NUMBER THEORY SEMINAR: | Alexander StasinskiDURHAM UNIVERSITY |
Representation zeta functions of simple Lie groups and reductive groups over local rings |

on Wednesday, March 04, 2015, at 3:00 PM in Science Center 507 | ||

A representation zeta function is a Dirichlet series whose nth coefficient is the number of n-dimensional representations of a given group. They were first introduced for compact semisimple Lie groups by E. Witten who also proved that their special values at positive even integers are rational multiples of powers of \pi. In a seminal paper, M. Larsen and A. Lubotzky studied representation zeta functions of several classes of linear groups, including certain simple Lie groups and linear groups over local rings. Zeta functions of these groups appear naturally as the local factors in the Euler product of representation zeta functions of arithmetic groups. One of the main objects of study is the abscissa of convergence of a representation zeta function, which governs the polynomial degree of the representation growth. Larsen and Lubotzky proved that the abscissa of Witten zeta functions of simple Lie groups is r/k, where r is the rank of the group and k is the number of positive roots. We will give a new proof of |

INFORMAL DYNAMICS AND GEOMETRY SEMINAR: | Alden WalkerUNIVERSITY OF CHICAGO |
Circle actions on the boundary of Schottky space |

on Wednesday, March 04, 2015, at 4:00 - 5:00 PM in Science Center 507 | ||

To a complex parameter c, we associate the two-generator iterated function system {z -> cz-1, z -> cz+1}. I'll describe how the IFS for certain parameters (those on the boundary of the connectedness locus) can give rise to circle actions. A finite amount of data encoded in these circle actions describes the set of cut points in the limit set of the IFS. In addition, these circle actions can be thought of as double covers of Lorenz maps and generalizations. This talk should be broadly accessible, and pictures will be provided. This is joint work with Danny Calegari. |

CENTER OF MATHEMATICAL SCIENCES AND APPLICATIONS RANDOM MATRIX & PROBABILITY THEORY SEMINAR: | Ivan CorwinClay Institute, Columbia, Institute Henri Poincare, Packard Foundation |
Stochastic quantum integrable systems |

on Wednesday, March 04, 2015, at 11:30 am - 1:00 pm in Science Center 232 | ||

We describe recent work involving interacting particle systems related to quantum integrable systems. This theory serves as an umbrella for exactly solvable models in the Kardar-Parisi-Zhang universality class, as well as provides new examples of such systems, and new tools in their analysis. |

THURSDAY SEMINAR: | Mike HopkinsHARVARD UNIVERSITY |
Constructing algebraic vector bundles on projective spaces |

on Thursday, March 05, 2015, at 3:30 - 5:30 pm in Science Center Hall E |

GAUGE THEORY, TOPOLOGY & SYMPLECTIC GEOMETRY SEMINAR: | Laura StarkstonUNIVERSITY OF TEXAS - AUSTIN |
Symplectic fillings of Seifert fibered spaces |

on Friday, March 06, 2015, at 3:30 - 4:30 PM in Science Center 507 | ||

Although the existence of a symplectic filling is well-understood for many contact 3-manifolds, complete classifications of all symplectic fillings of a particular contact manifold are more rare. Relying on a recognition theorem of McDuff for closed symplectic manifolds, we can understand this classification for certain Seifert fibered spaces with their canonical contact structures. Even in cases without complete classification statements, the techniques used can suggest constructions of symplectic fillings with interesting topology. These fillings can be used in cut-and-paste operations to construct examples of exotic 4-manifolds. |

CENTER OF MATHEMATICAL SCIENCES AND APPLICATIONS COLLOQUIUM: | Nikita NekrasovCenter for Geometry & Physics, STONY BROOK |
Crossed instantons and qq-characters |

on Friday, March 06, 2015, at 4:00 PM in Science Center Hall A | ||

I will describe a compactification of the moduli space of instantons in gauge theory on the union of two copies of Euclidean space R^4 intersecting transversely in R^8, and its orbifolds. The theory of qq-characters, deformations of the characters of Kac-Moody groups and q-characters of quantum affine algebras, is an application, leading to the solution of quantum many-body systems such as elliptic Calogero-Moser. |

BASIC NOTIONS SEMINAR: | Jacob LurieHARVARD UNIVERSITY |
Representation Theory Between Characteristic Zero and Characteristic p |

on Monday, March 09, 2015, at 3:00 PM in Science Center 507 | ||

Let G be a finite group. One can study representations of G over any field k: that is, vector spaces over k equipped with an action of G. In general, such representations behave very differently in characteristic zero (where all representations are completely reducible) and in characteristic p (where, if G is a p-group, there are no irreducible representations other than the trivial representation). In this talk, I will discuss representation theory over more exotic "fields" known as Morava K-theories, which in some sense interpolate between fields of characteristic zero and fields of characteristic p and share many pleasant features of both. No prior familiarity with Morava K-theory will be assumed. |

HARVARD/MIT ALGEBRAIC GEOMETRY SEMINAR: | Klaus HulekLeibniz University Hannover |
Stable cohomology of compactifications of Ag |

on Tuesday, March 10, 2015, at 3:00 PM in Science Center 507 | ||

A well known result of Borel says that the cohomology of Ag stabilizes. This was generalized to the Satake compactification by Charney and Lee. In this talk we will discuss whether the result can also be extended to toroidal compactifictaions. As we shall see this cannot be expected for the second Vornoi compactification, but we shall show that the cohomology of the perfect cone compactification does stabilize. We shall also discuss partial compactifications, in particular the matroidal locus. This is joint work with Sam Grushevsky and Orsola Tommasi. |

NUMBER THEORY SEMINAR: | Shrenik ShahCOLUMBIA UNIVERSITY |
Some strong twisted base changes for unitary similitude groups |

on Wednesday, March 11, 2015, at 3:00 PM in Science Center 507 | ||

Labesse, Morel, Skinner, and Shin have attached twisted base changes (on general linear groups) to regular cohomological cuspidal automorphic representations on certain unitary similitude groups. The expected local-global compatibility relations hold at places where the group is split or where the similitude group and representation are unramified. We improve this to include certain limited cases where the group is ramified. This allows us to construct some strong twisted base changes unconditionally for unitary similitude groups, which are needed in recent work of Skinner-Urban on the Bloch-Kato conjecture. We obtain new cases of the generalized Ramanujan conjecture in this setting. Our approach combines automorphic and p-adic methods. |

CENTER OF MATHEMATICAL SCIENCES AND APPLICATIONS COLLOQUIUM: | Yan GuoBROWN UNIVERSITY - Division of Applied Mathematics |
Absence of shocks in Euler-Maxwell system for two-fluid models in plasma |

on Wednesday, March 11, 2015, at 4:00 PM in Science Center Hall A | ||

As the cornerstone of two-fluid models in plasma theory, Euler-Maxwell (Euler-Poisson) system describes the dynamics of compressible ion and electron fluids interacting with their own self-consistent electromagnetic field. It is also the origin of many famous dispersive PDE such as KdV, NLS, Zakharov, ...etc. The electromagnetic interaction produces plasma frequencies which enhance the dispersive effect, so that smooth initial data with small amplitude will persist forever for the Euler-Maxwell system, suppressing possible shock formation. This is in stark contrast to the classical Euler system for a compressible neutral fluid, for which shock waves will develop even for small smooth initial data. A survey along this direction for various two-fluid models will be given during this talk. |

INFORMAL DYNAMICS AND GEOMETRY SEMINAR: | Sam TaylorYALE UNIVERSITY |
Hyperbolic extensions of free groups |

on Wednesday, March 11, 2015, at 04:00 PM in Science Center 507 | ||

Every subgroup $G$ of the outer automorphism group of a finite-rank free group $F$ naturally determines a free group extension $1\to F \to E_G \to G\to 1$. In this talk, I will discuss geometric conditions on the subgroup $G$ that imply its corresponding extension $E_G$ is hyperbolic. These conditions are in terms of the action of G on the free factor complex of $F$ and allow one to easily build new examples of hyperbolic free group extensions. This is joint work with Spencer Dowdall. |

CENTER OF MATHEMATICAL SCIENCES AND APPLICATIONS RANDOM MATRIX & PROBABILITY THEORY SEMINAR: | Ankur MoitraMASSACHUSETTS INSTITUTE OF TECHNOLOGY |
Tensor Prediction, Rademacher Complexity and Random 3-XOR |

on Wednesday, March 11, 2015, at 11:30 am - 1:00 pm in Science Center 232 | ||

Here we study the tensor prediction problem, where the goal is to accurately predict the entries of a low rank, third-order tensor (with noise) given as few observations as possible. We give algorithms based on the sixth level of the sum-of-squares hierarchy that work with roughly m = n^3/2 observations, and we complement our result by showing that any attempt to solve tensor prediction with fewer observations through the sum-of-squares hierarchy would run in moderately exponential time. In contrast, information theoretically roughly m = n observations suffice. This work is part of a broader agenda of studying computational vs. statistical tradeoffs through the sum-of-squares hierarchy. In particular, for linear inverse problems (such as tensor prediction) the natural sum-of-squares relaxation gives rise to a sequence of norms. Our approach is to characterize their Rademacher complexity. Moreover, both our upper and lower bounds are based on connections between this, and the task of strongly refuting random 3-XOR formulas, random matrix theory and the resolution proof system. This talk is based on joint work with Boaz Barak |

INFORMAL COMBINATORICS SEMINAR: | Anand LouisPRINCETON UNIVERSITY |
Hypergraph Heat Operators and Applications |

on Friday, March 13, 2015, at 1:00 - 2:00 PM in Science Center 530 | ||

Graph and hypergraph partitioning problems are a central topic of research in the study of algorithms and complexity theory, with connections to sampling algorithms, mixing time of Markov chains, metric embeddings, among others. There has been a lot of work studying the eigenvalues of a graph and relating it to combinatorial properties of the graph. It has remained open to define a suitable spectral model for hypergraphs whose spectra can be used to estimate various combinatorial properties of the hypergraph. I will define a heat operator for hypergraphs which generalizes the heat kernel for graphs. I will demonstrate the utility of this operator by showing generalizations of some known results from spectral graph theory to hypergraphs; for e.g. bounding various hypergraph expansion parameters via eigenvalues, bounding the diameter of the hypergraph, mixing time of this operator, etc. |

GAUGE THEORY, TOPOLOGY & SYMPLECTIC GEOMETRY SEMINAR: | Jingyu ZhaoCOLUMBIA UNIVERSITY |
Periodic Symplectic Cohomologies |

on Friday, March 13, 2015, at 3:30 PM in Science Center 507 | ||

Periodic cyclic homology group associated to a mixed complex was introduced by Goodwillie. In this talk, I will explain how to apply this construction to the symplecticcochain complex of a Liouville domain and obtain two periodic symplectic cohomology theories, which are called periodic symplectic cohomology and finitely supported periodic symplectic cohomology, respectively. The main result is that there is a localization theorem for the finitely supported periodic symplectic cohomology. |

BASIC NOTIONS SEMINAR: | Brian ConradSTANFORD UNIVERSITY |
The arithmetic Lefschetz Principle |

on Monday, March 23, 2015, at 3:00 PM in Science Center 507 | ||

In complex algebraic geometry there is a general technique called the "Lefschetz Principle" which roughly says that for certain appropriately-formulated algebro-geometric problems over a general field of characteristic 0, it is sufficient to solve the problem over the complex numbers (where one has access to analytic and topological methods). But this idea has relevance far beyond its traditional setting, and in fact underlies a wide array of powerful methods (sometimes called "spreading out") for relating problems in characteristic 0 (even over the complex field) or over the integers to problems over (many) finite fields, where new tools become available (point-counting, Frobenius, etc.). This has been widely used by experts since the 1960's but is not generally explained in the literature outside of EGA. We will illustrate the basic idea with a few interesting examples, not assuming prior familiarity with the usual Lefschetz Principle (or EGA). |

HARVARD/MIT ALGEBRAIC GEOMETRY SEMINAR: | Jesus Martinez-GarciaJOHNS HOPKINS |
On the moduli space of cubic surfaces and their anticanonical divisors |

on Tuesday, March 24, 2015, at 3:00 PM in Science Center 507 | ||

We study variations of GIT quotients of log pairs (X,D) where X is a hypersurface of some fixed degree and D is a hyperplane section. GIT is known to provide a finite number of possible compactifications of such pairs, depending on one parameter. Any two such compactifications are related by birational transformations. We describe an algorithm to study the stability of the Hilbert scheme of these pairs, and apply our algorithm to the case of cubic surfaces. Finally, we relate this compactifications with the (conjectural) moduli space of log K-semistable pairs. This is work in progress with Patricio Gallardo (University of Georgia). |

NUMBER THEORY SEMINAR: | Brian ConradSTANFORD UNIVERSITY |
A Lefschetz Principle in non-archimedean geometry |

on Wednesday, March 25, 2015, at 3:00 PM in Science Center 507 | ||

In analytic geometry over a non-archimedean field it is often convenient to work over a ground field that is algebraically closed, but that can create some difficulties because such fields have non-noetherian valuation ring. We explain a technique based on deformation theory and formal algebraic spaces to reduce certain problems in relative non-archimedean geometry to the case of a discretely-valued ground field, and give an application to de Rham cohomology, answering a question of P. Scholze. This is joint work with O. Gabber. |

CENTER OF MATHEMATICAL SCIENCES AND APPLICATIONS RANDOM MATRIX & PROBABILITY THEORY SEMINAR: | Alan HammondUC BERKELEY |
KPZ Line Ensemble: a marriage of integrability and probability |

on Wednesday, March 25, 2015, at 11:30 am - 1:00 pm in Science Center 232 | ||

The KPZ equation, introduced by Kardar, Parisi and Zhang, is a stochastic PDE that models randomly evolving interfaces that are subject to constraining forces such as surface tension. It is anticipated to be a universal object, in the sense that many microscopic models will share the KPZ equation as an accurate asymptotic description of their late time behavior. This view is supported by extensive numerical evidence, recent experimental evidence involving liquid crystal instabilities, and a limited but growing body of mathematically rigorous work. In recent work arXiv:1312.2600 with Ivan Corwin, we present a new technique for the analysis of the KPZ equation. The solution to the equation is represented as the lowest indexed curve in an N-indexedensemble of curves, which we call a KPZ line ensemble. Curves within the ensemble enjoy a natural invariance under resampling, the H-Brownian Gibbs property, which property has the effect of energetically penalizing, but not absolutely forbidding, the crossing of adjacently indexed curves. This property is inherited from the O'Connell Yor semi-discrete continuum random polymer ensemble after a limiting procedure is applied. The H-Brownian Gibbs property is an integrable one, in the sense that the precursor O'Connell-Yor ensemble is known to enjoy it by virtue of this ensemble's algebraic structure. However, it also offers a powerful probabilistic tool for the analysis of the KPZ equation. Since the solution of this equation is embedded in a KPZ line ensemble, we may analyse it using the H-Brownian Gibbs property, and in this way derive significant new estimates regarding the regularity and local structure of the KPZ solution. As I will aim to explain, these new estimates are valid uniformly in the time parameter for the KPZ equation, even after a natural rescaling of the equation is undertaken which accesses the fluctuation behavior of the KPZ evolution |