Farb and Friends

1. The lack of combinatorial proofs in this area, despite numeric evidence that the results hold for \( n\geq 8 \)
2. A variety of unsolved questions about the biases which seem utterly intractable and yet utterly interesting.
3. Linear independence of rational values of the digamma function.

• varieties are pseudomanifolds
• exotic spheres can show up in the topology of algebraic varieties!
• isolated singularities give rise to fibered knots!
• Lê's fibration theorem (which is one road to perverse sheaves)

• Finitely generated by Dehn twists
• Virtually surjects onto \( {\rm Sp}(2g,\mathbb{Z}) \)
• whose intersection with the Torelli group virtually surjects onto \( \Lambda^3 H/ H \)
• Contains framed mapping class group of large genus subsurfaces.

1. Real roots and signs of coefficients of polynomials parametrizing combinatorial objects
2. Combinatorics vs. topology of linear subspace arrangements
3. Controlling singularities on families of varieties

• 27 lines on a cubic surface
• 28 bitangents to a quartic curve
• Degree 2 del Pezzo surfaces
• Weyl groups and graph automorphisms
• Hodge and cyclic decompositions
• Lattices and gluing groups
• Moduli spaces and framings
• The Torelli theorem for K3 surfaces

Given a Jordan curve on the plane, can you always find 4 points that they form a rectangle, or even a square? What about other shapes? What about an imbedded 2 sphere in \(\mathbb{R}^3\), can you find similar statement? This presents work of Cole Hugelmeyer, Josh Greene and Andrew Lobb.

I will talk about a forgotten theorem of Coxeter, which gives a generalization of Gauss’s Pentogramma Mirificum. I will explain its role in the Goncharov Program and say a few words about its role in the proof of the theorem of Böhm on volumes of hyperbolic polytopes.

Let \(M\) be a closed aspherical manifold with the fundamental group \(\Gamma\). Suppose we are interested in the growth of topological invariants of a sequence of finite covers \(M_n\) of \(M\). These can be often read from the limit objects which are constructed using the sequence \(M_n\). One way of doing it is by considering the weak-star limits of normalized counting measures supported on the conjugates of the fundamental group of \(M_n\) inside the space of subgroups of \(\Gamma\). The limit is a random subgroup whose distribution is invariant by conjugation by \(\Gamma\) - we call them "invariant random subgroups”. I my talk I will explain how they can be applied to various problems in geometry and how they fit in the more general framework of unimodular measures on moduli spaces.

Given a rational function f(x) with complex coefficients, the iterates of f(x) determine a sequence of finite degree branched covers of the sphere. Finite branched covers of the sphere are essentially determined by their monodromy, so it stands to reason that the dynamics of f(x) should be encoded in this sequence of monodromy groups. But how exactly does that work? In this talk I'll introduce the notion of an iterated monodromy group, show how this leads us by the nose to the important concept of a post-critically finite map, and see how this relates to finite automata acting on infinite trees.

There are two kinds of sextic curves C in P^2 with exactly six ordinary cusps and no other singularities: one kind has \(\pi_1(\mathbb{P}^2 - C) = \mathbb{Z}/2 * \mathbb{Z}/3\) non-abelian, and the other has \(\pi_1(\mathbb{P}^2 - C) = \mathbb{Z}/6\) abelian. These two kinds of curves are distinguished by whether or not the six cusp points lie on a conic. This is very weird and has confused me for a long time. I'm going to talk about these sextics, some cool things, some weird things, and some things that still confuse me about all this.

I will discuss a notion of polynomial equations in groups motivated by a question about enlarging a group by adding a generator. Then I will explain why unitary groups are "algebraically closed".

The Prym construction associates a principally polarized abelian variety of dimension \(g-1\), referred to as the Prym variety of the cover, to a double cover \(f: Y \to X\) of a smooth genus \(g\) curve. I will discuss some interesting properties of this construction.

I'll discuss a fun (and the historically original) appearance of an outer automorphism of \(S_6\) by fussing around with hexagons on conic plane curves. Along the way we meet up with Steiner, Plücker, Salmon, Kirkman, Cayley, Bacharach, Braikenridge, Maclaurin, and of course Pascal.

The plane Cremona group \(\text{Cr}(2)\) (over \(\mathbb C\)) is the group of birational automorphisms of \(\mathbb{CP}^2\). There is a classification of the conjugacy classes of order \(2\) elements of \(\text{Cr}(2)\) into three types: de Jonquières, Geiser, and Bertini. In this talk, I will describe these three types of involutions explicitly.

It is well-known that there is no general solution by radicals for polynomials of degree at least 5...but are there other ways to solve polynomial equations? In this talk I will discuss the possibility of finding polynomial roots through iteration of rational functions. We will see, following McMullen, that iteration allows us to solve the general quintic (and not much else!)

Mednykh's formula counts homomorphisms from the fundamental group of a surface to a finite group \(G\) in terms of the representation theory of \(G\). We'll discuss two proofs of this formula, as well as related questions.

Recently Bruno Zimmermann posted a short paper on the Nielsen realization problem for the mapping class groups of the three-manifolds formed by taking finite connected sums of \(S^2 \times S^1\). This made me curious about what is known about these groups! I will survey some of what I found out and will also discuss some of the results and arguments in Zimmermann’s paper.

Consider the following statement: "The space of \(k\)-planes contained in varieties in \(A\) is in \(A\)." What can \(A\) be? For a certain \(A\), this will depend on a special property of varieties represented by points of \(A\).

It turns out that the (co)homology groups of an algebraic variety admit a number of amazing structures, which go beyond the homology of ordinary topological spaces. I will talk about several of these structures, how they reflect the "rigidity" of algebraic geometry, and about several deep analogies/conjectures connecting them.

Group actions on graphs reveal information about the group itself. We demonstrate this on SL(2,Z) and its various generalizations. For example, we show that mapping class groups of surfaces with positive genus is finitely generated by Dehn twists.

The stable torsion length in a group is the stable word length with respect to the set of all torsion elements. We show that the stable torsion length vanishes in crystallographic groups. We then give a linear programming algorithm to compute a lower bound for stable torsion length in free products of groups. Moreover, we obtain an algorithm that exactly computes stable torsion length in free products of finite abelian groups. The nature of the algorithm shows that stable torsion length is rational in this case. As applications, we give the first exact computations of stable torsion length for nontrivial examples. (Joint work with Lvzhou Chen.)

Every closed 3-manifold admits foliations, where the leaves are surfaces. For a given 3-manifold, which surfaces can appear as leaves? Kerékjártó and Richards gave a classification up to homeomorphism of noncompact surfaces, which includes surfaces with infinite genus and infinitely many punctures. In their 1985 paper "Every surface is a leaf", Cantwell–Conlon prove that for every orientable noncompact surface L and every closed 3-manifold M, M has a foliation where L appears as a leaf. We will discuss their paper and construction and the surrounding context.

I will give an overview of the Torelli theorems for curves and K3 surfaces.

A remarkably long list of mathematical structures can be classified by the simply laced Dynkin diagrams, i.e. those of types A, D and E (but not B or C). I will discuss just some of the connections I’ve learned between different structures sharing these classifications.

Consider the groups O(f, \mathbb{Z}) of integral matrices preserving an integral quadratic form f of signature (n, 1) for some n \geq 2. Such a group is called reflexive if its subgroup generated by reflections has finite index. In 1972, Vinberg proposed an algorithm to determine the reflexivity of these groups and applied it to the case n \leq 17. In this talk, we will give an overview of some basics of reflection groups and Vinberg's algorithm.

We will discuss a combinatorial plane geometry question that is currently best understood using a rather deep result on properties of algebraic surfaces.

The nth configuration space of a manifold M can be thought of as the space parameterizing n unlabelled particles in M, such that the particles are not allowed to collide with each other. Configuration spaces with summable labels are variants of configuration spaces where the particles now have names, and there is a prescribed rule for what happens when particles of different types collide (e.g. when a muon and a gluon collide they turn into a pion). Often, important spaces in algebraic topology are secretly configuration spaces with summable labels! I will talk about examples involving classifying spaces and mapping spaces, the Dold-Thom theorem and its generalizations.

Just how much information does the representation theory of a group carry about the group itself? Well that depends what you mean by "representation theory" or "group"...

What does the Dedekind eta function and the Hirzebruch signature defect have to do with each other? A priori, nothing much. But thanks to the fact that PSL(2,R) and SL(2,R) both act on the circle, the defect of log(eta) and the signature defect of the mapping torus pretty much differ by a constant factor, thanks to a link granted by the Euler class of Homeo^+(S^1).

We will focus on a result of Larsen--Lunts that gives a close connection between stable birational equivalence classes of smooth projective varieties and cut and paste relations. Following some background and a sketch of a proof, we will give applications to the overall structure of cut and paste relations and specific independence results.

Suppose a finite, cyclic group G acts on a 4-manifold. What can we say about the fixed set of this action? In this talk, I will discuss some conditions one can impose on the action (e.g. preserving the orientation or the spin structure) and their consequences on the topology of the fixed set.

Polyhedral complexes are 2-dimensional simplicial complexes which are created by gluing together regular Euclidean k-gons along their edges. A natural question is how many (simply connected) polyhedral complexes exist which satisfy some collection of local gluing criteria, up to isomorphism. This talk is an introduction to polyhedral complexes, and gives an overview of some work that has been done in this direction, specifically when the polyhedral complexes are (k,L)-complexes.

We give a brief overview of the first sections of McMullen's paper Braid Groups and Hodge Theory, exploring unitary representations of the braid group given by the monodromy of a family of hyperelliptic curves.

We will explore the relationship between some of the fundamental objects of algebraic geometry, divisors and line bundles, from a concrete, geometric perspective, using the language of Čech cohomology.

To better understand how a variety in affine or projective space "sits" in space, one can try to compute topological invariants of its complement. Such complements sometimes also parametrize the 'nice' or 'smooth' objects in a moduli space. I will discuss a method to compute the fundamental group of the complement of an algebraic curve (over ℂ) in the plane, and explicitly compute some examples. This is based on work originally due to Zariski and Van-Kampen, but the tools used, such as braid monodromy, are also useful in more complicated settings.