>Weak Property \(n\)R</i> if its inclusion as a component of such a link implies that the corresponding 4-manifold is the standard 4-sphere. In this talk, we will show that there is an infinite family of generalized square knots that have Weak Property \(n\)R. Along the way, we will place this result in the broader context of the Generalized Property R Conjectures, recover results of Gompf and Akbulut standardizing Cappell-Shaneson homotopy 4-spheres, and draw connections with the Slice-Ribbon Conjecture for fibered knots and the study of trisections of homotopy 4-spheres. This talk is based on joint work with Alex Zupan.<br><br><br>
<a name="miller"><b>Allison Miller</b>: Satellite operators and knot concordance<br><b>Abstract</b>: The classical satellite construction behaves nicely with respect to concordance, since if \(K\) and \(J\) are concordant then \(P(K)\) and \(P(J)\) are concordant for any pattern \(P\). It is therefore natural to ask about the properties of satellite-induced maps on the collection of knots modulo concordance.<br> I will briefly survey results in this area, focusing on differences between the smooth and topological categories. I will then describe Gompf and Miyazaki's construction of dualizable patterns, which induce invertible functions on the concordance group, and discuss joint work with Lisa Piccirillo which smoothly distinguishes certain dualizable pattern operators from any connected sum operator. <br><br><br>
<a name="mrowka"><b>Tom Mrowka</b>: TBA<!-- <br><b>Abstract</b>:--> <br><br><br>
<!-- <a name=""><b></b>: <br><b>Abstract</b>: <br><br><br> -->
<a name="ruberman"><b>Danny Ruberman</b>: The Lefschetz number of an involution in monopole homology<br><b>Abstract</b>: Let \(T\) be an involution on a rational homology 3-sphere \(Y\), describing it as a branched cover of the 3-sphere. Then \(T\) induces a map on the monopole Floer homology of \(Y\). I will present a formula for the Lefschetz number of this homomorphism, and give applications to problems in knot concordance and 4-dimensional topology. (Joint work with Jianfeng Lin and Nikolai Saveliev.)<br><br><br>
<a name="schwartz"><b>Hannah Schwartz</b>: Higher Order Corks and Exotic \(\mathbb{R}^4\)s<br><b>Abstract</b>: It was proved in the 1990's by Curtis-Freedman-Hsiang-Stong and Matveyev that any two homeomorphic, closed, simply-connected smooth 4-manifolds are related by removing and re-gluing a single compact contractible submanifold, called a cork. This talk will present joint work with Paul Melvin which generalizes this theorem to any finite list of such 4-manifolds. Although in the infinite case a strictly analogous theorem is not possible, we will also show that any infinite list of homeomorphic, closed, simply-connected smooth 4-manifolds can be generated by removing and re-gluing (along its end) a single submanifold homeomorphic to \(\mathbb{R}^4\).<br><br><br>
<a name="stipsicz"><b>András Stipcisz</b>: Upsilon invariants of knots<br><b>Abstract</b>: We review the extension of the definition of the Upsilon invariant of a knot in rational homology spheres
and show some simple computations and applications of this invariant for knot concordance questions.<br><br><br>
<a name="tosun"><b>Bulent Tosun</b>: Fillability of positive contact surgeries and Lagrangian disks<br><b>Abstract</b>: 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 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.<br><br><br>
<a name="yasui"><b>Kouichi Yasui</b>: Corks and exotic 4-manifolds represented by framed knots<br><b>Abstract</b>: A framed knot represents a 4-manifold by attaching a 2-handle to the 4-ball along the knot. In this talk, for any framing, we show that there exist infinitely many exotic pairs of 4-manifolds represented by framed knots. These are the simplest possible exotic 4-manifolds regarding handlebody structures, and the framing zero case gives counterexamples to the Akbulut-Kirby conjecture on knot concordance. To obtain these results, we introduce a new description of cork twists. We also show that there exist infinitely many symmetric links of Mazur type that do not yield corks, and furthermore that even if one symmetric link presentation of a contractible 4-manifold does not yield a cork, another presentation can yield a cork.<br><br><br>
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