The neckpinch is the easiest to visualize non-trivial Ricci flow singularity. Another singularity is the degenerate neckpinch, which seems to arise non-generically and exhibits a faster rate of curvature blow-up. The existence of both of these singularities was conjectured by Hamilton by 1995, and in the past decade their existence and asymptotic shape have been rigorously proven. Furthermore, solutions to Ricci flow emerging from the resulting singular data have been found, and these give the first rigorous examples of Ricci flow through singularities without surgery. In this talk we will describe the main features of the neckpinches, and explain some of the techniques used to find them. (This is a candidacy talk.)

Results of Lindenstrauss and McCarthy provide a description of the K-theory of endomorphisms , where is a discrete ring and is an -bimodule. This description is in terms of a "topological Witt vectors" construction , whose simpler form allows for the completion of otherwise intractable computations in algebraic K-theory. We will describe work done by the speaker to generalize these results to connective ring spectra, using a "density" argument and ideas from Goodwillie calculus. This is a thesis defense.

I will give first a brief overview on the selection problem for solutions of Hamilton--Jacobi equations, which leads to the theory of viscosity solutions. Then I will describe the cell/ergodic problem of interest and its interesting phenomena. Finally, I will state the corresponding selection problem, the main result, and explain some key ideas. This is a joint work with Hiroyoshi Mitake.

Considering congestion games with uncertain delays, we compute the inefficiency introduced in network routing by risk-averse agents. At equilibrium, agents may select paths that do not minimize the expected latency so as to obtain lower variability. A social planner, who is likely to be more risk neutral than agents because it operates at a longer time-scale, quantifies social cost with the total expected delay along routes. From that perspective, agents may make suboptimal decisions that degrade long-term quality. We define the price of risk aversion (PRA) as the worst-case ratio of the social cost at a risk-averse Wardrop equilibrium to that where agents are risk-neutral. For networks with general delay functions and a single source-sink pair, we show that the PRA depends linearly on the agents' risk tolerance and on the degree of variability present in the network. In contrast to the price of anarchy, in general the PRA increases when the network gets larger but it does not depend on the shape of the delay functions. To get this result we rely on a combinatorial proof that employs alternating paths that are reminiscent of those used in max-flow algorithms. For series-parallel (SP) graphs, the PRA becomes independent of the network topology and its size. As a result of independent interest, we prove that for SP networks with deterministic delays, Wardrop equilibria maximize the shortest-path objective among all feasible flows. Joint work with Nicolas Stier-Moses

I will give a brief introduction to Kac-Moody groups, infinite dimensional analogues of Chevalley groups, and Kazhdan's property (T) and then discuss joint work with Mikhail Ershov establishing property (T) for Kac-Moody groups over rings. We expand upon previous results by Dymara & Januszkiewicz establishing property (T) for Kac-Moody groups over finite fields and by Ershov, Jaikin, & Kassabov establishing property (T) for Chevalley groups over commutative rings to prove that given any indecomposable 2-spherical generalized Cartan matrix A there is an integer m (depending solely on A) such that if R is a finitely generated commutative unital ring with no ideals of index less than m then the Kac-Moody group over R associated to A has property (T).

This talk will survey recent results on Boolean stochastic geometry in high dimensional Euclidean spaces. A Boolean model in consists of a homogeneous Poisson point process in and of independent and identically distributed random closed sets of centered on each atom of this point process. The Shannon regime features a family of Boolean models indexed by , where the -th model has a Poisson point process of intensity and random compact sets with diameter of order , and lets tend to . A typical example is that where each random compact set is an ball of radius distributed like , with satisfying a large deviations principle. The main focus of the talk will be on the asymptotic behavior of classical Boolean stochastic geometry quantities, like volume fraction, percolation threshold or mean cluster size, in this Shannon regime. The analysis of this asymptotic regime is motivated by problems in information theory, and leads to new results on error exponents, which describe the rate of decay to 0 of the probability of decoding error in channel coding. Joint work with V. Anantharam.

We exhibit a construction in noncommutative nonnoetherian algebra that should be understood as a positive characteristic analogue of the Bernstein-Sato polynomial or b-function. Recall that the b-function is a polynomial in one variable attached to an analytic function f. It is well-known to be related to the singularities of f and is useful in continuing a certain type of zeta functions, associated with f. We will briefly recall the complex theory and then emphasize the arithmetic aspects of our construction.

When studying square-free monomial ideals, Commutative Algebra and Discrete Mathematics intersect. There is a natural one-to-one correspondence between square-free monomial ideals and clutters, which are also called simple hypergraphs. When the ideal is generated in degree two, the corresponding clutter is a graph. Using the correspondence, algebraic properties of ideals can be translated into graph theoretic or combinatorial properties and vice versa, allowing techniques from one field of mathematics to be used to answer questions from another. Exploiting the dual algebraic and combinatorial natures of square-free monomial ideals has proven to be a fertile source of mathematical results in recent years. Although the powers of the ideals are no longer square-free, the combinatorial structures can still be used to determine properties of the ideals. The focus of this talk will be algebraic properties of powers of square-free monomial ideals and their combinatorial translations. Results presented will focus on invariants such as depth, projective dimension, and associated primes.

The Hitchin component is a preferred component of the character variety consisting of homomorphisms from a surface group to the Lie group PSL_n(R) or, more generally, to a split real algebraic group. I will survey work of Labourie and Fock-Goncharov on the dynamics and geometric properties of those homomorphism that represent elements of the Hitchin component.

The Hitchin component is a preferred component of the character variety consisting of homomorphisms from a surface group to the Lie group PSL_n(R) or, more generally, to a split real algebraic group. I will present a global parametrization of the Hitchin component that is well adapted to the local geometry of geodesic laminations on the surface. This is joint work with Guillaume Dreyer.

Angular Synchronization is a method developed by Amit Singer in 2009 to recover a set of angles from observed noisy pairwise offsets. We will mention the two main approaches - eigenvectors and semidefinite programming - and in the context of the latter we will discuss a phenomenon known as rank recovery.

Graphene is a gapless semiconductor made of a sheet composed of a single layer of carbon atoms arranged into a honeycomb hexagonal lattice. In view of application in graphene-based electron devices, it is crucial to understand the basic transport properties of this material. A physically accurate model is given by a semiclassical transport equation whose scattering terms have been deeply analyzed recently. Due to the computational difficulties, the most part of the available solutions have been obtained by direct Monte Carlo simulations. The aim of this work is to use a numerical scheme based on the discontinuous Galerkin method for finding deterministic (non stochastic) solutions of the electron Boltzmann transport equation in graphene. The same method has been already successfully applied to a more conventional semiconductor material like Silicon or Gallium Arsenide. A n-type doping or equivalently a high value of the Fermi potential is considered. Therefore we neglect the inter band scatterings but retain all the main electron-phonon scatterings. Simulations in graphene nano-ribbons are presented and a comparison with stochastic simulations are presented. We will show that standard techniques for direct Monte Carlo simulations require important adjustments, and we propose a new stochastic scheme, which can be also validated by means of deterministic numerical solutions. Work in collaboration with Vittorio Romano.