Seminars Sorted by Series

We consider compactifications of the Betti, de Rham and Dolbeault realizations of the character variety. Starting from an example, we look at what can be said, mostly conjecturally, about the relationship between these spaces.

Decoupling inequalities in harmonic analysis permit to bound the Fourier transform of measures carried by hyper surfaces by certain square functions defined using the geometry of the hyper surface. The original motivation has to do with issues in...

Chern-Schwartz-MacPherson class is a functorial Chern class defined for any algebraic variety. I will give a geometric proof of a positivity conjecture of Aluffi and Mihalcea that Chern classes of Schubert cells and varieties in Grassmannians are...

One of the most effective methods for solving boundary value problems for basic elliptic equations of mathematical physics in a given domain is the method of layer potentials. Its essence is to reduce the entire problem to an integral equation on...

We consider elliptic curves whose coefficients are degree 2 polynomials in a variable $t$. We prove that for infinitely many values of $t$ the resulting elliptic curve has rank at least 1. All such curves together form an algebraic surface which is...

I'll review recent progress on properties of 3-manifold groups, especially following from geometric properties. Then I'll discuss some open questions regarding 3-manifold groups, including their profinite completions, torsion in covers, orderability...

What can you learn about a group from a presentation? Sometimes very little; almost every interesting problem about groups given by (finite) presentations is unsolvable in full generality. But if one asks about *typical* groups - so-called "random"...

A key result in spectral theory linking classical and quantum mechanics is the Quantum Ergodicity theorem, which states that in a system in which the classical is ergodic, almost all of the Laplace eigenfunctions become uniformly distributed in...

There are various notions of an ``optimal'' position for a knot in the 3-sphere. Beginning with Schubert's bridge position, I'll talk about some 1- and 2-parameter definitions of complexity for a knot, and explain some of the reasons they have been...

A conjecture of Read predicts that the coefficients of the chromatic polynomial of a graph form a log-concave sequence for any graph. A related conjecture of Welsh predicts that the number of linearly independent subsets of varying sizes form a log...

There is a natural action of the group $\mathrm{SL}(2,\mathbb R)$ of $2 \times 2$ matrices on the unit tangent bundle of the moduli space of compact Riemann surfaces. This action can be visualized using flat geometry models, which allows one to make...

Following the revolutionary work of Thurston and Perelman, the topology of 3-manifolds is deeply intertwined with their geometry. In particular, hyperbolic geometry, the non-Euclidean geometry of constant negative curvature, plays a central role. In...

After a brief introduction to the dynamics of the $\mathrm{GL}(2,\mathbb R)$ action on the Hodge bundle (the space of translations surfaces), we will give a construction of six new orbit closures and explain why they are interesting. Joint work with...

The Langlands program is an intricate network of conjectures, which are meant to connect different areas of mathematics, such as number theory, harmonic analysis and representation theory. One striking consequence of the Langlands program is the...

The problem of comparing the shapes of different surfaces turns up in different guises in numerous fields. I will discuss a way to put a metric on the space of smooth Riemannian 2-spheres (i.e. shapes) that allows for comparing their geometric...

The main object of study of this talk are matrices whose entries are linear forms in a set of formal variables (over some field). The main problem is determining if a given such matrix is invertible or singular (over the appropriate field of...

Mock modular forms were first formally defined in the literature by Zagier in 2007, though their roots trace back to the mock theta functions, curious power series described by Ramanujan in his last letter to Hardy in 1920. As the overarching theory...

This is joint work with Michael Magee. We combine concepts from random matrix theory and free probability together with ideas from the theory of commutator length in groups and maps from surfaces, and establish new connections between the two. More...

In this talk I will discuss some classical and new applications of the abc conjecture, its relation to conjectures about elliptic curves, and some (admittedly weak) unconditional partial results.

One way to construct new 3-manifolds is by surgery on a knot in the 3-sphere; that is, we remove a neighborhood of a knot, and reglue it in a different way. What 3-manifolds can be obtained in this manner? We provide obstructions using the Heegaard...

We will discuss some natural problems in arithmetic that can be reformulated in terms of orbits of certain "thin" (semi)groups of integer matrix groups.

Classically, the Gauss-Manin connection relates the de Rham cohomology of different smooth fibres of a map. Its point of origin is the fact that vector fields act trivially on de Rham cohomology (the Cartan homotopy formula). That fact has an...

I will give a brief overview of symplectic mapping class groups, then explain how one can use homological mirror symmetry to get information about them. This is joint work with Ivan Smith.

The law of quadratic reciprocity and the celebrated connection between modular forms and elliptic curves over $\mathbb Q$ are both examples of reciprocity laws. Constructing new reciprocity laws is one of the goals of the Langlands program, which is...

One of the main ideas that comes up in the proof of Fermat's Last Theorem is a way of "counting" 2-dimensional Galois representations over $\mathbb Q$ with certain prescribed properties. We discuss the problem of counting other types of Galois...

For certain applications of linear algebra, it is useful to understand the distribution of the largest eigenvalue of a finite sum of discrete random matrices. One of the useful tools in this area is the "Matrix Chernoff" bound which gives tight...

Computing the class number is a hard question. In 1956, Iwasawa announced a surprising formula for an infinite family of class numbers, starting an entire theory that lies behind this phenomenon. We will not focus too much on this theory (Iwasawa...

Representation theory over $\mathbb Z$ is famously intractable, but "representation stability" provides a way to get around these difficulties, at least asymptotically, by enlarging our groups until they behave more like commutative rings. Moreover...

Representations of open compact subgroups play a fundamental role in studying representations of $p$-adic groups and their covering groups. We give an overview of this subject, called the theory of types, in connection with harmonic analysis. We...

The Fukaya category of a symplectic manifold is a robust intersection theory of its Lagrangian submanifolds. Over the past decade, ideas emerging from Wehrheim--Woodward's theory of quilts have suggested a method for producing maps between the...

The tree amplituhedron $A(n,k,m)$ is the image in the Grassmannian $Gr(k,k+m)$ of the totally nonnegative part of $Gr(k,n)$, under a (map induced by a) linear map which is totally positive. It was introduced by Arkani-Hamed and Trnka in 2013 in...

I'll describe joint work with Sheel Ganatra and Nick Sheridan which rigorously establishes the relationship between different aspects of the mirror symmetry phenomenon for Calabi-Yau manifolds. Homological mirror symmetry---an abstract, categorical...

The Hofer-Zehnder capacity of a symplectic manifold is one of the early symplectic invariants: it is a non-negative real number, possibly infinite. Finiteness of this capacity has strong consequences for Hamiltonian dynamics, and it is an old...

Berkovich geometry is an enhancement of classical rigid analytic geometry. Mirror symmetry is a conjectural duality between Calabi-Yau manifolds. I will explain (1) what is mirror symmetry, (2) what are Berkovich spaces, (3) how Berkovich spaces...

Over the past two decades, information theory has reemerged within computational complexity theory as a mathematical tool for obtaining unconditional lower bounds in a number of models, including streaming algorithms, data structures, and...