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Matthew Harrison-Trainor, University of Illinois at Chicago

Back-and-forth games to characterize countable structures

Given two countable structures A and B of the same type, such as graphs, linear orders, or groups, two players Spoiler and Copier can play a back-and-forth games as follows. Spoiler begins by playing a tuple from A, to which Copier responds by playing a tuple of the same size from B. Spoiler then plays a tuple from B (adding it to the tuple from B already played by Copier), and Copier responds by playing a tuple from B (adding it to the tuple already played by Spoiler). They continue in this way, alternating between the two structures. Copier loses if at any point the tuples from A and B look different, e.g., if A and B are linear orders then the two tuples must be ordered in the same way. If Copier can keep copying forever, they win. A and B are isomorphic if and only if Copier has a winning strategy for this game.   Even if Copier does not have a winning strategy, they may be able to avoid losing for some (ordinal) amount of time. This gives a measure of similarity between A and B. A classical theorem of Scott says that for every structure A, there is an α such that if B is any countable structure, A is isomorphic to B if and only if Copier can avoid losing for α steps of the back-and-forth game, that is, when A is involved we only need to play the back-and-forth game for α many steps rather than the full infinite game. This gives a measure of complexity for A, called the Scott rank. I will introduce these ideas and talk about some recent results.

MC 5501

Nikolay Bogachev, University of Toronto

Commensurability classes and quasi-arithmeticity of hyperbolic reflection groups

In the first part of the talk I will give an intro to the theory of hyperbolic reflection groups initiated by Vinberg in 1967. Namely, we will discuss the old remarkable and fundamental theorems and open problems from that time. The second part will be devoted to recent results regarding commensurability classes of finite-covolume reflection groups in the hyperbolic space H^n. We will also discuss the notion of quasi-arithmeticity (introduced by Vinberg in 1967) of hyperbolic lattices, which has recently become a subject of active research. The talk is partially based on a joint paper with S. Douba and J. Raimbault.

MC 5417

Roberto Albesiano, University of Waterloo

A degeneration approach to Skoda’s division theorem

Let h1, …, hr be fixed holomorphic sections of E* ⊗ G → X, where E,G are holomorphic line bundles over a Stein manifold X. Is it always possible to write a holomorphic section g of G as a linear combination g = h1 ⊗ f1 + … + hr ⊗ fr , with f1, …, fr holomorphic sections of E? In 1972, H. Skoda proved a theorem addressing this question and giving L2 bounds on the minimal-L2-norm solution. I will sketch a new proof of a Skoda-type theorem inspired by a degeneration argument of B. Berndtsson and L. Lempert. In particular, we will see how to obtain L2 bounds on the solution (f1, …, fr) with minimal L2 norm by deforming a weight on the space of all linear combinations v1 ⊗ f1 + … + vr ⊗ fr to single out the linear combination h1 ⊗ f1 + … + hr ⊗ fr we are interested in.

MC 5417

Alex Cowan, University of Waterloo

Statistics of modular forms with small rationality fields

We present (i) a new database of weight 2 holomorphic modular forms, and (ii) a new statistical methodology for assessing probabilistic heuristics using arithmetic data. With this methodology we discover examples of non-random behavior and strange behavior in our dataset and beyond.  This is joint work with Kimball Martin.

MC 5479

Lucia Martin Merchan, University of Waterloo

About formality of compact manifolds with holonomy G2

The connection between holonomy and rational homotopy theory was discovered by Deligne, Griffiths, Morgan, and Sullivan, who proved that compact Kähler manifolds are formal. This led to the conjecture that compact manifolds with special and exceptional holonomy should also be formal. In this talk, I will discuss my recent preprint arXiv:2409.04362, where I disprove the conjecture for holonomy G2 manifolds.

MC 5479

Joey Lakerdas-Gayle, University of Waterloo

Fundamentals of Computability Theory 2

We will introduce priority arguments to construct interesting computability-theoretic structures, following Robert Soare's textbook.

MC 5403

Dicle Mutlu, McMaster University

Residually Dominated Groups

A dominated type refers to a type that is controlled by its restriction to certain sorts in the language. The concept was first introduced as stable domination for algebraically closed valued fields by Haskell, Hrushovski, and Macpherson, and was later extended to residue field domination in henselian fields of equicharacteristic zero in various studies. In the algebraically closed case, Hrushovski and Rideau-Kikuchi applied stable domination in the group setting, introducing stably dominated groups to study interpretable groups and fields in the theory. In this talk, we extend the notion of stably dominated groups to residually dominated groups in henselian fields of equicharacteristic zero, discussing how, in this setting, domination can be witnessed by a group homomorphism. This is joint work with Paul Z. Wang.

MC 5479

Jiahui Huang, University of Waterloo

Various de Rham cohomologies in algebraic geometry

De Rham's theorem states that the de Rham cohomology of a smooth manifold is isomorphic to its singular cohomology. Various generalizations of the de Rham cohomology exist in algebraic geometry. In this talk we will take a look at algebraic de Rham cohomology for singular varieties, Chiral de Rham cohomology for smooth schemes, and derived de Rham cohomology for derived stacks.

MC 5403

Candace Bethea, Duke University

The local equivariant degree and equivariant rational curve counting

I will talk about joint work with Kirsten Wickelgren on defining a global and local degree in stable equivariant homotopy theory. We construct the degree of a proper G-map between smooth G-manifolds and show a local to global property holds. This allows one to use the degree to compute topological invariants, such as the equivariant Euler characteristic and Euler number. I will discuss the construction of the equivariant degree and local degree, and I will give an application to counting orbits of rational plane cubics through 8 general points invariant under a finite group action on CP^2. This gives the first equivariantly enriched rational curve count, valued in the representation ring and Burnside ring. I will also show this equivariant enrichment recovers a Welchinger invariant in the case when Z/2 acts on CP^2 by conjugation.

MC 5417