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 Events
Mathematischen Instituts, Universität Münster
Thu 1 Nov 2012, 3:30pm
Number Theory Seminar
room MATH 126
The exterior square of a Lie algebra
room MATH 126
Thu 1 Nov 2012, 3:30pm-4:30pm

Abstract

The Lie bracket of a Lie algebra L induces a linear map L ^ L → L. When can the kernel of this map be generated by indecomposable vectors of the form y with [x,y] = 0? This seemingly elementary question does not seem to be tractable by elementary methods. For semisimple Lie algebras over the complex numbers Kostant has given a positive answer by means of representation theory. I will explain why a number theorist is interested in this question over fields of positive characteristics. Here representation theoretic arguments become very difficult. I will sketch a solution for the Lie algebra of matrices.
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University of Chicago
Fri 2 Nov 2012, 3:00pm
Department Colloquium
MATX 1100
Fractal properties of random curves arising in statistical physics
MATX 1100
Fri 2 Nov 2012, 3:00pm-4:00pm

Abstract

I will consider two models arising in statistical physics, loop-erased random walk and the self-avoiding random walk, and discuss the scaling limit which is curve of fractal dimension. I will focus on two dimensions where  we have a good description of this process and I will discuss recent progress in understanding the fine structure of the paths.  I also plan to discuss open problems in three dimensions.
 
This talk is intended for a general mathematics audience.  
 
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Ben Gurion University
Mon 5 Nov 2012, 3:00pm
Algebraic Geometry Seminar
ESB 2012
Residues and Duality for Schemes and Stacks
ESB 2012
Mon 5 Nov 2012, 3:00pm-4:00pm

Abstract

Let K be a regular noetherian commutative ring. I consider finite type commutative K-algebras and K-schemes. I will begin by explaining the theory of rigid residue complexes on K-algebras, that was developed by J. Zhang and myself several years ago. Then I will talk about the geometrization of this theory: rigid residue complexes on K-schemes and their functorial properties. For any map between K-schemes there is a rigid trace homomorphism (that usually does not commute with the differentials). When the map of schemes is proper, the rigid trace does commute with the differentials (this is the Residue Theorem), and it induces Grothendieck Duality.

Then I will move to finite type Deligne-Mumford K-stacks. Any such stack has a rigid residue complex on it, and for any map between stacks there is a trace homomorphism. These facts are rather easy consequences of the corresponding facts for schemes, together with etale descent. I will finish by presenting two conjectures, that refer to Grothendieck Duality for proper maps between DM stacks. A key condition here is that of tame map of stacks.

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UBC
Tue 6 Nov 2012, 3:00pm
Stochastic Dynamics Working Group
LSK 306
Competition of noise sources in delay dynamics.
LSK 306
Tue 6 Nov 2012, 3:00pm-4:00pm

Abstract

Transient or unstable behavior is often ignored in considering long time dynamics in the deterministic world. However, stochastic effects can change the picture dramatically, so that the transients can dominate the long range behavior.  This talk will  contrast the  effects of different noise sources in certain systems with delayed feedback, driving both detrimental and beneficial noise-stabilized behaviors that are composed of transients from the deterministic system. The analysis starts with a canonical model of delayed feedback in mechanical systems and is extended to consider questions about biological applications.

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WIS
Wed 7 Nov 2012, 3:00pm
Probability Seminar
ESB 2012
Asymptotic behavior of the Cheeger constant of super-critical percolation in the square lattice
ESB 2012
Wed 7 Nov 2012, 3:00pm-4:00pm

Abstract

Isoperimetry is a well studied subject that have found many applications in geometric measure theory (e.g. concentration of measure, heat-kernal estimates, mixing time, etc.)  Consider the super-critical bond percolation
on \mathbb{Z}^d (the d-dimensional square lattice), and \phi_n the Cheeger constant of the super-critical percolation cluster restricted to the finite box [-n,n]^d. Following several papers that proved that the leading
order asymptotics of  \phi_n is of the order 1/n, Benjamini conjectured a limit to n\phi_n exists. As a step towards this goal, Rosenthal and myself have recently shown that Var(n\phi_n)< C n^{2-d}. This implies
concentration of n\phi_n around its mean for dimensions d>2.
 
Consider the super-critical bond percolation on \mathbb{Z}^2 (the square lattice). We prove the Cheeger constant of the super-critical percolation cluster restricted to finite boxes scale a.s to a deterministic quantity.
This quantity is given by the solution to the isoperimetric problem on \mathbb{R}^2 with respect to a specific norm. The unique set which gives the solution, is the normalized Wulff shape for the same norm.
 
Joint work with Marek Biskup, Oren Louidor and Ron Rosenthal.

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UBC
Wed 7 Nov 2012, 3:00pm
Undergraduate Colloquium
MATH 104
The Transatlantic Cable
MATH 104
Wed 7 Nov 2012, 3:00pm-4:00pm

Abstract

UBC/UMC is the Undergraduate Mathematics Colloquium at UBC.

These talks are for undergraduates interested in mathematics and mathematical research. They are put on by professors, senior graduate students and visitors to the department. Graduate students are also invited to attend.

Title:
The Transatlantic Cable

Abstract:
A look at some fundamental partial differential equations through the lens of time.
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University of Arkansas
Wed 7 Nov 2012, 4:00pm
Harmonic Analysis Seminar
Math 126 (Note unusual date and time of event)
Box_b-heat kernel decay and the Fourier transform
Math 126 (Note unusual date and time of event)
Wed 7 Nov 2012, 4:00pm-5:00pm

Abstract

In this talk, I plan to discuss Gaussian decay of the Box_b-heat kernel on polynomial models in C^2 and explore how to recover exponential decay via a Fourier transform.
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UBC
Thu 8 Nov 2012, 11:00am
Algebraic Groups and Related Structures
Math 126
An introduction to Chow groups III
Math 126
Thu 8 Nov 2012, 11:00am-12:30pm

Abstract

 We will continue past discussions on the basic ideas of Chow theory. We review the basic properties that make the Chow functor to be an oriented Borel-Moore homology theory, and then we will discuss some further topics including their equivariant version, their operational version and their universal version.
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Eric Cytrynbaum
UBC
Thu 8 Nov 2012, 12:30pm
Mathematical Education
MATH 126
Lunch Series on Teaching and Learning: Efficient coordination and running of a large first year course - a survey of useful online tools
MATH 126
Thu 8 Nov 2012, 12:30pm-1:30pm

Abstract

In this talk, I'll present the tools that I've been using to run one of our department's multi-section first year calculus courses (Math 102). I'll discuss the how and why of what we have set up including two wikis (one public, one private), WeBWorK, and a third party forum called Piazza.com.
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Simon Fraser University
Thu 8 Nov 2012, 3:30pm
Number Theory Seminar
room MATH 126
L^4 norms, and a new autocorrelation record
room MATH 126
Thu 8 Nov 2012, 3:30pm-4:30pm

Abstract

Littlewood was interested in finding polynomials with ±1 coefficients (now called Littlewood polynomials) having a small ratio of L4 norm to L2 norm on the complex unit circle. This is equivalent to the problem of finding binary sequences with small mean-squared aperiodic autocorrelation, important in engineering and physics. The best known examples are polynomials whose coefficients are supplied by complex-valued characters of finite fields. As of 1988, the lowest known asymptotic ratio of norms was (7/6)1/4, and this was conjectured to be the lowest possible. In 2004, Borwein, Choi, and Jedwab analyzed a construction that reliably produces polynomials with a low ratio of norms, and they conjectured that these polynomials do indeed break the 1988 record. We prove that their conjecture is true, and the character sum methods we have devised settle further conjectures. (joint work with Jonathan Jedwab and Kai-Uwe Schmidt)
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Tim Rees
The University of Victoria
Tue 13 Nov 2012, 12:30pm
Scientific Computation and Applied & Industrial Mathematics
ESB 4133
The numerical solution of the Taylor-Goldstein equation
ESB 4133
Tue 13 Nov 2012, 12:30pm-1:30pm

Abstract

The Taylor-Goldstein (TG) equation governs the linear stability of internal gravity waves in an inviscid, stratified shear flow. It can be used to predict the most unstable modes in a flow, and is therefore useful in a wide range of geophysical applications. The TG equation is a singular eigenvalue problem, and attempting to solve it numerically for general velocity and stratification profiles is technically challenging. I will describe an ongoing project where the TG equation is used to understand processes in the atmospheric boundary layer. The focus of the talk will be on the numerical methods that are used to solve the TG equation, and some of the difficulties that arise in their application.

Note for Attendees

Pizza will be provided.
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ENS Ulm (Paris)
Tue 13 Nov 2012, 3:00pm
Algebraic Geometry Seminar
MATX 1102
On the birational geometry of the parameter space for codimension 2 complete intersections
MATX 1102
Tue 13 Nov 2012, 3:00pm-4:00pm

Abstract

Codimension 2 complete intersections in P^N have a natural parameter space that is a projective bundle over a projective space given by the data of the two equations. 

In this talk, we will be interested in the birational geometry of this parameter space. In particular, is it a Mori dream space?  If this is the case, is it possible to describe its MMP explicitly?  We will give motivations for these questions and answers in particular cases.

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University of Utah/Imperial College London
Tue 13 Nov 2012, 4:00pm
Algebraic Geometry Seminar
MATX 1102
On the Crepant Transformation Conjecture
MATX 1102
Tue 13 Nov 2012, 4:00pm-5:00pm

Abstract

Let X and X' be two smooth Deligne-Mumford stacks. We call dash arrow X-->X' a Crepant Transformation if there exists a third smooth Deligne-Mumford stack Y and two morphisms \phi:Y-> X,  \phi': Y-> X' such that the pullbacks of canonical divisors are equivalent, i.e. \phi^*K_{X}\cong \phi'^*K_{X'}.  The crepant transformation conjecture says that the Gromov-Witten theory of X and X' is equivalent if X-->X' is a crepant transformation. This conjecture was well studied in two cases:  the first one is the case when X and X' are both smooth varieties; the other is the case that there is a real morphism X-> |X'| to the coarse moduli space of X', resolving the singularities of X'.  In this talk I will present some recent progress for this conjecture, especially in the case when both  X and X' are smooth Deligne-Mumford stacks.

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UBC
Wed 14 Nov 2012, 3:00pm
Topology and related seminars
ESB 4127
Cohomology and K-theory of Crystallographic Groups I
ESB 4127
Wed 14 Nov 2012, 3:00pm-4:00pm

Abstract

We discuss the general problem of computing the cohomology and topological K-theory for classifying spaces of crystallographic groups. Integral computations will be provided for groups with prime order holonomy.
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Yan-Xia Ren
Peking University
Wed 14 Nov 2012, 3:00pm
Probability Seminar
ESB 2012
Strong law of large numbers for superprocesses
ESB 2012
Wed 14 Nov 2012, 3:00pm-4:00pm

Abstract

I will talk about our recent progress on strong law of large numbers for some  classes of superprocesses X corresponding to \partial_t u_t=A u_t+\beta u_t-\psi(u_t) in a domain D of {\bf R}^d, where A is the  generator of a diffusion or a stable process, and the branching mechanism \psi(x,\lambda)=\beta\lambda+a\lambda^2+\int_0^\infty (e^{-\lambda r}-1+\lambda r)n(x, {\rm d}r) satisfies \sup_{x\in D}\int_0^\infty (r\wedge r^2) n(x,{\rm
d}r)<\infty .

Recently many people  have established limit theorems for branching Markov processes or super-processes using the principal eigenvalue and ground state of the linear part of the characteristic equations. All the papers above assumed that the processes satisfy a second moment condition or  a (1+\theta)-moment condition with \theta>0. Asmussen and Hering (1976)  established a Kesten-Stigum L\log L type theorem for a class branching diffusion processes under a condition which is later called a positive regular property. We established Kesten-Stigum L\log L type theorems for superdiffusions and branching Hunt processes respectively.

Recently, we established strong law of large numbers for a class of superdiffusions in a domain D of {\bf R}^d with general branching mechanism, and for super-\alpha-stable processes in {\bf R}^d with \psi(x, \lambda)=-\beta\lambda+\eta\lambda^2, where \beta and \eta are positive constants. The main tool is the stochastic integral representation of superprocesses.

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UBC
Thu 15 Nov 2012, 11:00am
Algebraic Groups and Related Structures
Math 126
Computation of canonical dimensions
Math 126
Thu 15 Nov 2012, 11:00am-12:30pm

Abstract

 We compute the canonical dimension of a few objects, including the canonical dimension quadratic forms.
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University of Washington
Thu 15 Nov 2012, 12:30pm
Number Theory Seminar
room MATH 126
Warm-up talk: Modular forms and Galois representations
room MATH 126
Thu 15 Nov 2012, 12:30pm-1:30am

Abstract

This informal talk will provide background material for graduate students and others who will be attending the seminar talk "Level-lowering for Galois representations" later today. Three main background items for that seminar will be:
  1. Modular forms are analytic/geometric gadgets with lots of interesting arithmetic properties.
  2. Modular forms have a weight and a level.
  3. We can attach a Galois representation to an eigenform.
In this talk we will examine these three items in more detail.

Note for Attendees

Attendees are welcome to bring their lunch with them to this informal talk.
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University of Washington
Thu 15 Nov 2012, 3:30pm
Number Theory Seminar
room MATH 126
Level lowering for Galois representations
room MATH 126
Thu 15 Nov 2012, 3:30pm-4:30pm

Abstract

About 25 years ago, Ribet proved his famous level lowering result, which is an existence statement about congruences between modular forms of different levels. In this talk, I'll survey some recent progress towards giving a new proof of Ribet's result without any modularity assumptions. In place of a modular form, we start with a p-adic Galois representation, and in place of the level, we consider the conductor of this representation. We'll outline some ideas about how to show the existence of a second p-adic Galois representation of lower conductor which is congruent to the first.
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Vince Chan
UBC
Mon 19 Nov 2012, 3:00pm
Harmonic Analysis Seminar
Math 126
Necessary dimension to guarantee a given angle in a set
Math 126
Mon 19 Nov 2012, 3:00pm-3:50pm

Abstract

An interesting problem in various areas involves examining the size of a set in a geometric way, namely if it contains a given pattern. If we focus on finite patterns in Euclidean space, the first non-trivial case is a 3 point pattern, and since a similar copy of a triangle preserves its angles, a natural question might be: For a fixed ambient dimension and fixed angle, what is the minimal dimension for which any set of higher dimension is guaranteed to contain three points which form the specified angle? Or conversely, what is the maximal dimension for which there exists a set of that dimension which does not "contain" the specified angle? In this talk, we discuss a few new bounds on this number. Time permitting, we will also discuss a generalization of this notion, where the set avoids all angles close to the specified angle.

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Max Planck
Mon 19 Nov 2012, 3:00pm
Algebraic Geometry Seminar
ESB 2012
Ramified Satake Isomorphisms
ESB 2012
Mon 19 Nov 2012, 3:00pm-4:00pm

Abstract

I will explain how to associate a Satake-type isomorphism to certain characters of the compact torus of a split reductive group over a local field. I will then discuss the geometric analogue of this isomorphism. (Joint work with Travis Schedler). 
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Université Paul Sabatier, Toulouse, France
Tue 20 Nov 2012, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012 (in the new PIMS building)
On analytical properties of Alexandrov spaces
ESB 2012 (in the new PIMS building)
Tue 20 Nov 2012, 3:30pm-4:30pm

Abstract

 In this talk, I will discuss some analytical aspects in the study of a finite dimensional Alexandrov space. Loosely speaking, the question I will consider is: to what extent does an Alexandrov space resemble a Riemannian manifold? In the first part of the talk, I will recall the background of Alexandrov's theory of metric spaces with curvature bounded from below, including results on the topology of these spaces.
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Ed Richmond
UBC
Tue 20 Nov 2012, 4:00pm
Discrete Math Seminar
MATH 126
Factorial Schur functions and majorized sums of Hermitian matrices
MATH 126
Tue 20 Nov 2012, 4:00pm-5:00pm

Abstract

One remarkable application of Schur functions and Littlewood-Richardson coefficients is their close connection to the eigenvalue problem on sums of hermitian matrices. The eigenvalue problem asks: Given three sequences of real numbers, do there exist hermitian matrices A+B=C with eigenvalues given by the three sequences? This problem has a generalization to eigenvalues of majorized sums of hermitian matrices where we replace "A+B=C" with "A+B>C".

In this talk, I discuss joint work with D. Anderson and A. Yong where we show that the eigenvalue problem on majorized sums is related to factorial Schur functions in the same way that classical Schur functions are related to eigenvalue problem on usual sums of Hermitian matrices. One consequence of this connection is a generalization of the celebrated saturation theorem to structure constants corresponding to factorial Schur functions.


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UBC
Wed 21 Nov 2012, 3:00pm
Topology and related seminars
ESB 4127
Cohomology and K-theory of Crystallographic Groups II
ESB 4127
Wed 21 Nov 2012, 3:00pm-4:00pm

Abstract

This is continuation from the previous week:

We discuss the general problem of computing the cohomology and topological K-theory for classifying spaces of crystallographic groups. Integral computations will be provided for groups with prime order holonomy.
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Cambridge
Wed 21 Nov 2012, 3:00pm
Probability Seminar
ESB 2012
Counting self-avoiding walks
ESB 2012
Wed 21 Nov 2012, 3:00pm-4:00pm

Abstract

How small/large can be the connective constant of a regular graph? We give sharp inequalities for transitive graphs, and we explain how to prove strict inequalities as the graph varies. (joint work with Zhongyang Li)
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UBC
Thu 22 Nov 2012, 11:00am
Algebraic Groups and Related Structures
Math 126
Further Calculation in canonical dimension of quadrics
Math 126
Thu 22 Nov 2012, 11:00am-12:30pm

Abstract

 We continue to go over the proof of the canonical dimension of quadrics.
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Rutgers University
Thu 22 Nov 2012, 3:30pm
Number Theory Seminar
room MATH 126
Spherical varieties and the Langlands program
room MATH 126
Thu 22 Nov 2012, 3:30pm-4:30pm

Abstract

The study of periods of automorphic forms, and their relations with L-functions, has for a long time been regarded as a field separate from the mainstream of the Langlands program, a collection of fortunate coincidences allowing us to get our hands on difficult arithmetic objects. The work of Jacquet and others, however, has continuously emphasized the relation of periods to functoriality: the nonvanishing of H-period integrals of G-automorphic forms (where H is a spherical subgroup of G) should detect functorial lifts from some other group to G. Building on the theory of spherical varieties developed by Brion, Knop, Luna, Vust and others, Gaitsgory and Nadler attached a dual group to every spherical variety G. This can be used to recast periods in the language of the Langlands program, with many classical aspects of the program recovered as special cases when G=H. I will give an overview of this, mostly conjectural, program.
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John Calabrese
Oxford University
Mon 26 Nov 2012, 3:00pm
Algebraic Geometry Seminar
ESB 2012
Donaldson-Thomas invariants and birational transformations
ESB 2012
Mon 26 Nov 2012, 3:00pm-4:00pm

Abstract

 
I'll discuss two results regarding how DT invariants (of smooth and projective Calabi-Yau threefolds) change under birational modifications. The first deals with flops and the second is related to the McKay correspondence and work of Jim Bryan and David Steinberg.
 
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Cenk Sahinalp
Simon Fraser University
Tue 27 Nov 2012, 12:30pm
Scientific Computation and Applied & Industrial Mathematics
ESB 4133
Algorithms for the discovery of normal genomic variants and complex rearrangements in cancer via high throughput sequencing
ESB 4133
Tue 27 Nov 2012, 12:30pm-1:30pm

Abstract

 In this talk we will focus on some of the genome and transcriptome analysis software developed at the Lab for Computational Biology at SFU. These programs are all based on combinatorial optimization formulations for which exact or provably approximate polynomial time solutions exist - either for the original problem or its dual formulation.  Some of the problems can be solved through standard integer linear program solvers while others require specialized (approximate) solutions to known NP-hard problems.  The software we will cover include VariationHunter, a program to identify structural differences between a next generation sequenced genome and a reference genome, CommonLAW, a program to compare two or more next generation sequenced genomes through the help of a reference genome, deFuse, Comrad and nFuse, programs to detect gene fusions through the use of genomic or transcriptomic data or both, dissect, a program to associate assembled transcripts to a genome, and CLIIQ, a program to simultaneously identify and quantify novel splice variants.
 
Bio: S. Cenk Sahinalp is a Professor of Computing Science at Simon Fraser University, Canada. He received his B.Sc. degree in Electrical Engineering from Bilkent University and his Ph.D. in Computer Science from the University of Maryland at College Park. He did his postdoctoral work at Bell Labs, Murray Hill and spent the next two years on the faculty of U. Warwick, while holding a visiting position at U. Penn. Before moving to SFU he was on the faculty at Case Western Reserve. Sahinalp is an NSF Career Awardee, a Canada Research Chair, a Michael Smith Foundation Scholar and an NSERC DAS Awardee. He was recently named University of Maryland distinguished CS alumnus. Sahinalp has (co)chaired some of the leading conferences in computer science and bioinformatics such as RECOMB 2011 in Vancouver and serves on the editorial boards of several leading journals. His lab's recent research on computational genomics, in particular algorithms for high throughput sequence data and network biology have received several best paper awards, are used by major international research consortia, and have been highlighted by some leading scientific journals and media outlets.
 

Note for Attendees

Pizza will be provided.
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Panel : Carmen Bruni, Vince Chan, Mike Lindstrom, Tatchai Titichetrakun
UBC
Tue 27 Nov 2012, 12:30pm SPECIAL
Mathematical Education
MATH 126
Lunch Series : TAAP Course Presentation
MATH 126
Tue 27 Nov 2012, 12:30pm-1:30pm

Abstract

The participants of the TA Accreditation Program will present some concrete tools they have gotten from participating at this course.
This presentation is intended for the whole department, both faculties curious about TAs professional development and grad students wanting to know more about this program. For more info : http://blogs.ubc.ca/mathtaap


Note for Attendees

Pizza and pop will be served as usual.
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Berkeley
Wed 28 Nov 2012, 3:00pm
Probability Seminar
ESB 2012
Robust Gaussian noise stability
ESB 2012
Wed 28 Nov 2012, 3:00pm-4:00pm

Abstract

Given two Gaussian vectors that are positively correlated, what is the probability that they both land in some fixed set A? Borell proved that this probability is maximized (over sets A with a given volume) when A is a
half-space. We will give a new and simple proof of this fact, which also gives some stronger results. In particular, we can show that half-spaces uniquely maximize the probability above, and that sets which almost
maximize this probability must be close to half-spaces.

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UBC
Thu 29 Nov 2012, 11:00am
Algebraic Groups and Related Structures
Math 126
Canonical dimension of Brauer-Severi varieties
Math 126
Thu 29 Nov 2012, 11:00am-12:30pm

Abstract

We'll discuss how Karpenko and Merkurjev computed the canonical p-dimension of certain products of Brauer Severi varieties. They related these varieties to subgroups of the Brauer group, and used the theory of Grothendieck groups and Chow groups.
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Amir Ghadermarzi
UBC
Thu 29 Nov 2012, 3:30pm
Number Theory Seminar
room MATH 126
On families of Thue equations
room MATH 126
Thu 29 Nov 2012, 3:30pm-4:30pm

Abstract

 
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Dennis Timmers
Fri 30 Nov 2012, 4:00pm SPECIAL
One Time Event
Graduate Student Center, Room 200
Doctoral Exam
Graduate Student Center, Room 200
Fri 30 Nov 2012, 4:00pm-6:00pm

Details

We study systems consisting of interacting spin particles which can have a positive or negative spin. We consider an Ising model and a type of Widom-Rowlinson (WR) model. The interactions between spin particles are regulated by Kac potentials which carry a parameter g. It is known that in the Kac limit as g tends to zero models with Kac potentials become mean field theory. Mean field theory is known to exhibit phase transitions. The focus of this work is to prove phase transitions not only in the Kac limit but also near the Kac limit i.e., for g small but strictly positive.

Placing the Ising and WR model in a rectangular box with side-length L and periodic boundary conditions defines finite volume Gibbs measures. The infinite volume Gibbs state n is the limit of the finite volume Gibbs measures as L tends to infinity. A particle system exhibits a phase transition if n is a mixture of ergodic states.

The main achievement of this thesis is the development of a new method to prove phase transitions. We first apply the Kac-Siegert transformation which reformulates the particle system by introducing an external field. The spin-spin interactions are replaced by interactions of the spin particles with the external field. The main idea of this dissertation is to study the mean field. In principle it should be easier to work with the mean field because, as we will show, it is approximately Gaussian. By a new expansion around mean field theory we prove that for g strictly positive but small the infinite volume Gibbs state for the external field, for both the Ising and the WR model, is a mixture of two ergodic states. It is shown that this implies that the infinite volume Gibbs state for both the Ising and WR model is a mixture of two ergodic states. One Gibbs state predominantly has positive spin particles, the other Gibbs state predominantly has negative spin particles. The new expansion is related to the Glimm Jaffe Spencer expansion around mean field theory.
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