Events

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

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

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

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

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

Gregory G. Smith, Queen's University

Sums of Squares and Projective Geometry

A multivariate real polynomial is nonnegative if its value at any real point is greater than or equal to zero.  These special polynomials play a central role in many branches of mathematics including algebraic geometry, optimization theory, and dynamical systems.  However, it is very difficult, in general, to decide whether a given polynomial is nonnegative.  In this talk, we will review some classical methods for certifying that a polynomial is nonnegative.  We will then present novel certificates in some important cases. This talk is based on joint work with Grigoriy Blekherman, Rainer Sinn, and Mauricio Velasco.

MC 5501

Agniva Dasgupta, University of Texas at Dallas

Second Moment of GL(3) L-functions

In this talk, I will discuss our recent result, joint with Wing Hong Leung and Matthew Young, on the second moment of GL(3) L-functions on the critical line. Moments of L-functions on the critical line have been studied for over a hundred years now, and still remains a very active field of research in number theory. The second moment of GL(3) L-functions has proved to be especially difficult, and only in the last couple of years have we seen some progress on this. Building on top of these works, we are able to obtain a strong upper bound for this moment. This allows us to deduce some nice corollaries including an improvement on the error term in the Rankin-Selberg problem, and on certain subconvexity bounds for GL(3) x GL(2) and GL(3) L-functions. As a byproduct of the method of proof, we also obtain an improved estimate for an average of shifted convolution sums of GL(3) Fourier coefficients. 

MC 5479

AJ Fong, University of Waterloo

Toric geometry without (most of) the geometry

Toric varieties are great because they lend themselves to combinatorial description and study. I will introduce some of the combinatorial objects involved, leaning heavily on and subsequently specialising to smooth toric surfaces as simple examples. If time permits, I will demonstrate the power of this approach by showing a result on Galois representations of toric surfaces.

MC 5403

Joey Lakerdas-Gayle, University of Waterloo

Fundamentals of Computability Theory 3

We will continue learning about priority constructions, now using the finite injury method, following Robert Soare's textbook.

MC 5403

Paul Cusson, University of Waterloo

The Kodaira embedding theorem and background material

The Kodaira embedding theorem is a crucial result in complex geometry that forms a nice bridge between differential and algebraic geometry, giving a necessary and sufficient condition for a compact complex manifold to be a smooth projective variety, that is, a complex submanifold of a complex projective space. The material and proof will follow the exposition in Griffiths & Harris's classic textbook.

MC 5479