This Week in Mathematics


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Today:

Mathematics Colloquium [url]
Cosmological singularities and the topology of Cauchy surfaces
- Eric Ling, University of Copenhagen
Time: 3:05 Room: Lov 101
Abstract/Desc: The celebrated singularity theorems of Hawking and Penrose demonstrate that the existence of singularities, in the sense of timelike or null geodesic incompleteness, is an inevitable feature of generic spacetimes modeling the big bang or black holes. Hawking's original cosmological singularity theorem relied on a curvature condition, called the strong energy condition, which is known to be violated in models with a positive cosmological constant. In this talk, I will show how Penrose's singularity theorem can be adapted to prove a singularity theorem applicable to this regime: if a four-dimensional spacetime satisfying the null energy condition contains a compact Cauchy surface that is expanding in all directions, then the spacetime is past null geodesically incomplete unless the Cauchy surface is topologically a spherical space. The proof makes use of a well-known result in geometric measure theory along with the positive resolutions of the Poincare conjecture and the virtual positive first Betti number conjecture. I will illustrate the theorem with some examples and conclude by analyzing its rigidity under null geodesic completeness.

Entries for this week: 6

Monday January 12, 2026

Mathematics Colloquium [url]
Cosmological singularities and the topology of Cauchy surfaces
- Eric Ling, University of Copenhagen
Time: 3:05 Room: Lov 101
Abstract/Desc: The celebrated singularity theorems of Hawking and Penrose demonstrate that the existence of singularities, in the sense of timelike or null geodesic incompleteness, is an inevitable feature of generic spacetimes modeling the big bang or black holes. Hawking's original cosmological singularity theorem relied on a curvature condition, called the strong energy condition, which is known to be violated in models with a positive cosmological constant. In this talk, I will show how Penrose's singularity theorem can be adapted to prove a singularity theorem applicable to this regime: if a four-dimensional spacetime satisfying the null energy condition contains a compact Cauchy surface that is expanding in all directions, then the spacetime is past null geodesically incomplete unless the Cauchy surface is topologically a spherical space. The proof makes use of a well-known result in geometric measure theory along with the positive resolutions of the Poincare conjecture and the virtual positive first Betti number conjecture. I will illustrate the theorem with some examples and conclude by analyzing its rigidity under null geodesic completeness.

Tuesday January 13, 2026

Topology seminar
Multiplicative comparison of K-Theories and homology
- Ettore Aldrovandi, FSU
Time: 3:05PM Room: LOV 232
Abstract/Desc: K-Theory is a machine taking certain types of categories as input and producing spectra (or spaces) as its output. There are two notable versions, Segal and Waldhausen’s K-theory, depending on the specific kind of input, with a precise comparison between the two, due to A.-M. Bohmann and A. Osorno. We fully extend Bohmann and Osorno’s comparison to include a fourth object, spectra in chain complexes. This is joint work with Brandon Doherty and Arash Karimi.

Wednesday January 14, 2026

Biomathematics Journal Club
Network Science and the Effects of Music Preference on Functional Brain Connectivity: From Beethoven to Eminem
- Richard Bertram, FSU
Time: 5:00 Room: Dirac Library

Thursday January 15, 2026

Financial Math
Organizational Meeting
Time: 3.05 Room: LOV 231

Algebra seminar
Formal groups of elliptic curves
- Amod Agashe, FSU
Time: 3:05pm Room: LOV 232
Abstract/Desc: In this two talk series, we will start by recalling elliptic curves and give the explicit construction of the formal group associated to an elliptic curve. While the construction is explicit, it does not make it clear if the construction is a special case of something more general. We will then describe the formal group of a specific Lie group as motivation for the notion of a formal group, and show the analogy with the case of elliptic curves. After that (most likely in the second talk) we will describe the general abstract construction of the formal group associated to an algebraic group (which includes elliptic curves), and then explain how the explicit construction of the formal group of an elliptic curve described earlier is a special case of this more general construction. Finally, we will explain how a perhaps better way of looking at a formal group is via the formal group scheme associated to an elliptic curve. We will assume background in basic algebraic geometry (e.g., local rings of varieties), but nothing beyond that. The talk is mostly expository in nature, and there will be some interesting algebraic geometry during the talk.

Friday January 16, 2026

Mathematics Colloquium [url]
Prescribed Mean Curvature Surfaces in Riemannian Manifolds
- Liam Mazurowski, Lehigh University
Time: 3:05 Room: Lov 101
Abstract/Desc: Prescribed mean curvature surfaces arise as critical points of functionals involving the area of a surface and the volume it encloses. They model equilibrium interfaces in physical settings and appear naturally in general relativity, capillarity, materials science, and related areas. While stable examples can often be found via direct minimization, many geometrically interesting surfaces are inherently unstable and require different techniques to detect. In this talk, I will describe some new min-max approaches for finding prescribed mean curvature surfaces that go beyond the classical setting of smooth, compact ambient manifolds. These methods apply in situations where additional challenges arise such as the presence of constraints, singularities, or a lack of compactness. I will explain how these methods lead to new existence results for prescribed mean curvature surfaces, and discuss several open problems motivated by these developments.

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Department of Mathematics

This Week in Mathematics