The Evolving Classroom

Speaker: 

Christopher Jankowski

Institution: 

New York University

Time: 

Tuesday, January 20, 2015 - 4:00pm

Location: 

Rowland Hall 306

Online lectures and online classes are changing the landscape of math education. At New York University, we are creating a hybrid Calculus 1 course which will combine interactive online content with an in-class component involving lectures and problem sessions. We relate the structure of this course to that of other non-traditional calculus classes, and we discuss some potential advantages of this class over the standard lecture format.

Sage labs for Math 173AB: Introduction to Cryptology

Speaker: 

Christopher Davis

Institution: 

University of Copenhagen

Time: 

Thursday, January 8, 2015 - 4:00pm

Location: 

Rowland Hall 306

Cryptology provides a real-world application of many topics in number theory: integer factorization, primality testing, quadratic reciprocity, and elliptic curves, just to name a few. For these applications to cryptology, it is important to know whether or not a given procedure can be performed quickly. How does one convey to students that, for example, primality testing is relatively fast while integer factorization is relatively slow? We will present labs designed for UC Irvine Math 173AB: Introduction to Cryptology. These labs use Sage to introduce relevant cryptology topics, and at the same time they enable students to work with numbers at the limits of what their computers can handle computationally. For such numbers, the difference between a "fast" algorithm and a "slow" algorithm is striking, and as a result, students learn a key principle justifying the security of many modern cryptosystems.

ABP Estimate and Minkowski Integral Formulae

Speaker: 

Xiangwen Zhang

Institution: 

Columbia University

Time: 

Monday, January 12, 2015 - 4:00pm

Host: 

Location: 

Rowland Hall 306

The Alexandrov-Bakel'man-Pucci (ABP) estimate is one of
the most beautiful applications of geometric ideas in PDE and it is the
backbone of the regularity theory of fully nonlinear elliptic PDE. I will
start from the classical ABP estimate and then talk about its general-
ization on Riemannian manifolds, obtained in joint work with Yu Wang.
As applications, I will present results about the Harnack inequalities for
non-divergent PDE on manifolds and also an ABP approach to the clas-
sical Minkowski and Heintze-Karcher inequalities. In the second part of
the talk, I will give a brief overview of the classical Minkowski integral
formulae which are related to the divergence structure of some elliptic
operators. I will present the spacetime analogue of this type formula
I obtained with co-authors. Motivated by the problems from general
relativity, we consider the co-dimension two submanifolds in Lorentzian
spacetimes and establish some new Minkowski formulae in this setting.
 

Torsion and Galois Representations

Speaker: 

Davide Reduzzi

Institution: 

University of Chicago

Time: 

Monday, January 26, 2015 - 4:00pm

Host: 

Location: 

Rowland Hall 306

The absolute Galois group of a number field is a mysterious
object, that one can try to understand by means of its representations. It
is known that holomorphic cuspidal modular forms are, in a suitable sense,
a source of many p-adic Galois representations. More generally, it is
conjectured that also torsion classes in the coherent cohomology of
Shimura varieties have attached Galois representations, with prescribed
local properties. I will give an introduction to these themes, and present
results obtained in collaboration with Matthew Emerton and Liang Xiao
toward the conjecture.

The Log-Sobolev Inequality for Unbounded Spin Systems.

Speaker: 

Georg Menz

Institution: 

Stanford University

Time: 

Tuesday, January 13, 2015 - 4:00pm

Host: 

Location: 

Rowland Hall 306

The log-Sobolev inequality (LSI) is a very useful tool for analyzing
high-dimensional situations. For example, the LSI can be used for
deriving hydrodynamic limits, for estimating the error in stochastic
homogenization, for deducing upper bounds on the mixing times of Markov
chains, and even in the proof of the Poincaré conjecture by
Perelman. For most applications, it is crucial that the constant in the
LSI is uniform in the size of the underlying system. In this talk, we
discuss when to expect a uniform LSI in the setting of unbounded spin
systems. We will also explain a connection to the KLS conjecture.

Low-Rand Recovery: From Convex to Nonconvex Methods

Speaker: 

Xiaodong Li

Institution: 

The Wharton School at the University of Pennsylvania

Time: 

Friday, January 9, 2015 - 4:00pm

Host: 

Location: 

Rowland Hall 306

Low-rank structures are common in modern data analysis and signal processing, and they usually
play essential roles in various estimation and detection problems. It is challenging to recover the underlying low-rank structures reliably from corrupted or undersampled measurements. In this talk, we will introduce convex and nonconvex optimization methods for low-rank recovery by two examples.

The first example is community detection in network data analysis. In the literature, it has been formulated as a low-rank recovery problem, and then SDP relaxation methods can be naturally applied. However, the statistical advantages of convex optimization approaches over other competitive methods, such as spectral clustering, were not clear. We show in this talk that the methodology of SDP is robust against arbitrary outlier nodes with strong theoretical guarantees, while standard spectral clustering may fail due to a small fraction of outliers. We also demonstrate that a degree-corrected version of SDP works well for a real-world network dataset with a heterogeneous distribution of degrees.

Although SDP methods are provably effective and robust, the computational complexity is usually high and there is an issue of storage. For the problem of phase retrieval, which has various applications and can be formulated as a low-rank matrix recovery problem, we introduce an iterative algorithm induced by nonconvex optimization. We prove that our method converges reliably to the original signal. It requires far less storage and has much higher rate of convergence compared to convex methods

Zeros in Families of Polynomial Equations

Speaker: 

Nathan Kaplan

Institution: 

Yale University

Time: 

Tuesday, January 6, 2015 - 4:00pm

Host: 

Location: 

Rowland Hall 306

We will discuss several counting problems in number theory.  What is the probability that a random degree d monic polynomial with integer coefficients is irreducible? How many degree d algebraic number fields have discriminant at most X?  For a given field, how many orders does it contain of discriminant at most X?  We will also briefly discuss some statistical questions about rational points in families of elliptic curves.

We will then transition to talking about similar problems over finite fields.  In particular, we will focus on questions about rational points in families of curves and surfaces over a fixed F_q.  For example, if we take two plane cubic curves what is the probability that they intersect in exactly 9 F_q-rational points?

A Special Lagrangian Type Equation for Holomorphic Line Bundles

Speaker: 

Adam Jacob

Institution: 

Harvard University

Time: 

Wednesday, January 28, 2015 - 4:00pm

Host: 

Location: 

Rowland Hall 306

Consider a holomorphic line bundle L over a compact Kahler manifold. Motivated by mirror symmetry, I will define an equation on L that is the line bundle analogue of the special Lagrangian equation, which can be studied even when the base is not a Calabi-Yau manifold. I will show solutions are unique global minimizers of a positive functional. To address existence, I will introduce a line bundle analogue of the Lagrangian mean curvature flow, and prove convergence in certain cases. This is joint work with S.-T. Yau.

Interpolation Problems in Algebraic Geometry

Speaker: 

Jack Huizenga

Institution: 

University of Illinois at Chicago

Time: 

Friday, January 16, 2015 - 4:00pm

Host: 

Location: 

Rowland Hall 306

Classical Lagrangian interpolation states that one can always prescribe n+1 values of a single variable polynomial of degree n. This result paves the way for many beautiful generalizations in algebraic geometry. I will discuss a few of these generalizations and their relevance to important questions in mathematics. I will then discuss recent connections between interpolation problems and the birational geometry of Hilbert schemes of points and moduli spaces of vector bundles.

 

Fast Direct Methods for Structured Matrices

Speaker: 

Kenneth Ho

Institution: 

Stanford University

Time: 

Tuesday, January 27, 2015 - 4:00pm

Host: 

Location: 

Rowland Hall 306

Many linear systems arising in practice are governed by rank-structured matrices. Examples include PDEs, integral equations, Gaussian process regression, etc. In this talk, we describe our recent work on fast direct algorithms that exploit such structure. These methods are of particular interest due to their exceptional robustness and high capacity for information reuse. Our main technical achievement is a linear-complexity matrix factorization as a generalized LU decomposition. This factorization permits fast multiplication/inversion and furthermore supports rapid updating. We anticipate that such techniques will be game-changing in environments requiring the analysis of many right-hand sides or the solution of many closely related systems, such as in protein design or other inverse problems. Similar applications abound in computational statistics and data analysis.

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