Mathematics Dept.
  Events
Stanford
Thu 5 Jan 2017, 4:00pm SPECIAL
Probability Seminar
ESB 2012
Persistence of Gaussian stationary processes
ESB 2012
Thu 5 Jan 2017, 4:00pm-5:00pm

Abstract

Consider a real Gaussian stationary process, either on Z or on R. That is, a stochastic process, invariant under translations, whose finite marginals are centred multi-variate Gaussians. The persistence of such a process on [0,T] is the probability that it remains positive throughout this interval.

The relation between the decay of the persistence as T tends to infinity and the covariance function of the process has been investigated since the 1950s with motivations stemming from probability, engineering and mathematical physics. Nonetheless, until recently, good estimates were known only for particular cases, or when the covariance kernel of the process is either non-negative or summable.

In the talk we discuss a new spectral point of view on persistence which greatly simplifies its analysis. This is then used to obtain better bounds in a very general setting.

Joint work with Naomi Feldheim and Shahaf Nitzan.
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Stanford University
Fri 6 Jan 2017, 3:00pm
Department Colloquium
ESB 2012
Lattice models of magnetism: from magnets to antiferromagnets
ESB 2012
Fri 6 Jan 2017, 3:00pm-4:00pm

Abstract

The Ising model, and its generalisation, the Potts model, are two  classical graph-colouring models for magnetism and antiferromagnetism. Albeit their simple formulation, these models were instrumental in explaining many real-world magnetic phenomena and have found various applications in physics, biology and computer science.

While our understanding of these models as modeling magnets has been constantly improving since the early twentieth century, little progress was made in treatment of Potts antiferromagnets.

In the talk, after a historical introduction to magnets, antiferromagnets and their modeling, we will describe how application of modern combinatorial and probabilistic methods resulted in recent breakthroughs in the rigorous analysis of Potts antiferromagnets.

Note for Attendees

Tea and cookies will be served at 2:45 p.m.
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Bonn
Mon 9 Jan 2017, 1:30pm
Harmonic Analysis Seminar
Math 126
Entangled multilinear forms and applications
Math 126
Mon 9 Jan 2017, 1:30pm-2:30pm

Abstract

We discuss L^p estimates for some multilinear singular integral forms and their applications to sharp quantitative norm convergence of ergodic averages with respect to two commuting transformations, quantitative cancellation estimates for the simplex Hilbert transform, and a question on side lengths of corners in dense subsets of the Euclidean space.
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University of Illinois at Chicago
Mon 9 Jan 2017, 3:00pm
Algebraic Geometry Seminar
MATX 1102
Mordell-Weil Groups of Hitchin Systems
MATX 1102
Mon 9 Jan 2017, 3:00pm-4:00pm

Abstract

In this talk, I will discuss work giving in many cases a complete description of the group of rational sections of the relative Jacobian of a linear system of curves on a surface. By specializing to the case of spectral curves, we are able to determine very explicitly the group of sections of the Hitchin fibration. We will also discuss work in progress to extend this work to principal G-Higgs bundles for more general groups G.
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McGill University
Mon 9 Jan 2017, 3:00pm
Department Colloquium
ESB 2012
Random discrete structures: Phase transitions, scaling limits, and universality
ESB 2012
Mon 9 Jan 2017, 3:00pm-4:00pm

Abstract

The aim of this talk is to give an overview of some recent results in two interconnected areas:

*a) **Random graphs, random trees, and complex networks: *The last decade of the 20th century saw significant growth in the availability of empirical data on networks, and their relevance in our daily lives. This stimulated activity in a multitude of fields to formulate and study models of network formation and dynamic processes on networks to understand real-world systems.

One major conjecture in probabilistic combinatorics, formulated by statistical physicists using non-rigorous arguments and enormous simulations in the early 2000s, is as follows: for a wide array of random graph models on n vertices and degree exponent \tau>3, typical distance both within maximal components in the critical regime as well as on the minimal spanning tree on the giant component in the supercritical regime scale like n^{\frac{\tau\wedge 4 -3}{\tau\wedge 4 -1}}. In other words, the degree exponent determines the universality class the random graph belongs to. The mathematical machinery available at the time was insufficient for providing a rigorous justification of this conjecture.

More generally, recent research has provided strong evidence to believe that several objects, including
(i) components under critical percolation,
(ii) the vacant set left by a random walk, and
(iii) the minimal spanning tree,
constructed on a wide class of random discrete structures converge, when viewed as metric measure spaces, to some random fractals in the Gromov-Hausdorff-Prokhorov sense, and these limiting objects are universal under some general assumptions. We report on recent progress in proving these conjectures.

*b) Stochastic geometry:*  In contrast, less precise results are known in the case of spatial systems. We discuss a recent result concerning the length of spatial minimal spanning trees that answers a question raised by Kesten and Lee in the 90's, the proof of which relies on a variation of Stein's method and a quantification of the classical Burton-Keane argument in percolation theory.

Based on joint work with Louigi Addario-Berry, Shankar Bhamidi, Nicolas Broutin, Sourav Chatterjee, Remco van der Hofstad, and Xuan Wang.
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Colin Macdonald, Department of Mathematics, UBC
Tue 10 Jan 2017, 12:30pm
Scientific Computation and Applied & Industrial Mathematics
ESB 4133 (PIMS Lounge)
Parallel high-order time-stepping with revisionist integral deferred correction
ESB 4133 (PIMS Lounge)
Tue 10 Jan 2017, 12:30pm-1:30pm

Abstract

RIDC (revisionist integral deferred correction) methods are a class of time integrators well-suited to parallel computing.  RIDC methods can achieve high-order accuracy in wall-clock time comparable to forward Euler.  The methods use a predictor and multiple corrector steps.  Each corrector is lagged by one time step; the predictor and each of the correctors can then be computed in parallel.  This presentation introduces RIDC methods and demonstrates their effectiveness on some test problems.
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McGill
Tue 10 Jan 2017, 4:00pm SPECIAL
Probability Seminar
ESB 2012
Intrinsic geometry of critical discrete structures
ESB 2012
Tue 10 Jan 2017, 4:00pm-5:00pm

Abstract

Motivated by the presence of empirical data on a wide array of real-world networks, there has been an explosion in the number of random graph models proposed to explain various phenomenon observed in real-world systems including power law degree distribution and small world phenomenon. A major general theme in this area, since the time of Erdos and Renyi, has been understanding the properties of components in the critical regime. In the past five years, significant progress has been made in establishing scaling limits of critical random graphs and various constructs on random graphs when they are viewed as metric measure spaces, and understanding the general universality principles underlying such scaling limits. Further, striking connections between these questions and some of the central models of stochastic coalescence, and random interlacements and vacant set left by random walks have emerged. In this talk, we survey some recent results in this area and discuss general methodology for proving such results.
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UBC Mathematics
Tue 10 Jan 2017, 5:00pm SPECIAL
Coach House, Green College, UBC
What is a Probabilist?
Coach House, Green College, UBC
Tue 10 Jan 2017, 5:00pm-6:30pm

Details

 Everyone understands the statement: "The probability of rain tomorrow is .90, or 90%. For a probabilist, many questions have answers of this type. The problem is to find that number.

Note for Attendees

 reception to follow talk
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Wed 11 Jan 2017, 2:45pm SPECIAL
PIMS Lounge (ESB 4133)
PIMS Afternoon Tea (resumes for Term 2)
PIMS Lounge (ESB 4133)
Wed 11 Jan 2017, 2:45pm-3:10pm

Details

The tea runs until Wednesday, April 5th, 2017.
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Microsoft Research
Thu 12 Jan 2017, 4:00pm SPECIAL
Probability Seminar
ESB 2012
High-dimensional random geometric graphs
ESB 2012
Thu 12 Jan 2017, 4:00pm-5:00pm

Abstract

I will talk about a random geometric graph model, where connections between vertices depend on distances between latent d-dimensional labels. We are particularly interested in the high-dimensional case when d is large. Upon observing a graph, we want to tell if it was generated from this geometric model, or from the Erdos-Renyi model. We show that there exists a computationally efficient procedure to do this which is almost optimal (in an information-theoretic sense). The key insight is based on a new statistic which we call "signed triangles". To prove optimality we compute the total variation distance between Wishart matrices and the Gaussian Orthogonal Ensemble. This is joint work with Sebastien Bubeck, Jian Ding, Ronen Eldan, and Jacob Richey.
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Microsoft Research
Fri 13 Jan 2017, 3:00pm
Department Colloquium
ESB 2012
Statistical inference in networks and genomics
ESB 2012
Fri 13 Jan 2017, 3:00pm-4:00pm

Abstract

From networks to genomics, large amounts of data are increasingly available and play critical roles in helping us understand complex systems. Statistical inference is crucial in discovering the underlying structures present in these systems, whether this concerns the time evolution of a network, an underlying geometric structure, or reconstructing a DNA sequence from partial and noisy information. In this talk I will discuss several fundamental detection and estimation problems in these areas.

I will present an overview of recent developments in source detection and estimation in randomly growing graphs. For example, can one detect the influence of the initial seed graph? How good are root-finding algorithms? I will also discuss inference in random geometric graphs: can one detect and estimate an underlying high-dimensional geometric structure? Finally, I will discuss statistical error correction algorithms for DNA sequencing that are motivated by DNA storage, which aims to use synthetic DNA as a high-density, durable, and easy-to-manipulate storage medium of digital data
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Brown
Mon 16 Jan 2017, 3:00pm SPECIAL
Institute of Applied Mathematics
ESB 2012
Functional interpretation for transverse arches of human foot
ESB 2012
Mon 16 Jan 2017, 3:00pm-4:00pm

Abstract

Fossil record indicates that the emergence of arches in human ancestral feet coincided with a transition from an arboreal to a terrestrial lifestyle. Propulsive forces exerted during walking and running load the foot under bending, which is distinct from those experienced during arboreal locomotion. I will present mathematical models with varying levels of detail to illustrate a simple function of the transverse arch. Just as we curve a dollar bill in the transverse direction to stiffen it while inserting it in a vending machine, the transverse arch of the human foot stiffens it for bending deformations. A fundamental interplay of geometry and mechanics underlies this stiffening -- curvature couples the soft out-of-plane bending mode to the stiff in-plane stretching deformation. In addition to presenting a functional interpretation of the transverse arch of the foot, this study also identifies a classification of flat feet based on the skeletal geometry and mechanics.

Note for Attendees

Reception before the talk in ESB 4133 (PIMS lounge). This is the annual IAM alumni lecture. 
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San Francisco State University
Mon 16 Jan 2017, 3:00pm
Algebraic Geometry Seminar
MATX 1102
Double Ramification Cycles and Tautological Relations
MATX 1102
Mon 16 Jan 2017, 3:00pm-4:00pm

Abstract

Tautological relations are certain equations in the Chow ring of the moduli space of curves.  I will discuss a family of such relations, first conjectured by A. Pixton, that arises by studying moduli spaces of ramified covers of the projective line.  These relations can be used to recover a number of well-known facts about the moduli space of curves, as well as to generate very special equations known as topological recursion relations.  This is joint work with various subsets of S. Grushevsky, F. Janda, X. Wang, and D. Zakharov.

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San Francisco State University
Mon 16 Jan 2017, 4:15pm
Algebraic Geometry Seminar
MATX 1102
Genus-One Landau-Ginzburg/Calabi-Yau Correspondence
MATX 1102
Mon 16 Jan 2017, 4:15pm-5:15pm

Abstract

First suggested by Witten in the early 1990's, the Landau-Ginzburg/Calabi-Yau correspondence studies a relationship between spaces of maps from curves to the quintic 3-fold (the Calabi-Yau side) and spaces of curves along with 5th roots of their canonical bundle (the Landau-Ginzburg side). The correspondence was put on a firm mathematical footing in 2008 when Chiodo and Ruan proved a precise statement for the case of genus-zero curves, along with an explicit conjecture for the higher-genus correspondence. In this talk, I will begin by describing the motivation and the mathematical formulation of the LG/CY correspondence, and I will report on recent work with Shuai Guo that verifies the higher-genus correspondence in the case of genus-one curves.

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Ben Adcock
Department of Mathematics, SFU
Tue 17 Jan 2017, 12:30pm
Scientific Computation and Applied & Industrial Mathematics
ESB 4133 (PIMS Lounge)
Sparse polynomial approximation of high-dimensional functions
ESB 4133 (PIMS Lounge)
Tue 17 Jan 2017, 12:30pm-1:30pm

Abstract

Many problems in scientific computing require the approximation of smooth, high-dimensional functions from limited amounts of data.  For  instance, a typical problem in uncertainty quantification involves identifying the parameter dependence of the output of a computational model. Complex physical systems involve models with multiple parameters, resulting in multivariate functions of many variables. Although the amount of data may be large, the curse of dimensionality
essentially prohibits collecting or processing sufficient data to approximate the unknown function using classical techniques.

In this talk, I will give an overview of the approximation of smooth, high-dimensional functions using sparse polynomial expansions.  I will focus on the application of techniques from compressed sensing to this problem, and discuss the extent to which such approaches overcome the curse of dimensionality. If time, I will also discuss several extensions, including dealing with corrupted and/or unstructured data, the effect of model error and incorporating additional information such as gradient data. I will also highlight several challenges and open problems.

This is joint work with Casie Bao, Simone Brugiapaglia and Yi Sui (SFU).
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Kseniya Garaschuk and Eric Cytrynbaum
University of the Fraser Valley and UBC
Tue 17 Jan 2017, 12:30pm
Lunch Series on Teaching & Learning
Irving K. Barber Learning Ctr Room 461
Collaborative exams in large university courses
Irving K. Barber Learning Ctr Room 461
Tue 17 Jan 2017, 12:30pm-1:30pm

Abstract

As we use more and more group work in our classes, should we consider introducing it into our assessments? One model that has been used are so-called two-stage assessments, where students first complete and turn in the questions individually and then, working in small groups, answer the same questions again. This technique was first introduced in the UBC Faculty of Science in 2009 and is now being used in at least 20 science courses. 
 
In this session, we will discuss a study of feasibility and effectiveness of two-stage quizzes as introduced into two mathematics courses at UBC with a total of 834 students. We examine both short and long term retention resulting from introducing group assessments, we analyze results of collaborative learning based on question type and group composition. Finally, we present student and instructor feedback as well as discuss future directions of implementation and research. 
 
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UBC Math
Tue 17 Jan 2017, 4:00pm SPECIAL
Probability Seminar
ESB 2012
Boundary Harnack principle for diffusions
ESB 2012
Tue 17 Jan 2017, 4:00pm-5:00pm

Abstract

 The boundary Harnack principle (BHP) is a fundamental tool to understand the behaviour of positive harmonic functions near the boundary of a domain. For instance, the BHP implies a concrete description of the Martin boundary of a domain in geometric terms. Other applications of BHP include Carleson estimate, Fatou's theorem, and heat kernel estimates for diffusions killed upon exiting a domain. In this talk, I will discuss a recent extension of BHP that provides new examples of diffusions satisfying BHP even in  R^n.
 
This is joint work with Martin Barlow.
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UBC Math
Wed 18 Jan 2017, 3:00pm SPECIAL
Department Colloquium
ESB 2012 (note special day)
Stability of elliptic Harnack inequality
ESB 2012 (note special day)
Wed 18 Jan 2017, 3:00pm-4:00pm

Abstract

Harnack inequalities have proved to be a powerful tool in PDE (regularity estimates), geometry (geometric flows) and probability (heat kernel estimates). In the early 1990s Grigor'yan and Saloff-Coste gave a characterisation of the parabolic Harnack inequality (PHI). This characterisation implies that PHI is stable under perturbations (quasi-isometries). In this talk, I will provide an introduction to Harnack inequalities and discuss the stability of elliptic Harnack inequality. 

This is joint work with Martin Barlow.

Note for Attendees

Tea and cookies will be served before this special colloquium.
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Hao Shen
Columbia University
Thu 19 Jan 2017, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
seminar has been cancelled.
CANCELLED: Scaling limits of open ASEP and ferromagnetic Glauber dynamics
seminar has been cancelled.
Thu 19 Jan 2017, 3:30pm-4:30pm

Abstract

 
We discuss two recent scaling limit results for discrete dynamics converging to stochastic PDEs. The first is the asymmetric simple exclusion process in contact with sources and sinks at boundaries, called Open ASEP.  We prove that under weakly asymmetric scaling the height function converges to the KPZ equation with Neumann boundary conditions. The second is the Glauber dynamics of the Blume-Capel model (a generalization of Ising model), in two dimensions with Kac potential. We prove that the averaged spin field converges to the stochastic quantization equations. The main purpose of this talk is to discuss the general issues one needs to address when passing from discrete to continuum, the common challenge in the proofs of such scaling limit theorems, and how we overcome these difficulties in the two specific models. (Based on joint works with Ivan Corwin and Hendrik Weber)
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Hao Shen
Columbia University
Fri 20 Jan 2017, 3:00pm
Department Colloquium
seminar has been cancelled.
CANCELLED: Singular Stochastic Partial Differential Equations - How do they arise and what do they mean?
seminar has been cancelled.
Fri 20 Jan 2017, 3:00pm-4:00pm

Abstract

 
Systems with random fluctuations are ubiquitous in the real world. Stochastic PDEs are default models for these random systems, just as PDEs are default models for deterministic systems. However, a large class of such stochastic PDEs were poorly understood until very recently: the presence of very singular random forcing as well as nonlinearities render it challenging to interpret what one even means by a ``solution". The recent breakthroughs by M. Hairer, M. Gubinelli and other researchers including the speaker not only established solution theories for these singular SPDEs, but also led to an explosion of new questions. These include scaling limits of random microscopic models, development of numerical schemes, ergodicity of random dynamical systems and a new approach to quantum field theory. In this talk we will discuss the main ideas of the recent solution theories of singular SPDEs, and how these SPDEs arise as limits of various important physical models.
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Toronto
Mon 23 Jan 2017, 3:00pm SPECIAL
Institute of Applied Mathematics
ESB 2012
Matrix regulation of stem cell fate
ESB 2012
Mon 23 Jan 2017, 3:00pm-4:00pm

Abstract

Stem cell fate is regulated by cues from the cellular microenvironment, including biophysical and biochemical cues presented by the extracellular matrix. Matrix regulation of cell fate has broad implications from disease to regeneration. In this talk, I will discuss our work aimed at determining how biophysical and biochemical cues from the matrix act to drive the fate and function of mesenchymal stem cells. In particular, I will discuss how the pathological differentiation of heart valve progenitor cells is modulated by the extracellular matrix and how we are using microdevices to dissect how matrix mechanical and biochemical cues act in concert to regulate mesenchymal stem cell differentiation and tissue regeneration.

Note for Attendees

Reception preceding the talk in ESB 4133 (the PIMS lounge). This is part of the IAM/PIMS distinguished colloquium series. 
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Rennes / PIMS
Mon 23 Jan 2017, 3:00pm
Algebraic Geometry Seminar
MATX 1102
On Grothendieck rings in real geometry
MATX 1102
Mon 23 Jan 2017, 3:00pm-4:00pm

Abstract

The study of Grothendieck rings of varieties in the context of real algebraic geometry has begun since the apparition of motivic integration. Several such rings are of interest, depending notably on the class of functions we are interested in. We will discuss recent progress in the cases of real algebraic varieties and of arc-symmetric sets.

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Max Planck Institute Bonn
Tue 24 Jan 2017, 3:00pm SPECIAL
Diff. Geom, Math. Phys., PDE Seminar
MATH 126
Chiral differential operators and the curved beta-gamma system
MATH 126
Tue 24 Jan 2017, 3:00pm-4:00pm

Abstract

Chiral differential operators (CDOs) are a vertex algebra analog of the associative algebra of differential operators. Originally introduced by mathematicians, Witten explained how CDOs arise as the perturbative part of the curved beta-gamma system with target X. I will describe recent work with Gorbounov and Williams in which we construct the BV quantization of this theory and use a combination of factorization algebras and formal geometry to recover CDOs. At the end, I hope to discuss how the techniques we developed apply to a broad class of nonlinear sigma models, including source manifolds of higher dimension.
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Center for Geometry and Physics/ Pohang Univ. of Sci. and Tech. Korea.
Tue 24 Jan 2017, 4:10pm SPECIAL
MATH 1102 Topology/Algebra/Math Physics seminar [We will start at 4:10pm]
A quantization of the unipotent fundamental group
MATH 1102 Topology/Algebra/Math Physics seminar [We will start at 4:10pm]
Tue 24 Jan 2017, 4:10pm-5:10pm

Details

We construct two homotopy functors from the category of homotopy commutative algebras to the category of pro-unipotent group and the category of completed commutative Hopf algebras such that

(1) the group is isomorphic to the pro unipotent fundamental group of smooth connected and based manifold (M,*) and
(2) the Hopf algebra is that of homotopy functionals defined by Chen’s iterated path integrals  over closed and based loops on M

if the homotopy commutative algebra is quasi-isomorphic to the algebra of differential forms on M.

These constructions can be quantized in appropriate sense after interpreting them as doing “classical field theory” such that the pro unipotent fundamental group is the structure in the space of all "classical expectations" and the completed Hopf algebra is the algebra of  “classical observables”.  The quantization involves a generalized deformation quantization of homotopy commutative algebra into topologically free homotopy associative algebra along the direction of compatible homotopy Lie algebra.
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UBC
Wed 25 Jan 2017, 1:45pm
Mathematical Biology Seminar
PIMS, ESB 4th floor
Multi-scale modeling of vesicular release at neuronal synapses.
PIMS, ESB 4th floor
Wed 25 Jan 2017, 1:45pm-2:45pm

Abstract

Binding of molecules, ions or proteins to small target sites is a generic step of cell activation. This process relies on rare stochastic events where a particle located in a large bulk has to find small and often hidden targets. I will present in this talk a hybrid discrete-continuum model that takes into account both a stochastic regime governed by rare events and a continuous regime in the bulk, in the context of vesicular release at chemical synapses.

In a first part, I computed the mean time for a Brownian particle to arrive at a narrow opening defined as the small cylinder joining two tangent spheres. This models the binding of calcium ions on the SNARE complex, a process that triggers vesicular release. Using this result, I developed a model to study how vesicles and calcium channels organization shape such process.
In a second part, I will present a model for the pre-synaptic terminal built using the results described above. This model was formulated in an initial stage as a reaction-diffusion problem in a confined microdomain, where Brownian particles have to bind to small target sites. I coarse-grained this model into a system of mass action equations coupled to a set of Markov equations, which allows to obtain analytical results and to realize fast stochastic simulations.
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Max Planck Institute Bonn
Wed 25 Jan 2017, 3:00pm SPECIAL
Department Colloquium
ESB 2012 (note special day)
From Feynman diagrams to commutative diagrams
ESB 2012 (note special day)
Wed 25 Jan 2017, 3:00pm-4:00pm

Abstract

Factorization algebras are local-to-global objects that play a role in quantum field theory akin to the role of sheaves in geometry: they conveniently organize complicated information. In the talk I will introduce this notion, give some concrete examples, and then explain how factorization algebras mediate between QFT and higher algebra. An important example will be Chern-Simons theory; ongoing work with Costello and Francis recovers quantum groups with formal parameter by combining Koszul duality with Feynman diagrams.

Note for Attendees

Tea and cookies will be served in the PIMS Lounge ESB 4133 at 2:45 p.m. before this special colloquium.
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University of Chicago
Wed 25 Jan 2017, 4:15pm SPECIAL
Topology and related seminars
ESB 4133 (PIMS Lounge)
Quantitative Nullcobordism and the (in)effectiveness of algebraic topology.
ESB 4133 (PIMS Lounge)
Wed 25 Jan 2017, 4:15pm-5:15pm

Abstract

Topology is full of ineffective arguments constructing objects and equivalences by algebra.

One of the great early achievements of algebraic topology was the work of Thom, followed by Milnor and Wall, on cobordism theory, which describes when a compact smooth (oriented) manifold is the boundary of some compact manifold with boundary. This method is typical of the problems that arise in the use of algebraic methods and is an early example of one of the dominant philosophies of geometric topology. The question we study is to what extent the complexity of a manifold can be used to bound, when it exists, the minimum necessary complexity of something that it bounds.

The goal of this talk is to explain generally some of the issues of making topology less ineffective.

We shall show that there are polynomial size nullcobordisms in a suitable sense. This is joint work with Greg Chambers, Dominic Dotterer and Fedor Manin.
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Harold Williams
University of Texas at Austin
Thu 26 Jan 2017, 3:30pm SPECIAL
Diff. Geom, Math. Phys., PDE Seminar
MATH 102
Cluster Theory of the Coherent Satake Category
MATH 102
Thu 26 Jan 2017, 3:30pm-5:00pm

Abstract

We discuss recent work showing that in type A_n the category of equivariant perverse coherent sheaves on the affine Grassmannian categorifies the cluster algebra associated to the BPS quiver of pure N=2 gauge theory. Physically, this can be understood as a statement about line operators in this theory, following ideas of Gaiotto-Moore-Neitzke, Costello, and Kapustin-Saulina -- in short, coherent IC sheaves are the precise algebro-geometric counterparts of Wilson-'t Hooft line operators. The proof relies on techniques developed by Kang-Kashiwara-Kim-Oh in the setting of KLR algebras. A key moral is that the appearance of cluster structures is in large part forced by the compatibility between chiral and tensor structures on the category in question (i.e. by formal features of holomorphic-topological field theory). This is joint work with Sabin Cautis.

Note for Attendees

Refreshments will be served at 3:15pm in the MATH 125 Lounge Area.
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Harold Williams
University of Texas at Austin
Fri 27 Jan 2017, 3:00pm SPECIAL
Department Colloquium
ESB 2012
Canonical Bases and Physical Mathematics
ESB 2012
Fri 27 Jan 2017, 3:00pm-4:00pm

Abstract

The discovery of so-called canonical (or crystal) bases of quantum groups by Lusztig and Kashiwara was one of the major milestones of Lie theory during the past three decades. The larger theory of canonical bases that grew out of their work (and others') has continually revealed surprising connections to other areas of mathematics such as algebraic and symplectic geometry. In this talk we sample some of these connections, for example to the geometry of character varieties of surfaces, and explain how ideas from theoretical physics provide an organizing framework for many mathematically disparate results. A central role is played by the flexible combinatorial language of cluster algebras, which provides a useful lingua franca for the subject.

Note for Attendees

Refreshments will be served at 2:45 p.m. before this special colloquium.
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Martin Fraas
Institute for Theoretical Physics, KU Leuven
Mon 30 Jan 2017, 3:00pm
Department Colloquium
ESB 2012
Adiabatic Theory: From Spinning Top to Gapped Matter
ESB 2012
Mon 30 Jan 2017, 3:00pm-4:00pm

Abstract

Adiabatic theory studies dynamical systems featuring separation of motion into slow and fast components. When the system is driven these components start mixing. The basic premise of adiabatic theory is that this mixing is small for slow driving. Furthermore, the resulting driven motion is irreversible in the slow component but reversible in the fast component. In this talk, I first illustrate this phenomena on a spinning top and state the classical adiabatic theorem of quantum mechanics. Then, I will show extensions of the theory to abstract Banach spaces and stochastic equations, culminating with an adiabatic theorem for extended systems where the separation holds uniformly in the volume of the system. An important corollary of this theorem is a proof of the Kubo formula for gapped interacting systems.
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University of Washington
Mon 30 Jan 2017, 3:00pm
Algebraic Geometry Seminar
Math 126
Fiber Powers and Uniformity
Math 126
Mon 30 Jan 2017, 3:00pm-4:00pm

Abstract

In 1997 Caporaso, Harris and Mazur, motivated by uniformity results in Diophantine Geometry, proposed a conjecture about fibered powers of families of varieties of general type. In particular they conjecture that, if X -> B is a family whose general fiber is a variety of general type, then for large n, the n-th fiber power of X over B dominates a variety of general type. The conjecture has been proved by Abramovich and used to deduce interesting results on the distribution of rational points on projective varieties. I will discuss recent work, joint with Kenny Ascher (Brown), generalizing this to pairs (X,D) of a projective scheme and a divisor, and the new challenges that arise when one tries to obtain analogous results for the distribution of integral points in quasi projective varieties.
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Martin Fraas
Institute for Theoretical Physics, KU Leuven
Tue 31 Jan 2017, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012
On Products of Correlated Matrices Originating in the Statistical Structure of Quantum Mechanics
ESB 2012
Tue 31 Jan 2017, 3:30pm-4:30pm

Abstract

Statistics of measurement outcomes in quantum mechanics is described by a map that associates a matrix to each possible measurement outcome. The probability of this outcome is then given by a trace of this matrix. Probability of a sequence of measurement outcomes is computed in the same way from a product of associated matrices. In this talk I will describe two results related to this setting. A theorem giving optimal conditions for uniqueness of the associated invariant measure on the projective sphere, and a theorem describing large deviation theory in the case when the matrices commute. The latter problem received lots of recent attention following experiments of S.~Haroche.
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Tue 31 Jan 2017, 4:00pm
ESB 4127
Talk practice
ESB 4127
Tue 31 Jan 2017, 4:00pm-5:00pm

Details


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