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Robert Haslhofer, University of Toronto

Mean curvature flow through singularities

A family of surfaces moves by mean curvature flow if the velocity at each point is given by the mean curvature vector. Mean curvature flow first arose as a model of evolving interfaces in material science and has been extensively studied over the last 40 years. In this talk, I will give an introduction and overview for a general mathematical audience. To gain some intuition we will first consider the one-dimensional case of evolving curves. We will then discuss Huisken's classical result that the flow of convex surfaces always converges to a round point. On the other hand, if the initial surface is not convex we will see that the flow typically encounters singularities. Getting a hold of these singularities is crucial for most striking applications in geometry, topology and physics. In particular, we will see that flow through conical singularities is nonunique, but flow through neck singularities is unique. Finally, I will report on recent work with various collaborators on the classification of noncollapsed singularities in R^4.

MC 5501

John Yin, Ohio State University

A Chebotarev Density Theorem over Local Fields

I will present an analog of the Chebotarev Density Theorem which works over local fields. As an application, I will use it to prove a conjecture of Bhargava, Cremona, Fisher, and Gajović. This is joint work with Asvin G and Yifan Wei.

MC 5479

Brent Nelson, Michigan State University

Uniqueness of almost periodic states on hyperfinite factors

Murray and von Neumann initiated the study of "rings of operators" in the 1930's. These rings, now known as von Neumann algebras, are unital *-algebras of operators acting on a Hilbert space that are closed under the topology of pointwise convergence. Elementary examples include square complex matrices and essentially bounded measurable functions, but the smallest honest examples come from infinite tensor products of matrix algebras. These latter examples are factors—they have trivial center—and are hyperfinite—they contain a dense union of finite dimensional subalgebras. Highly celebrated work of Alain Connes from 1976 and Uffe Haagerup from 1987 showed that these infinite tensor products are in fact the unique hyperfinite factors. Haagerup eventually provided several proofs of this uniqueness, and one from 1989 included as a corollary a uniqueness result for so-called periodic states. This result only holds for some infinite tensor products of matrix algebras and is known to fail for certain other examples, but in recent joint work with Mike Hartglass we show that it can be extended to the remaining examples when periodicity is generalized to almost periodicity. In this talk, I will discuss these results beginning with an introduction to von Neumann algebras that assumes no prior knowledge of the field.

MC 5501

Spiro Karigiannis

Organizational Meeting

We will meet to plan out the Differential Geometry Working Seminar for the Fall 2024 term.

MC 5479

Andy Zucker

Topological groups with tractable minimal dynamics

For Polish groups, there are several interesting dividing lines in how complicated their minimal flows can be. While metrizability of the universal minimal flow is the most obvious, a theorem of Ben Yaacov, Melleray, and Tsankov suggests the broader class of Polish groups whose universal minimal flows have a comeager orbit. In joint work with Gianluca Basso, we find natural extensions of these classes to general topological groups, obtaining the classes of topological groups with ``concrete minimal dynamics'' or ``tractable minimal dynamics,'' respectively. Both classes admit a wide variety of non-trivial characterizations. In particular, the class of groups with tractable minimal dynamics is the largest class of topological groups admitting any form of KPT correspondence, allowing us to show that this class is absolute between models of set theory.

MC 5479

Mathias Sonnleitner, University of Passau

Covering completely symmetric convex bodies

A completely symmetric convex body is invariant under reflections or permutations of coordinates. We can bound its metric entropy numbers and consequently its mean width using sparse approximation. We provide an extension to quasi-convex bodies and present an application to unit balls of Lorentz spaces, where we can provide a complete picture of the rich behavior of entropy numbers. These spaces are compatible with sparse approximation and arise from interpolation of Lebesgue sequence spaces, for which a similar result is by now classical. Based on joint work with J. Prochno and J. Vybiral.

MC5403

Scott Wilson & Joana Cirici

Higher-homotopical BV-structures on the differential forms of symplectic and complex manifolds.

In 1985 Koszul showed that the differential forms of a symplectic manifold have an additional second order operator; part of what is now called a differential BV-algebra. Subsequent work by Getzler, Barannikov-Kontsevich, and Manin describe this structure as a (genus zero) cohomological field theory on the de Rham cohomology, i.e. an action of the compactified moduli space of (genus zero) Riemann surfaces with marked points. Such structures, also known as (formal) Frobenius manifolds, or hypercommutative algebras, have numerous connections with the A-model and mirror symmetry.

In this talk I'll explain a natural generalization of this to (almost) symplectic and complex manifolds using a higher-homotopical notion of BV-algebras. This relies on generalizations of the Kahler identities to these cases. I'll explain the setup, establish the existence of the higher-homotopy BV-structure, and give some explicit examples of almost symplectic and complex manifolds where these higher operations on cohomology are non-zero. Some examples suggest a relationship with ABC-Massey products, defined for complex manifolds.

MC 5417

Faisal Romshoo

The Ebin Slice Theorem

The Ebin Slice Theorem shows the existence of a "slice" for the action of the group of diffeomorphisms $\textrm{Diff}(M)$ on the space of Riemannian metrics $\mathcal{R}(M)$ for a closed smooth manifold $M$. We will see a proof of the existence of a slice in the finite-dimensional case and if time permits, we will go through the generalization of the proof to the infinite-dimensional setting.

MC 5417

Jashan Bal

Non-Archimedean Polish Groups

We consider the class of Polish non-Archimedean groups, those groups admitting a base at the identity of clopen subgroups. We give a complete characterization of these groups as those groups isomorphic to automorphism groups of countable, first-order structures. Time permitting, we will also discuss van Dantzig's theorem. References include Section 2.4 of Gao's IDST along with Chapter 1 of Becker and Kechris's DST of PGA.

MC 5403

William Gollinger

The Adams Spectral Sequence: Construction of the Adams Spectral Sequence

Now that we are equipped with the context of the stable homotopy category we can perform our construction. We will introduce resolutions of spectra and show how a resolution produces a spectral sequence. Identifying the terms of the resulting spectral sequence is unapproachable without additional assumptions, and we define an Adams Resolution to satisfy some homotopically exhaustive conditions. Using these conditions we can identify our E_2 page in terms of the Ext functor, and the E_\infty page in terms of the (p-completed) homotopy groups.

MC 5417