Roman Aranda-Cuevas
L-invarinats in dimension four
The study of 3-manifold topology via complexes of curves has been a recurrent theme in low-dimensional topology. In 2001, Hempel used the curve complex of a surface to define an integer complexity for Heegaard splittings, which measures how “far apart” the two sides of the Heegaard splitting are by computing the distance between compressing disks on each side. In the last twenty years, versions of Hempel’s distance have been used to make progress on important problems in 3-manifold topology. In this talk we will discuss interpretations of Hempel distance for 4-dimensional smooth objects such as compact 4-manifolds and surfaces in 4-space. The talk is motivated by the following fact proven by many authors: Closed 4-manifolds can be described as loops of Morse functions on a surface, loops in the cut complex, loops in the pants complex, or as multisections.
———————————————
Yuri Berest
Quasi-flag manifolds, moment graphs and homotopy Lie groups
In this talk, we will introduce a new class of topological G-spaces generalizing the classical flag manifolds G/T of compact connected Lie groups. These spaces --- which we call the m-quasi-flag manifolds F_m(G,T) --- are topological realizations of the rings Q_m(W) of m-quasi-invariant polynomials of finite reflection groups. Many properties and geometric structures related to the classical flag manifolds can be naturally extended to quasi-flag manifolds. Some these structures (e.g., those related to equivariant K-theory and elliptic cohomology) will be briefly discussed in this talk.
The spaces F_m(G,T) can be obtained from G/T in a natural way --- by a topological `gluing' construction that we call the m-simplicial thickening. This construction can be applied to more general spaces than G/: for example, one can start with a partial flag manifold G/P or an arbitrary GKM manifold M that carries a G-action. Another, perhaps even more intriguing generalization, arises when we pass to the p-local setting (p a fixed prime) and replace compact Lie groups with p-compact groups (aka homotopy Lie groups). In this case, we obtain topological realizations of algebras of quasi-invariants of non-Coxeter (complex) reflection groups.
———————————————
Joseph Breen
On Lagrangian sliceness
A Legendrian knot is Lagrangian slice if it bounds a properly embedded Lagrangian disk in the symplectic 4-ball. One may further ask for a Lagrangian slice disk which is regular or decomposable, two conditions that are oft-studied in symplectic topology. At present, we do not have a complete characterization of Lagrangian sliceness, or an understanding of whether the additional regularity/decomposable conditions are nontrivial and/or equivalent; these questions form a symplectic analogue of the slice-ribbon conjecture. In this talk I will discuss recent work and, time permitting, joint work-in-progress with A. Zupan on progress towards characterizing types of Lagrangian sliceness.
———————————————
Rime Chatterjee
Classification of knots vs. links in contact manifolds
A knot in a contact manifold is Legendrian if it is everywhere tangent to the contact planes. The classification problem in Legendrian knot theory is lot finer than its topological counterpart. The problem gets even trickier when we start considering links. In this talk, I'll survey some of the results in this area and then discuss the classification problem for cable links of twist knots. Part of this is joint work with John Etnyre , Hyunki Min and Tom Rodewald.
———————————————
John Etnyre
Symplectic rational homology ball fillings of Seifert fibered spaces
There has been a lot of work towards studying when a small Seifert fibered space bounds a rational homology ball. Not only is this inherently interesting, but it is also related to generalizing rational blowdowns that have been effectively used to build exotic 4-manifolds. In this talk, we will discuss various constructions, highlight the differences between the smooth and symplectic category, and completely characterize, for most small Seifert fibered spaces, whether or not they bound symplectic rational homology balls. This is joint work with Burak Ozbagci and Bülent Tosun.
———————————————
Slava Krushkal
Cornered skein lasagna theory
I will discuss an extension of the skein lasagna theory to 4-manifolds with codimension 2 corners, its behavior under gluing, and applications to trisections of 4-manifolds.
(Joint work with Sarah Blackwell and Yangxiao Luo)
———————————————
Tye Lidman
Cosmetic surgeries and Chern-Simons invariants
Dehn surgery is a fundamental construction in low-dimensional topology where one removes a neighborhood of a knot from the three-sphere and reglues with a twist to obtain a new three-manifold. The Cosmetic Surgery Conjecture predicts two different surgeries on the same non-trivial knot always gives different three-manifolds. We discuss how techniques from gauge theory can help approach this problem. This is joint work with Ali Daemi and Mike Miller Eismeier.
———————————————
Maggie Miller
Branched covers of non-compact manifolds
It’s a classical theorem that every closed, oriented 3-manifold admits a 3-fold branched covering over S^3 with branch set a link. I will discuss an analogue for non-compact 3-manifolds. This is joint work with Mark Hughes and Alexandra Kjuchukova.
———————————————
Aaron Pixton
Tautological relations via counting graphs
How many ways are there to cut a compact orientable surface of genus h into pairs of pants? Equivalently, how many connected trivalent graphs are there with 2h-2 vertices? I'll start by answering this question and then explain how the answer is related to the structure of the tautological ring, a subring of the rational cohomology ring of the moduli space of curves (or of the mapping class group). After explaining how to (conjecturally) write down every tautological relation in terms of graph-counting, I'll say a little about a generalization to the universal Jacobian over the moduli space of curves.
———————————————
Laura Schaposnik
Higgs bundles, spectral data, and applications.
Higgs bundles (introduced by N. Hitchin in 1987) are pairs of
holomorphic vector bundles and holomorphic 1-forms taking values in
the endomorphisms of the bundle. The moduli space of Higgs bundles
carries a natural Hyperkahler structure, through which we can study
Lagrangian subspaces (A-branes) or holomorphic subspaces (B-branes)
with respect to each structure. Notably, these A and B-branes have
gained significant attention in string theory.
We shall begin the talk by first introducing Higgs bundles for complex
Lie groups and the associated Hitchin fibration, and recalling how to
realize Langlands duality through spectral data. We shall then look at
a natural construction of families of subspaces which give different
types of branes. Finally, by means of spectral data, we shall relate these
subspaces to the study of 3-manifolds and surface group representations.
We shall conclude with some conjectures related to Langlands duality.
———————————————
Hiro Lee Tanaka
Excellent Morse functions (joint with Lisa Traynor)
With spectral Floer's rise, a thirst for good examples; we found infinite.
———————————————
Alex Waldron
Lojasiewicz inequalities for the Dirichlet energy of maps in the critical dimension
I will discuss a general Lojasiewicz inequality for self-maps of the 2-sphere that I proved in 2023 as well as a Gaussian-weighted version that appeared this year. These lead respectively to a unique convergence theorem for harmonic map flow between 2-spheres and to a proof of continuity of the body map and no necks at finite-time singularities. The latter is the fundamental case of a conjecture of Topping.
———————————————
Joshua Wang
Free loop spaces and link homology
The Khovanov homology groups of torus knots T(n,m) are known to stabilize as m goes to infinity with n fixed. In this talk, we make the observation that when n = 2, the stable limit happens to be isomorphic to the homology of the free loop space of the 2-sphere. Our main result suggests that this is not merely a coincidence: we prove that the k-colored sl(N) homology of T(2,m) stabilizes to the homology of the free loop space of the complex Grassmannian Gr(k,N). We also relate the space of closed geodesics on the Grassmannian to the k-colored sl(N) homologies of the individual torus knots T(2,m).
———————————————
Yuanqi Wang
On the model metric near the exceptional divisors of a Borcea-Voisin Calabi-Yau 3-fold.
Let S be a K3 surface with a non-symplectic Nikulin involution $\sigma$, and E be an elliptic curve. A crepant resolution of the compact orbifold $\frac{E\times S}{\sigma }$ is a Borcea-Voisin Calabi-Yau 3-fold. This is a generalization of Kummer K3 surfaces. Near the small enough $(-2)-$curves on the Kummer K3’s, the model metric is the Eguchi-Hanson metric. However, the model metric near the exceptional divisor on the Borcea-Voisin Calabi-Yau might appear mysterious in general. In this talk, we will discuss a Kummer-type gluing of ALG Ricci flat K\”ahler three folds into the 3-orbifold. This is based on an ongoing joint project with Yu-Shen Lin and Ronan Conlon.
———————————————
Ian Zamke
The link surgery formula: past, present and future.
The link surgery formula of Manolescu and Ozsvath is a powerful tool for computing Heegaard Floer homology. Work of Manolescu, Ozsvath and Thurston proves that the surgery formula can be used to algorithmically compute the Heegaard Floer homology of all closed 3-manifolds. The description they give (using grid diagrams) is too large for practical computations, though it illustrates the robustness of the surgery formula. In this talk, we will survey recent applications and perspectives of the surgery formula. These include a combinatorial description of the link Floer homology of algebraic links and a description of L-space satellite operators on knot Floer homology. We will focus on conceptual aspects of the theory (as opposed to algebraic or technical) and also discuss several topological applications. Parts of the talk are joint with Borodzik, Liu, Chen and Zhou. Time permitting we will discuss future directions and applications of the surgery formula.
———————————————
Alex Zupan
Weak reduction in the settings of Heegaard splittings and trisections
Casson and Gordon proved that if Y admits a weakly reducible Heegaard splitting S, then either S is reducible, or Y contains an essential surface, and the concept of weak reducibility became central to the idea of thin position in dimension three. In this talk, we discuss a natural adaptation of weak reducibility to the setting of 4-manifold trisections, and we prove that if X admits a weakly reducible genus-3 trisection, then either X reduces as a connected sum of simpler 4-manifolds, or X can be obtained by surgery on a loop in S^1 x S^3. This result partially resolves an open question of Meier on the classification of genus-3 4-manifolds. This is joint work with Román Aranda.