Abstracts and Lecture Videos
Richard Canary
Title: Hitchin representations of Fuchsian groups
Abstract: The Hitchin component of representations of a closed surface group into \(\mathrm{SL}(d,R)\) is one of the primary examples of a Higher Teichmuller space (a component of the space of representations of a surface group into a Lie group which consists entirely of discrete, faithful representations). We will survey this theory and then discuss a theory of cusped Hitchin representations of geometrically finite Fuchsian groups into \(\mathrm{SL}(d,R)\). These cusped Hitchin representations arise naturally by "pinching" classical Hitchin representations. We show that cusped Hitchin representations are cusped Borel Anosov and establish counting and equidistribution results.
The long term goal of this project is to develop a metric theory of the augmented Hitchin component which generalizes the fact that augmented Teichmuller space is the metric completion of Teichmuller space with the Weil-Petersson metric. (This is joint work with Harry Bray, Nyima Kao and Giuseppe Martone and with Tengren Zhang and Andy Zimmer).
Moira Chas
Title: Tantalizing patterns created by curves on surfaces
Abstract: Consider an orientable surface \(S\) with negative Euler characteristic, a minimal set of generators of the fundamental group of \(S\), and a hyperbolic metric on \(S\). Each unbased homotopy class \(C\) of closed oriented curves on \(S\) determines three numbers: the minimal geometric self-intersection number, the geometric length, and the word length (that is, the minimal number of letters needed to express \(C\) as a cyclic reduced word in the generators and their inverses). Also, the set of free homotopy classes of closed directed curves on \(S\) (as a set) is the vector space basis of a Lie algebra discovered by Goldman. This Lie algebra is closely related to the intersection structure of curves on \(S\). These three numbers, as well as the Goldman Lie bracket of two classes, can be explicitly computed (or approximated) using a computer. We will discuss the algorithms to compute or approximate these numbers, and how these computer experiments led to counterexamples to existing conjectures, to formulate new conjectures and (sometimes) to subsequent theorems.
(These results are joint work with different collaborators; mainly Arpan Kabiraj, Steven Lalley and Rachel Zhang)
Ralph L. Cohen
Title: Floer homotopy theory, old and new
Abstract: In 1995 the speaker, Jones, and Segal introduced the notion of “Floer homotopy theory”. The proposal was to attach a (stable) homotopy type to the geometric data given in a version of Floer homology. More to the point, the question was asked, “When is the Floer homology isomorphic to the (singular) homology of a naturally occurring spectrum defined from the properties of the moduli spaces inherent in the Floer theory?” Years passed before this notion found some genuine applications to symplectic geometry and low dimensional topology. However in recent years several striking applications have been found, and the theory has been developed on a much deeper level. In this expository talk I will sketch both the interesting algebraic topology and symplectic topology involved in the theory and talk about recent applications by Lipshitz-Sarkar, Manolescu, and others.
Gregory Falkovich
Title: Mathematical Aspects of Turbulence
Abstract: I shall review two unsolved mathematical problems related to turbulence. The first one is the broken scale invariance and an anomalous scaling in direct turbulent cascades. The second one is an emerging conformal invariance in inverse turbulent cascades.
Edson de Faria
Title: Asymptotic holomorphic dynamics and renormalization
Abstract: The purpose of this talk is to present a version of the Fatou-Julia-Sullivan theorem for infinitely renormalizable, asymptotically holomorphic polynomial-like maps, as well as a topological straightening theorem in this setting. As a consequence of these results, we deduce that such maps have no wandering domains, and that their Julia sets are locally connected. The talk is based on recent joint work with Trevor Clark and Sebastian van Strien.
Ruth Lawrence-Naimark
Title: Infinity structures, BV formalism and discrete models of continuum vector calculus
Abstract: We will discuss different algebraic structures which arise when we move from the differential graded algebra of continuum differential forms to the discrete setting, considered as a tower of different scales interlinked by some form of renormalization operator.
In each case, we find that at least one of the properties: commutativity, associativity, Leibniz property or preserving homology, must be dropped and the resulting algebraic object will be a form of infinity structure which may in addition come with quantitative estimates on the deviation from the dropped property.
This is based on work with Dennis Sullivan and my students Nir Gadish, Itay Griniasty and Maor Siboni.
Camillo de Lellis
Title: Anomalous dissipation for the forced Navier-Stokes equations
Abstract: Consider smooth solutions to 3d Navier-Stokes
While the balance of the energy is
it is a tenet of the theory of fully developed turbulence that in a variety of situations the left hand side should typically be independent of \(\varepsilon\): the mechanism is not supposed to be ignited by high oscillations in the initial data, which would trigger an "immediate" dissipation through the viscosity, but it is rather thought to be an effect of the quadratic nonlinearity. It is on the other hand very challenging to produce rigorous examples. If the bounds on \(u_0^\varepsilon\) are too strong, the well-posedness theory for Euler obstructs the anomalous dissipation up until the first blow-up time of Euler. With sufficiently coarse bounds Euler can be shown to have a variety of dissipative solutions, but it is very difficult to prove the convergence of Navier-Stokes to any of them. In a recent joint work with Elia Brué we study the forced version of the equation and we can prove rigorously that, as soon as the regularity of the force drops below the known thresholds for the well-posedness of classical Euler, it is in fact possible to show anomalous dissipation.
Hee Oh
Title: Rigidity of Kleinian groups
Abstract: Discrete subgroups of \(\mathrm{PSL}(2,\mathbb{C})\) are called Kleinian groups. Mostow rigidity theorem (1968) says that Kleinian groups of finite co-volume (=lattices) do not admit any faithful discrete representation into \(\mathrm{PSL}(2,\mathbb{C})\) except for conjugations. I will present a new rigidity theorem for finitely generated Kleinian groups which are not necessarily lattices and explain how this theorem compares with Sullivan’s rigidity theorem (1981).
This talk is based on joint work with Dongryul Kim.
Manuel Rivera
Title: Bialgebras and loop spaces
Abstract: A bialgebra is a vector space equipped with a multiplication map and a comultiplication map satisfying certain compatibilities. Bialgebras are ubiquitous throughout mathematics and they appear in different flavors with the multiplication and comultiplication maps satisfying different types of compatibility equations. Some prominent examples are Hopf bialgebras, Frobenius bialgebras, Lie bialgebras, and infinitesimal bialgebras. In this talk, I will discuss the construction and significance of different bialgebra structures associated to spaces of loops in a geometric space.
The first discussion will focus on a chain level lift of the classical Hopf bialgebra structure on the homology of the based loop space of a topological space. The construction of such a lift will use a version of Adams’ cobar construction applied to a chain coalgebra model for the underlying space. The cobar construction may be used to describe coalgebraic models for the homotopy theory of (non-simply connected) spaces extending seminar work of Quillen, Sullivan, Goerss, and Mandell. One of the upshots is an explanation of the sense in which the coalgebraic structure of the chains on a space completely determines the fundamental group.
The second discussion will focus on bialgebra structures inspired by Chas and Sullivan’s string topology operations on families of free loops on a manifold. I will describe a construction which allows us to incorporate two major intersection type operations, known as the Chas-Sullivan loop product and the Goresky-Hingston loop coproduct, into a single bialgebra structure where the multiplication and comultiplication are "infinitesimally" compatible. The resulting structure turns out to have a surprising homological algebra interpretation.
Nathan Seiberg
Title: Quantum Field Theory, Separation of Scales, and Beyond
Abstract: We will review the role of Quantum Field Theory (QFT) in modern physics. We will highlight how QFT uses a reductionist perspective as a powerful quantitative tool relating phenomena at different length and energy scales. We will then discuss various examples motivated by string theory and lattice models that challenge this separation of scales and seem outside the standard framework of QFT. These lattice models include theories of fractons and other exotic systems.
Nathalie Wahl
Title: What does string topology know about the manifold it lives on?
Abstract: By classical Morse theory, the homology of the free loop space \(\mathcal{L}M\) on a Riemannian manifold \(M\) is build out of closed geodesics in \(M\). It however, as such, only depends on the homotopy type of \(M\). String topology, as introduced 20 years ago by Chas and Sullivan, can be thought of as a refinement of the homology of \(\mathcal{L}M\), that remembers the extra information of how loops can sometimes be concatenated or cut. I'll explain in my talk in what sense the original hope that string topology knows more than the homotopy type of \(M\) has finally become a reality, and indicate how it can teach us something about an older invariant, namely Reidemeister torsion.
Amie Wilkinson
Title: Dynamical asymmetry is \(\mathcal{C}^1\)-typical