Seminars Sorted by Series

Combinatorial discrepancy asks the following question: Given a ground set U and a collection S of subsets of U, how do we color each element in U red or blue so that each subset in S has almost an equal number of each color? A straightforward idea...

We will explore some counting problems for number fields in which the conjectures formulated first by Malle and refined by Bhargava play a central role. In the process we will see some standard techniques in analytic number theory. 

For a low-dimensional manifold, one often tries to understand its intrinsic topology and geometry through its submanifolds, in particular of co-dimension 1. To be interesting and to give some information, such a submanifold should interact with the...

I’ll describe graph complexes, introduced by Kontsevich in the context of mathematical physics. I’ll survey its connections to geometry and topology, highlighting its relation to the cohomology of moduli spaces of curves. 

We give a gentle introduction to rational and Du Bois singularities in algebraic geometry. Through examples, we will see how birational geometry comes into play with the theory of differential operators. Time permitting, we discuss the sheaf...

Persistence modules offer a way to analyze how features, such as connected components or holes, evolve as a space is gradually changed. One can think of a persistence module as a sequence of vector spaces, each corresponding to a particular stage of...

In the 1970s, Viro's method paved an important path in the study of the topology of real algebraic varieties and became a precursor to tropical geometry. This method involves subdividing an integer polytope and using the information from each of its...

Open book decompositions provide a topological decomposition of a given manifold. We focus on dimension three. While the definition seems to be purely topological, it encodes information about fibered knots, surface dynamics, contact structures of...

CAT(0) cube complexes are cell complexes whose cells are cubes, whose naturally defined metric is non-positively curved in some precise sense. 

They can be equivalently defined in a variety of ways, which a priori looks very different. They naturally...

Tropical enumerative geometry is a branch of combinatorial algebraic geometry that aims to count algebraic objects (usually curves on some surface passing through a number of points) by turning them into combinatorial objects, called tropical curves...

I will give an overview of the classical study of local complex dynamics in one dimension, and the more recent study in several complex variables; with an emphasis on the `neutral’ case, that is when the local behavior is neither attracting nor...

Two graphs are isomorphic if they are the same up relabelling the vertices. Two matrices are equivalent if they are the same up to elementary row and column operations. Tensor isomorphism generalises these basic notions in graph theory and linear...

The tropical Grassmannian Trop G(2,n) and its positive part are combinatorial objects revealing fascinating connections between tropical geometry and particle physics. In particular, they play a relevant role in the CHY integral formulation of...

In the 1850s, Kummer discovered some striking congruences mod powers of a prime number p between values of the Riemann zeta function at negative odd integers.  This was part of his attempt to understand structural aspects of certain algebraic...

Property (T) was defined by Kazhdan in the 1960s, who used it to prove two conjectures of Selberg on lattices in high-rank Lie groups. Shortly after that, Margulis used it to construct expander graphs.

Property $\tau$ is a baby version of property (T...

The Shannon entropy of a discrete random variable quantifies the number of bits of information conveyed by sampling that variable. Although originally introduced in the context of information theory, techniques relying on Shannon entropy have been...

In the 1930s, Einstein, Podolsky and Rosen devised the "EPR paradox", which shed light on a peculiar phenomenon in the mathematical modeling of quantum mechanics:  Very far apart particles can exhibit correlated behaviour, which seemed to suggest a...

Harmonic functions on groups are connected to many properties of the groups: algebraic, geometric, analytic, and probabilistic.
For some groups (or spaces), it can be a challenge even to determine whether harmonic functions of certain types—such as...

Given two families of loops on a closed smooth manifold, one can concatenate the loops at the intersections points of these families to obtain a new family of loops. This is the Chas–Sullivan product on the homology of the free loop space of a...