4/26/2021 – Wern Yeong: Survey of techniques and progress toward the Green-Griffiths-Lang and Kobayashi conjectures on hyperbolicity
The Green-Griffiths-Lang and Kobayashi conjectures make connections between concepts in geometry and number theory, e.g. curvature, positivity, entire curves and rational points. These conjectures can be thought of as generalizations of the behavior of Riemann surfaces depending on their geometric genus, which is the starting point of this talk. Then we discuss some techniques and progress toward these conjectures, and their algebraic analogs. Aimed for a general math audience.
4/19/2021 – Pavel Mnev: Topological quantum mechanics, Stasheff’s associahedra and homotopy transfer of algebraic structures
I will explain the setup of topological quantum mechanics and how its natural extension to spacetimes being metric trees leads to the construction of a family of differential forms In on the moduli space of metric trees (a.k.a. Stasheff’s associahedron). Periods of these differential forms give the Kontsevich-Soibelman sum-over-trees formula for the A∞-algebra structure on the cohomology of a differential graded algebra (e.g. Massey operations on de Rham cohomology). Higher associativity relations for the A∞-infinity structure correspond in this construction to the factorization property of the differential forms In on the compactification strata of the moduli space.
4/12/2021 – Juanita Pinzon Caicedo: Instantons and Knot Concordance
Knot concordance can be regarded as the study of knots as boundaries of surfaces embedded in spaces of dimension 4. Specifically, two knots K0 and K1 are said to be smoothly concordant if there is a smooth embedding of the annulus S1 × [0, 1] into the “cylinder” S3 × [0, 1] that restricts to the given knots at each end. Smooth concordance is an equivalence relation, and the set C of smooth concordance classes of knots is an abelian group with connected sum as the binary operation. The algebraic structure of C, the concordance class of the unknot, and the set of knots that are topologically slice but not smoothly slice are much studied objects in low-dimensional topology. Gauge theoretical results on the nonexistence of certain definite smooth 4-manifolds can be used to better understand these objects. In particular, the study of anti-self dual connections on 4-manifolds can be used to shown that (1) the group of topologically slice knots up to smooth concordance contains a subgroup isomorphic to ℤ∞, and (2) satellite operations that are similar to cables are not homomorphisms on C.
4/5/2021 – Nikolai Konovalov: Quillen F-isomorphism Theorem
Group cohomology is notoriously hard and yet interesting invariant. In 1971, D. Quillen suggested a way to approximate (up to nilpotents) the mod p cohomology ring of a group in terms of cohomology rings of its elementary p-subgroup. In my talk, I am going to briefly discuss the proof of his theorem, possible corollaries, and if time permits, one interesting generalization in terms of the Steenrod operations.
3/29/2021 – Gabor Szekelyhidi: Gromov-Hausdorff limits of Kahler manifolds
I will give an overview of the Cheeger-Colding theory of Gromov-Hausdorff limits of Riemannian manifolds with Ricci curvature bounds, and the more recent results of Donaldson-Sun on the additional structure one obtains in the Kahler case.
2/22/2021 – Ethan Reed: A Proof of Stillman’s Conjecture Using Ultraproducts
Following the work of Erman, Sam, and Snowden, we give a proof of Stillman’s conjecture. As in the work of Ananyan and Hochster, we first reduce to the “existence of small algebras”. This reduces to proving a specific manifestation of the Ananyan-Hochster Principle, which we prove using ultraproducts.
1/25/2021 – Elia Portnoy (MIT): Distortions of Embedded Knots
Is it true that any knot can be embedded in ℝ3 so that the ratio between the intrinsic distance and induced distance is not too large? Pardon first showed that the answer was surprisingly, No! Later Gromov and Guth constructed knots with arbitrarily large distortion using hyperbolic geometry of 3-manifolds. I will sketch some ideas in Gromov and Guth’s construction.
1/13/2021 – Richard Birkett: Berkovich Space
Berkovich spaces have proven themselves a great tool in algebraic geometry and dynamics in recent years. This talk serves as a crash course or introduction. I will begin with the opaque algebraic definition and provide a perspective of how Berkovich space is actually an incredible geometric lens for seeing many varieties at once. Only basic algebra or geometry needed to follow the talk, although familiarity with divisors on a plane provide important motivation.
1/4/2021 – John Siratt: The Strength of Büchi’s Decidability Theorem
Although second order logic is not decidable in general, there are interesting theories in fragments of second order logic that are decidable. Büchi’s Decidability Theorem for the monadic second order theory of the natural numbers with the less-than-or-equal-to relation is one such decidability result. Recent work by Kolodziejczyk, Michaelewski, Pradic, and Skrzypczak, has shown that the logical strength of Buchi decidability over RCA0 is equivalent to induction over Σ2 formulas. We will present some needed background including some automata theory and reverse mathematics, then sketch a modern proof of Buchi decidability. This proof can be easily modified to establish one direction of the strength result. Finally, we will give a short overview of the authors’ proof showing the other direction.