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Rachael Alvir, Department of Pure Mathematics, University of Waterloo

"Computable Structure Theory I"

In this talk we give a basic introduction to computable structure theory. 

MC 5479

Changho Han, Department of Pure Mathematics, University of Waterloo

"Extending the torelli map to alternative compactifications of the moduli space of curves"

It is well-known that the Torelli map, that turns a smooth curve of genus g into its Jacobian (a principally polarized abelian variety of dimension g), extends to a map from the Deligne—Mumford moduli of stable curves to the moduli of semi-abelic varieties by Alexeev. Moreover, it is also known that the Torelli map does not extend over the alternative compactifications of the moduli of curves as described by the Hassett—Keel program, including the moduli of pseudostable curves (can have nodes and cusps but not elliptic tails). But it is not yet known whether the Torelli map extends over alternative compactifications of the moduli of curves described by Smyth; what about the moduli of curves of genus g with rational m-fold singularities, where m is a positive integer bounded above? As a joint work in progress with Jesse Kass and Matthew Satriano, I will describe moduli spaces of curves with m-fold singularities (with topological constraints) and describe how far the Torelli map extends over such spaces into the Alexeev compactifications.

MC 5417

AJ Fong, Department of Pure Mathematics, University of Waterloo

"Affine Schemes"

We will introduce affine schemes, the building blocks of schemes and a generalisation of affine varieties, and discuss the interesting and nontrivial geometry that can happen in them. We will briefly describe some sheaf theory in the process. This talk closely follows section I.1 of Eisenbud-Harris.

MC 5417

Benoit Charbonneau, Department of Pure Mathematics, University of Waterloo

"Deformed Hermitian-Yang-Mills equation"

The Deformed Hermitian-Yang-Mills equation has been an intense topic of study in the recent past. I will describe the equation, the concept of central charge pertinent in this story, and various conjectures and progress that has been made.

MC 5403

Jason Bell, Department of Pure Mathematics, University of Waterloo

"Sparse subsets of the reals"

We look at the first-order theory of the real numbers augmented by a predicate X that is in some natural sense self-similar with respect to a positive integer base. We show that there is a dichotomy: either we can define a Cantor set in our structure or our expansion of the reals is interdefinable with the real numbers augmented by a set of the form {1/r, 1/r^2, 1/r^3, …} for some integer r>=2.  In the latter case, this is equivalent to the structure having NIP and NTP_2.  This is joint work with Alexi Block Gorman.

MC 5479

Rohini Ramadas, University of Warwick

"Complex dynamics and algebraic geometry"

The field of complex dynamics began in the early 1900s with the study of iterating polynomials with complex coefficients. It gained momentum in the 1980s with important results on the structure of the Mandelbrot set by Douady-Hubbard and others, and connections established by Thurston, Sullivan and others with surface topology and hyperbolic 3-manifolds. The last decade has seen many breakthroughs achieved via new tools from number theory, measure theory and algebraic geometry. I will discuss some of my results, proved via degeneration techniques from algebraic geometry. The talk will highlight the rich interplay between topology on one hand and algebraic geometry/number theory on the other hand.

M3 3127

Lena Ji, University of Michigan

"Rationality of algebraic varieties over non-algebraically-closed fields"

The most basic algebraic varieties are projective spaces, and their closest relatives are rational varieties. These are varieties that agree with affine space on a dense open subset, and hence have a coordinate system on this open subset. Thus, rational varieties are the easiest varieties to understand. Historically, rationality problems have been of great importance in algebraic geometry: for example, Severi was interested in finding rational parametrizations for moduli spaces of Riemann surfaces (algebraic curves). Over the complex numbers, techniques from geometry and topology can be used to extract invariants useful for rationality questions. Over fields that are not algebraically closed (such as the rational numbers), the arithmetic of the field adds additional subtleties to the rationality problem. When the dimension of the variety is at most 2, there are effective criteria to determine rationality. However, in higher dimensions, there are no such known criteria. In this talk, I will first give a survey of some results on rationality of algebraic varieties. Then I will explain results on rationality obstructions for higher-dimensional varieties that involve the arithmetic of the field.

M3 3127

Xiao Zhong, Department of Pure Mathematics, University of Waterloo

"Capacity Theory, equilibrium distribution and potential functions"

This is a continuation of the series of talks from the last semester.  This talk will basically follow Chapter 6 of Baker-Rumely's Book: "Potential Theory and Dynamics on the Berkovich Projective Line". We will introduce the Capacity Theory and equilibrium distribution. Then we will study the potential function attached to the equilibrium distribution.

MC 5417

Uri Bader, The Weizmann Institute

"Cohomology of Arithmetic Groups, Higher Property T and Spectral Gap"

Groups of matrices with integer entries, aka arithmetic groups, are prominent objects of mathematics. From a geometric point of view, they appear as the fundamental groups of locally symmetric space. Topological invariants of such spaces could be seen as group invariants and vice versa. In my talk I will relate this useful link between topology and arithmetics with the theory of unitary representations. More precisely, I will focus on the cohomology of arithmetic groups with unitary coefficients, presenting a recent joint work with Roman Sauer which completely clarifies the theory in small degrees. By the end of the talk I will discuss the relation of the above with the phenomenon of spectral gap and state various related conjectures. I will make an effort to present the subject to a general audience.

MC 5501

Adi Glücksam, Northwestern University

"Multi-fractal spectrum of planar harmonic measure"

In this talk, I will define various notions of the multi-fractal spectrum of harmonic measures and discuss finer features of the relationship between them and properties of the corresponding conformal maps. Furthermore, I will describe the role of multifractal formalism and dynamics in the universal counterparts. This talk is based on a joint work with I. Binder.

MC 5501