11/15/2021 – Nikolai Konovalov: Polynomial functors and Steenrod algebra
The Steenrod algebra is an useful and classical tool in algebraic topology which appears naturally as the algebra of cohomology operations. However, there is a completely topology-free way to construct it. Namely, one could consider the category of polynomial functors from the category of vector spaces over 𝔽p to itself. Then this category embeds fully faithfully into the category of modules over the Steenrod algebra. In this result and also discuss the image of the embedding.
11/9/2021 – Eric Riedl: Geometric Manin’s Conjecture in Fano Threefolds
Manin’s Conjecture predicts the number of rational points on a variety with bounded height. Like many conjectures in Number Theory, it appears far out of reach to prove in general. However, we can obtain evidence for this conjecture by considering the analogous version for varieties defined over the function field of a curve, namely, Geometric Manin’s Conjecture. In this talk, we describe why Geometric Manin’s Conjecture is the analogue of the number theory conjecture, then we talk about some results toward proving Geometric Manin’s Conjecture for Fano Threefolds.
10/11/2021 – Nick Salter: Life After Galois
The famous Abel-Ruffini theorem asserts that there is no formula for expressing the roots of a general fifth-degree polynomial using only radicals. Rather than being the definitive end to a story, it turns out that there are many further aspects of root-finding left for contemporary (and indeed future) mathematicians to discover. In this talk, I will discuss some more modern perspectives on root-finding that showcase the roles played by topology and by dynamics.
10/6/2021 – Sarah Petersen: The RO(C2)-Homology of C2-Equivariant Eilenberg-Maclane Spaces
This talk describes work in progress computing the H𝔽2 homology of the C2-equivariant Eilenberg-Maclane spaces associated to the constant Mackey functor 𝔽2. We expand a Hopf ring argument of Ravenel-Wilson computing the mod p homology of non-equivariant Eilenberg-Maclane spaces to the RO(C2)-graded setting. An important tool that arises in this equivariant context is the twisted bar spectral sequence which is quite complicated, lacking an explicit E2 page and having arbitrarily long equivariant degree shifting differentials. We avoid working directly with these differentials and instead use a computational lemma of Behrens-Wilson along with norm and restriction maps to complete the computation.
9/20/2021 – Luan Minh Doan: Segal-Bargmann spaces, coherent state transforms, and the large-N limit problem
The Segal–Bargmann transforms, or coherent state transforms, have been interesting objects of study in mathematical physics. In the operator-theoretic settings, a coherent state transform can be described as a unitary linear map from a Hilbert space of square summable functions L2(MN, dρN) to a Hilbert space of entire functions HL2(MNℂ, dμN), where MN is a configuration space (of dimension N), MNℂ is the complexified version of MN, and ρN and μN are appropriate heat kernel measures. We will look at the case MN = ℝN and briefly discuss the case where MN is a compact Lie group or a compact symmetric group. We will also see some examples in which there are convergence phenomena of both the Hilbert spaces and the transforms when N is large.
9/13/2021 – Richard Birkett: Dynamically Stabilising Birational Surface Maps: Two Methods
Given a rational map f : X ⇢ X, we have a natural pullback operator on curves f* : Pic(X) → Pic(X)$, and understanding the sequence of actions (fn)* is fundamental to understanding the dynamics of f. It is important to stress however that pullback need not be functorial, meaning (fn)* may not be the same as (f*)n for n ≠ 1. When this does hold for every n ∈ ℕ, we call f algebraically stable. Without algebraic stability (fn)* is nearly intractable, furthermore it is difficult to construct certain natural invariant divisors, currents and measures of the dynamics of f.
The situation is better when the map f is birational. It was shown by Diller and Favre (2001) that if f : X ⇢ X is a birational map then there exists a surface Y and a birational morphism π : Y ⇢ X which lifts f to an algebraically stable g : Y ⇢ Y. I will discuss these interesting issues with pullbacks, then sketch an old and a new elementary method to lift a birational map to an algebraically stable one.
9/8/2021 – Connor Malin: Demystifying buzzwords: real homotopy theory is derived algebra
In this talk, I aim to demystify two intimidating notions: homotopy theory and derived mathematics. After describing what these things really are, I will show their versatility and apply them to studying classical algebraic topology.