# Mathematics

## Unsolved Combinatorial Games Richard K. Guy liked and others he would have liked

Richard started the Unsolved Problems in Combinatorial Games Column. I'll consider some of his favourites, talk about some developments, and add a few reminiscences.

## A recurrence lemma and its applications and extensions

I will discuss a lemma which is usually attributed to Atkinson, is very useful and apparently has been rediscovered multiple times. The lemma says that if (X, B, mu, T) is an ergodic system and f is in L^1(mu) with mean zero, then the Birkhoff sums of f do not diverge almost surely. Moreover the sums switch signs in a wide sense, infinitely many times. I will give the proof and discuss some applications of this result. Then I will discuss higher dimensional analogues, where the situation is significantly more complicated.

## Equidistribution of geodesic flow pushes via exponential mixing.

Pick a point at random in a finite volume hyperbolic surface and simultaneously flow in all directions from it. For the typical starting point these expanding circles will equidistribute and this talk will present a (more general) argument of Margulis establishing this fact.

## Conformal field theories and quantum phase transitions: an entanglement perspective

Quantum phase transitions occur when a quantum system undergoes a sharp change in its ground state, e.g. between a ferro- and para-magnet. I will present a remarkable set of transitions, called quantum critical, that are described by conformal field theories (CFTs). I will focus on 2 and 3 spatial dimensions, where the conformal symmetry is powerful yet less constraining than in 1 dimension. We will probe these scale-invariant theories via the structure of their quantum entanglement. The methods will include large-N expansions, the AdS/CFT duality from string theory, and large-scale numerical simulations. Finally, we’ll see that certain quantum Hall states, which are topological in nature, possess very similar entanglement properties. This hints at broader principles that relate very different quantum states.

For other events in this series see the quanTA events website.

## Characterizing handle-ribbon knots

Kauffman conjectured that a knot K is slice if and only if it bounds a genus-g Seifert surface containing a g-component slice link as a cut system. It’s very easy to show that a knot is ribbon if and only if it bounds a genus-g Seifert surface containing a g-component unlink as a cut system. Alex Zupan and I proved something in the middle of these statements: a knot is handle-ribbon (aka strongly homotopy-ribbon, aka something I will define in the talk) if and only if it bounds a genus-g Seifert surface containing a g-component R link L as a cut system—meaning that zero-surgery on L yields #_ g S^1 × S^2 . This gives a 3-dimensional definition of a 4-dimensional property. I’ll talk about these 3.5D knot properties and maybe how we use these techniques to extend a statement of Casson and Gordon. (The work in this talk is joint with Alexander Zupan from the University of Nebraska–Lincoln.)

## Weak factorization and transfer systems

Transfer systems are discrete objects that encode the homotopy theory of N∞ operads, i.e., the operads whose algebras are homotopy commutative monoids with a class of equivariant transfer (or norm) maps. They have a rich combinatorial structure defined in terms of the subgroup lattice of the group of equivariance, G. Indeed, if G is a cyclic p-group, there are Catalan-many transfer systems that assemble into the Tamari lattice (i.e., associahedron). In this talk, I will show that when G is finite Abelian, transfer systems are in natural bijection with weak factorization systems on the poset category of subgroups of G. This leads to a novel involution on the lattice of transfer systems, generalizing an observation of Balchin–Bearup–Pech–Roitzheim for cyclic groups of squarefree order. I will conclude with an enumeration of saturated transfer systems and comments on the Rubin and Blumberg–Hill saturation conjecture.

This is joint work with Angélica Osorno and a team of Reed College undergraduates: Evan Franchere, Usman Hafeez, Peter Marcus, Weihang Qin, and Riley Waugh (the Electronic Collaborative Mathematics Research Group, or eCMRG).

## Representation stability and configurations of disks in a strip

Representation stability, formalized in 2012 by Church, Ellenberg, and Farb, is a property exhibited by the homology of the configuration space of points in the plane: even as the number of points goes to infinity, the jth homology is generated by cycles in which at most 2j of the points move. What about the configuration space of disks of width 1 in an infinite strip of width w? This disks in a strip space behaves more like the no-k-equal configuration space of the line, where k-1 but not k points may be collocated; we show that the homology of this no-k-equal space exhibits generalized representation stability as defined by Sam–Snowden and Ramos. The method is to compute homology combinatorially using discrete Morse theory. Unlike other examples of homology with generalized representation stability, here the asymptotic behavior depends on the degree of homology.

## Symmetric knots and the equivariant 4-ball genus

Given a knot K in the 3-sphere, the 4-genus of K is the minimal genus of an orientable surface embedded in the 4-ball with boundary K. If the knot K has a symmetry (e.g., K is periodic or strongly invertible), one can define the equivariant 4-genus by only minimising the genus over those surfaces in the 4-ball which respect the symmetry of the knot. I'll discuss ongoing work with Keegan Boyle trying to understand the equivariant 4-genus.

## Enumerative geometry via the A^1-degree

Morel's $A^1$ -degree in $A^1$-homotopy theory is the analog of the Brouwer degree in classical topology. It takes values in the Grothendieck-Witt ring $GW(k)$ of a field $k$, that is the group completion of isometry classes of non-degenerate symmetric bilinear forms. We can use the $A^1$ -degree to count algebro-geometric objects in $GW(k)$, giving an $A^1$-enumerative geometry over non-algebraically closed fields. Taking the rank and the signature recovers classical counts over the complex and the real numbers, respectively. For example, the count of lines on a smooth cubic surface enriched in $GW(k)$ has rank 27 and signature 3.

## Random discrete surfaces

A triangulation of a surface is a way to divide it into a finite number of triangles. Let us pick a random triangulation uniformly among all those with a fixed size and genus. What can be said about the behaviour of these random geometric objects when the size gets large? We will investigate three different regimes: the planar case, the regime where the genus is not constrained, and the one where the genus is proportional to the size. Based on joint works with Baptiste Louf, Nicolas Curien and Bram Petri.

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