# Colloquium 2015-2016

The Colloquium is held on Mondays at 4pm in Deady 208.

### Spring Quarter, 2016

- April 11,
**Malabika Pramanik** (University of British Columbia)

Configurations in sets big and small
**Abstract**: Does a set of positive Lebesgue measure contain an affine copy of your favourite pattern, say a line of specially arranged points, the vertices of a polyhedron or a geometric sequence on a spiral? Would the answer change if the set is Lebesgue-null, but is still large in some quantifiable sense? Such problems, involving identification of prescribed configurations, have been vigorously pursued both in the discrete and continuous setting, often with spectacular results. Yet many deceptively simple questions remain open. I will survey the literature in this area, emphasizing some of the landmark results that focus on different aspects of the problem.

- April 18,
**Lenny Ng** (Duke)

Studying topology through symplectic geometry
**Abstract**: Symplectic geometry has recently emerged as a key tool in the study of low-dimensional topology. One approach, championed by Arnol’d, is to examine the topology of a smooth manifold through the symplectic geometry of its cotangent bundle, building on the familiar concept of phase space from classical mechanics. I’ll describe a way to use this approach to construct a rather powerful invariant of knots called “knot contact homology”, and discuss its properties, including a connection to string theory and mirror symmetry.

- April 25,
**Ciprian Manolescu** (UCLA)

The triangulation conjecture
**Abstract**: The triangulation conjecture stated that any n-dimensional topological manifold is homeomorphic to a simplicial complex. It is true in dimensions at most 3, but false in dimension 4 by the work of Casson and Freedman. In this talk I will explain the proof that the conjecture is also false in higher dimensions. This result is based on previous work of Galewski-Stern and Matumoto, who reduced the problem to a question in low dimensions (the existence of elements of order 2 and Rokhlin invariant one in the 3-dimensional homology cobordism group). The low-dimensional question can be answered in the negative using a variant of Floer homology, Pin(2)-equivariant Seiberg-Witten Floer homology.

- May 2,
**Nathan Kaplan** (UC Irvine)

Arithmetic Statistics of Elliptic Curves
**Abstract**: How many points does a random elliptic curve have? The Mordell-Weil theorem says that the set of rational points on an elliptic curve defined over Q is a finitely generated abelian group. It is conjectured that the average value of the rank of this group 1/2, but the data we have does not match up well with this theoretical prediction. We will discuss joint work with Balakrishnan, Ho, Spicer, Stein, and Weigandt, in which we construct a new database of rank information for curves ordered by height.

There are only finitely many isomorphism classes of elliptic curves over a fixed finite field and we can give precise answers to statistical questions about the distribution of their groups of rational points. For example, what is the probability that the number of points is divisible by 5? What is the probability that the group of rational points is cyclic? We will discuss joint work with Petrow in which we compute moments for the rational point count distribution for elliptic curves containing a specified subgroup and give several applications.

- May 9,
**Farshid Hajir** (UMASS)

Cohen-Lenstra conjectures on the distribution of class groups and their generalizations
**Abstract**: Class groups of number fields are important invariants in algebraic number theory; they exhibit highly irregular behavior that has mystified number theorists for generations, going all the way back to the publication, in 1801, of Gauss’ Disquisitiones Arithmaticae. In the 1980s, Henri Cohen and Hendrik Lenstra developed a fascinating set of conjectures for the average behavior of class groups. In this talk, I will describe the Cohen-Lenstra conjectures and some non-abelian extensions of them in the case of imaginary quadratic fields. This is joint work with Nigel Boston and Michael Bush.

- May 24,
**Ken Ono** (Emory University)

**Special Time/Location:** Tuesday at 4PM in Fenton 105

New theorems at the interface of representation theory and number theory
**Abstract**: The speaker will discuss recent work on Moonshine and the Rogers-Ramanujan identities. The Rogers-Ramanujan identities are two peculiar identities which express two infinite product modular forms as number theoretic q-series. These identities give rise to the Rogers-Ramanujan continued fraction, whose values at CM points are algebraic integral units. In recent work with Griffin and Warnaar, the speaker has obtained a comprehensive framework of identities for infinite product modular forms in terms of Hall-Littlewood q-series. This work characterizes those integral units that arise from this theory. In a related direction, the speaker revisits the classical Moonshine Theorem which asserts that the coefficients of the modular j-functions are dimensions of virtual characters for the Monster, the largest of the simple sporadic groups. There are 194 irreducible representations of the Monster, and it has been a longstanding open problem to determine the distribution of these representations in Moonshine. In joint work with Griffin and Duncan, the speaker has obtained exact formulas for these distributions.

### Winter Quarter, 2016

- January 4,
**Paul Johnson** (University of Sheffield)

Lattice Points and Simultaneous Core Partitions
**Abstract**: For an integer t, t-core partitions are a class of partitions that appear naturally in representation theory, number theory, and geometry. More recently, in connection to rational Catalan combinatorics, there has been active study into partitions that are simultaneously a-core and b-core.

After a gentle introduction to core partitions, we will explain our recent work connecting simultaneous core partitions with the geometry of lattice points. This connection allows us to use Ehrhart theory to prove a conjecture of Armstrong about the average size of simultaneous core partitions.

- January 11,
**Michael Thaddeus** (Columbia University)

The multiplicative Horn problem
**Abstract**: In the unitary group U(n), what is the set of k-tuples of conjugacy classes having representatives whose product is the identity? This was the multiplicative Horn problem, now solved thanks to the work of Belkale, Knutson-Tao, Klyachko, and others. Their solution is thoroughly satisfying: it characterizes the set of such k-tuples as a convex polyhedron whose faces are determined by an explicit recursion. Yet even a problem which is completely solved may still live and breathe. This colloquium aims to demonstrate how.

- February 1,
**Anthony Varilly-Alvarado** (Rice University)

Elliptic curves, torsion subgroups, and uniform bounds for Brauer groups of K3 surfaces
**Abstract**: Elliptic curves are smooth plane curves defined by a homogeneous equation of degree three that come with a marked point. Results on elliptic integrals going back to Euler show that one can endow such a curve with an abelian group structure, making the marked point the origin of this group. Mordell showed in 1922 that if E is an elliptic curve defined by an equation over the rational numbers Q, then the group of points E(Q) is finitely generated. Surprisingly, there are only 15 possibilities for the torsion subgroup of E(Q). This is a spectacular theorem of Mazur from 1977. I will explore this circle of ideas for a higher dimensional analogue of elliptic curves: K3 surfaces. Unlike “abelian surfaces”, K3 surfaces have no group structure, so even understanding what the analogue of E(Q) should be is tricky. I will explain how the Brauer group of K3 surface comes to the rescue, argue for a conjecture along the lines of Mazur’s theorem, and explain the impact this would have in our understanding of K3 surfaces.

- February 8,
**Doug Lind** (University of Washington)

An invitation to algebraic actions
**Abstract**: Actions of groups like Z or Z^d using automorphisms of compact abelian groups have served as a source of inspiration and rich examples in dynamics for over fifty years. More recently, the study of similar actions of groups such as the discrete Heisenberg group have revealed a host of new phenomena and connections to areas such as von Neumann algebras. Using concrete examples that anyone can understand, I will sketch some of the previous theory, and indicate some of the many fascinating open problems remaining. This is necessarily just a sampler, in the spirit of Halmos’s dictum that math talks should “attract and inform”.

- February 15,
**Emanuele Macri** (Northeastern University)
- February 29,
**Glenn Stevens** (Boston University)

### Fall Quarter, 2015

- September 28,
**Valentino Tosatti ** (Northwestern)

The Kahler-Ricci flow and its singularities
**Abstract**: I will give an introduction to the study of Ricci flow on compact Kahler manifolds, and explain how its behavior reflects the structure of the complex manifold. I will then describe a result (joint with T.Collins) which gives a geometric description of the set where finite-time singularities occur, answering a conjecture of Feldman-Ilmanen-Knopf and Campana.

- October 5,
**Radu Dascaliuc ** (OSU)

**Joint Oregon/OSU Colloquium**

PDE, stochastic cascades, explosions, and the issue of symmetry breaking
**Abstract**: I will talk about a probabilistic cascade structure that can be naturally associated with certain partial differential equations and how it can be used to study well-posedness questions.

In the context of the still unsolved uniqueness problem for the 3D Navier-Stokes equations, our aim will be to see how the explosion properties of such cascades help establish a connection between the uniqueness of symmetry-preserving (self-similar) solutions and the uniqueness of the general problem.

Based on the joint work with N. Michalowski, E. Thomann, and E. Waymire.

- October 12, No Colloquium
- October 19,
** Ivan Corwin ** (Columbia University, Clay Mathematics Institute)

A drunk walk in a drunk world
**Abstract**: In a simple symmetric random walk on Z a particle jumps left or right with 50% chance independently at each time and space location. What if the jump probabilities are taken to be random themselves (e.g. uniformly distributed between 0% and 100%). In this talk we will describe the effect of this random environment on a random walk, in particular focusing on a new connection to the Kardar-Parisi-Zhang universality class and to the theory of quantum integrable systems. No prior knowledge or background will be expected.

- November 2,
**Chris Skinner** (Princeton)

Points and p-adics
**Abstract**: Let p be a prime number. This talk will leisurely recall the p-adic numbers and their past roles in proofs of results about solutions in integers or rational numbers (e.g., zeros of linear recurrences, rational points on curves), building up to recent results establishing mod p and p-adic criteria for elliptic curves over the rationals to have rank one.

- November 9,
**David Zureick-Brown ** (Emory)

Diophantine and tropical geometry
**Abstract**: Diophantine geometry is the study of integral solutions to a polynomial equation. For instance, for integers a,b,c >= 2 satisfying 1/a + 1/b + 1/c > 1, Darmon and Granville proved that the individual generalized Fermat equation x^a + y^b = z^c has only finitely many coprime integer solutions. Conjecturally something stronger is true: for a,b,c >= 3 there are no non-trivial solutions.

I’ll discuss various other Diophantine problems, with a focus on the underlying intuition and conjectural framework. I will especially focus on the uniformity conjecture, and will explain new ideas from tropical geometry and our recent partial proof of the uniformity conjecture.

- November 30,
**Rachel Ollivier** (UBC)

Remarks on the local Langlands conjectures
**Abstract**: The Langlands program, initiated in the 1960s, is a set of conjectures predicting a unification of number theory and the representation theory of groups. More precisely, the Langlands correspondence provides a way to interpret results in number theory in terms of group theory, and vice versa.

In this talk we sketch a few aspects of the local Langlands correspondence using elementary examples. We then comment on some questions raised by the emerging “mod p” Langlands program.

**Previous years:** 2013 2012 2011 2010