Suppose a function $\{1,\dots,N\} \to \mathbb R$ has the property that when we take discrete derivatives $k$ times, the result is identically zero. It is fairly well-known that this is equivalent to being a polynomial of degree $k-1$. It's not too unnatural to ask: what does the function look like if, instead, the iterated derivative is required to be zero just a positive proportion of the time? Such \emph{approximate polynomials} have a richer structure, related to nilpotent Lie groups.

On an unrelated note: given an $n \times n$ chessboard, how many ways are there to arrange $n$ queens on it, so that no two attack each other?

I'll outline how both these questions are connected to what's known as \emph{higher order Fourier analysis}, and explain more generally what higher order Fourier analysis is and what it can be used for (other than potentially placing queens on chessboards).

The minimal superpermutation problem asks how many symbols a string must have before it contains all $n!$ permutations of $\{1,\ldots,n\}$ as substrings. The Haruhi problem asks for the most efficient way to watch the anime ``The Melancholy of Haruhi Suzumiya'' in every way possible. Remarkably, these two problems turn out to be equivalent. Even more remarkably, the anime community has made more progress on this problem than the mathematicians! In this talk we will discuss an improved lower bound to the minimal superpermutation problem that was recently discovered on 4chan, as well as the anime theoretic motivation for considering this problem. As time permits we will also briefly discuss De Brujin sequences. This talk will be entirely self contained and assume no prior knowledge of anime.

Given a unitary Modular Tensor Category M, the reconstruction program asks for the construction of a rational conformal field theory such that its representation category is isomorphic to M. This is a low dimensional version of Dolpicher-Roberts/Deligne theory which is much richer due to nontrivial representations of braid group, and is strongly motivated by recent development in subfactor theory. In this talk we will describe questions around this program and present recent progress.

A drawing of a graph on a surface is {\em independently even} if every pair of nonadjacent edges in the drawing crosses an even number of times. The strong Hanani-Tutte theorem states that a graph admitting an independently even drawing in the plane must be planar.

The {\em genus} $g(G)$ of a graph $G$ is the minimum $g$ such that $G$ has an embedding on the orientable surface $M_g$ of genus $g$. The {\em $\mathbb{Z}_2$-genus} of a graph $G$, denoted $g_0(G)$, is the minimum $g$ such that $G$ has an independently even drawing on the orientable surface of genus $g$. Clearly, every graph $G$ satisfies $g_0(G) \leq g(G)$, and the strong Hanani-Tutte theorem states that $g_0(G) = 0$ if and only if $g(G) = 0$. Although there exist graphs $G$ for which the values of $g(G)$ and $g_0(G)$ differ, several recent results suggest that these graph parameters are closely related. We provide further evidence of their similarity.

For complete bipartite graphs $K_{n,m}$ with $n \geq 3$, we prove the following:
$$
g_0(K_{n,m}) \geq \lceil \frac{1}{2} \left( \lceil \frac{(n-2)(m-2)}{2} \rceil - (n-3) \right) \rceil
$$
The value of $g(K_{n,m})$ was determined by Ringel in 1965, and equals
$\lceil \frac{(n-m)(m-2)}{4} \rceil$.

Joint work with J. Kyncl.

The classical Schur(-Weyl) duality relates the representation theory of general linear Lie algebras and symmetric groups. Drinfeld and Jimbo independently introduced quantum groups in their study of exactly solvable models, which leads to a quantization of the Schur duality relating quantum groups of general linear Lie algebras and Hecke algebras of symmetric groups.

In this talk, I will explain the generalization of the (quantized) Schur duality to other classical types, algebraically, geometrically, and categorically. This new duality leads to a theory of canonical bases arising from quantum symmetric pairs generalizing Lusztig’s canonical bases on quantum groups.

The classical Borsuk-Ulam theorem states that a
continuous map from a n-dimensional sphere to m-dimensional sphere
which preserves the antipodal Z/2-actions only exists when m is
greater than or equal to n. One can ask a similar question, by
replacing the antipodal Z/2-action with an action of the Lie group
Pin(2).

On a seemingly unrelated side, the Geography Problem of 4-manifolds
asks which simply connected topological 4-manifolds admits a smooth
structure. By the celebrated works of Kirby-Siebenmann, Freedman,
Donaldson, Seiberg-Witten and Furuta, there is a surprising connection
between these two questions. In this talk, I will:

1. Explain this beautiful connection between the two problems.

2. Present a solution to the Pin(2)-equivariant Borsuk–Ulam problem.

3. State its application to the Geography Problem. In particular, a
partial result on the famous 11/8-conjecture.

4. Describe the ideas of our proof, which uses Pin(2)-equivariant
stable homotopy theory.

This talk is based on a joint work with Mike Hopkins, XiaoLin Danny
Shi and Zhouli Xu. No familiarity of homotopy theory or 4-dimensional
topology will be assumed.

In this talk I will introduce two different notions of solutions to the Plateau Problem, called Area and Size minimizers, due respectively to Federer-Fleming and Almgren. The fundamental difference between them is wether multiplicity/orientation plays a role or not, and they were originated respectively to describe integral homology class and soap films. I will then explain how different types of singularities arise in both formulation and some recent progress made on the structure of the singular set and of minimizers near singularities. If time permits I will also explain some possible future developments.

Branching-style models were proposed in the 1960's as a logician's
tool to study combinations of tenses and modalities, as in ``it is
still possible that it will rain in SD tomorrow'' but ``it is already
settled that last Summer was hot in SD''. A current theory of
Branching Space-Times (BST), put forward by N.~Belnap in 1992, is an
axiomatic framework that aims to describe how indeterminism plays out
in a spatio-temporal world. To this end it postulates a set of
relativistic space-times, any two of which are pasted together in some
particular region. Although a BST structure is continuous, it is
possible to discretise it, by focusing on particular objects, known as
transitions, and interpreted as places at which chancy actions
happen. A discretised structure, defined as a partially ordered set of
transitions, recovers then much, but not all information about the
initial structure. As these ideas are reminiscent of the Causal Set
Program, I will end up the talk by discussing some connections between
the two frameworks.

In this talk I will present a new method for studying the regularity of minimizers of some variational problems, including in particular some classical free-boundary problems. Using as a model case the so-called Obstacle problem, I will explain what regularity of the free-boundary means and how we obtain it by using a new tool, called (Log) -epiperimetric inequality. This technique is very general, and much like Caffarelli's 'improvement of flatness' for regular points, it allows for a uniform treatment of singularities in many different free-boundary problems. Moreover it is able to deal with logarithmic regularity, which in the case of the Obstacle problem is optimal due to an example of Figalli-Serra. If time permits I will explain how such an inequality is linked to the behavior of a gradient flow at infinity.

We discuss a new method to bound 5-torsion in class groups using
elliptic curves. The most natural ``trivial'' bound
on the n-torsion is to bound it by the size of the entire class group, for
which one has a global class number formula. We explain how to
make sense of the n-torsion of a class group intrinsically as a ``dimension
0 selmer group'', and by embedding it into an appropriate Elliptic curve we
can bound its size
by the Tate-Shafarevich group which we can bound using the BSD conjecture.

This fits into a general paradigm where one bounds ``dimension 0 selmer
groups'' by embedding into global objects, and using class number formulas.

Minimal surfaces are critical points of the area functional on the space of surfaces. Thus, it is natural to try to construct them via Morse theory. However, there is a serious issue when carrying this out, namely the occurrence of ``multiplicity.'' I will explain this issue and recent joint work with C. Mantoulidis ruling this out for generic metrics.

Because of their connections with the representation theory of the symmetric group, Hopf algebras have been used to study the enumerative properties of many combinatorial objects. In this talk we will give an introduction to this topic with the intention of defining Hopf monoids, a generalization that ends up being a better set up for the study of objects like graphs, matroids, posets, and generalized permutahedra.

I'll discuss joint work with M. Eichmair and V. Moraru in which we prove a natural minimal surface analogue of the splitting theorem for 3-manifolds with non-negative scalar curvature.