Schedule

March 23: Daniel Spector

Title: Some techniques in $W^{1,p}(\Omega)$

Abstract

March 30:  Alex Rand

Title: Finite Element Interpolation on Triangular Meshes

An important part of the theory of the finite element method (for approximating solutions
to partial differential equations) involves approximation of Sobolev functions by functions in
finite dimensional subspaces. One such interpolation question is given below.

Question: Let T be a triangulation of a set $\Omega \in R^2$. Given a function $u : \Omega \rightarrow R$, can
we find a function $u_T$ which is continuous on  $\Omega$, affine on each triangle $t \in T$ , and a “good”
approximation of u?

The answer to the question depends on three things: the smoothness of the function
u, the triangles in T , and the norm in which we are measuring the interpolation error. In
this talk, we will discuss how the interpolation error depends on each of these things and
generalize these error bounds to the case of piecewise polynomial interpolants.

Thursday, April 9, 12:30 pm, PH 125B: Peter Lumsdaine

Title:  Topology and algebra.

In this talk I will aim to explain, accessibly, the saying that  
"Algebra is topology, done backwards".

This paradigm (made precise as various dualities of categories) has  
been very fruitful in both geometric topology and algebra, providing  
the jumping-off point for algebraic geometry, non-commutative geometry/ 
toplogy, K-theory, and many other modern approaches to geometry.

The idea is often surrounded by much formidable technical material,  
but at its core is quite simple and beautiful.  I will prove one  
instance (Ge'fand Duality, between locally compact spaces and  
commutative C*-algebras), and use this to illustrate how such  
dualities work in general.

April 13:   Chris Potter

Title: Monte Carlo Integration and Metropolis Algorithms

The name "Monte Carlo" often brings to mind the rolling of dice, and it is from this
association that Monte Carlo methods draw their name.  Generally speaking, any
method of solving a problem using a sequence of random numbers (a faster and more
flexible substitute for die rolls) can be termed a Monte Carlo method.  However, out
of this enormous set of processes we will be focusing on a small subset: Monte Carlo
integration and Metropolis algorithms.

In the case of Monte Carlo integration, we will show some results characterizing the
conditions under which the indeterminacy of this method allows it to outperform
traditional quadrature methods in efficiency and/or accuracy.  We will also discuss
Metropolis algorithms, which are used to tackle more general problems involving
objects that can be modeled as Markov chains.  Finally, we will apply a Metropolis
technique to simulate the behavior of an idealized ferromagnetic crystal subjected
to variation in temperature and external magnetic field.

April 20:  Brian Seguin

Title:  How Should We Think About Space?
        A Motivation for the Principle of Material Frame-Indifference

Many people talk about "space" without giving the word much thought. Of course,
there is no such thing as physical space but only various frames of reference. 
After talking about the historical and psychology reasons for the introduction of
the concept of physical space I will discuss how one can avoid it and give the
precise mathematical structure necessary to model frames of reference.  I will then
show how this way of thinking leads to restrictions on constitutive laws, i.e. the
laws that govern the behavior of materials.



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Schedule

October 8: Anne Yust

Title: Probabilistic Modeling of Gene Interactions

Our ultimate goal is to obtain a reasonable model of the gene
interactions within the immune system. From the data we have which
contains time plots of the growth of specific proteins produced by
classified immune system genes, we want to determine the inhibiting
and catalyzing interactions between those genes. We are specifying
these interactions as stochastic in nature. The probabilities we are
attempting to determine are defined by the likelihood that a gene
affects another gene, positively or negatively. In our attempt to
realize these interactions, we have produced various models that show
protein production. Here we try to solve the forward problem: given a
specified number of probabilities, create a time plot of the protein
productions. These models consider single genes producing one protein,
simple two gene interactions producing two proteins, and random $n$
gene interactions producing $n$ proteins. After creating these models,
we use them to solve the backward problem: given a time plot of
protein productions, find the inhibiting and catalyzing interactions
between specific genes and their associated probabilities.

October 15:  Fritz Obermeyer

Title:   Meaning in Mathematics

  What mathematical statements are meaningful?
  What mathematical statements are true?
  Conservatively, only proven facts are true,
  only what can be deduced from assumptions.
  And assumptions come from science,
  the logic of what "might be true" and hasn't yet been falsified.
  Just as Godel's 1st Incompleteness Theorem limits how much can be proven,
  an analogous theorem limits what can be scientifically tested,
  i.e. how much can be physically meaningful.

  In this talk, I will formulate the incompleteness theorems
  and discuss their relation to math and resp. science.
  I will also suggest some heuristics for guessing what "might be true"
  (step 1 in the scientific method) and prove that they are all doomed to fail.

October 22: Peter Lumsdaine

Title:  Intensional Type Theory and Higher-Dimensional Category Theory

Old philosophical example:

"A human is a rational animal."
"A human is a featherless biped".

These two definitions describe the same class of objects --- they have the same
"extension" --- but have different meaning, different "intension".  This sort of
logical distinction is an old and subtle one, and turns up in many different places.
Mathematically, it can be formalised in Intensional Martin-L\"of Type Theory. 
The syntax of this is very well-understood (it has been implemented as the basic
framework for various proof assistants, e.g. Coq), but the semantics less so: most
models destroy the intension/extension distinction.

Higher-Dimensional Category Theory emerged in the 80's and 90's from homological
algebra and algebraic topology, and is now a very active research area in its own
right; recently, it has been realised that higher categories give natural models of
type theory that are truly intensional.  In this talk, I will explain each of these
sides (type theory and higher categories) a bit, and describe some of the
connections between them.

Thanks to the topological origins of higher-dimensional category theory, there
will be lots of pictures.

October 29: Cristina Canepa
 
 Title:  Introduction to credit risk with examples from recent financial events
 
 I will give an introductory presentation on the general topic of credit risk  with
specific glimpses from actual financial crisis.
  What are the models used to price a CDS(credit default swap0,a CDO(collateralized
debt obligation) or any credit derivative?
The general framework is given by two types of models: structural models and
reduced-form models.The former ones are more intuitive and they give nice
closed-form solutions; however ,they don't fit the data. Reduced-form models are less
intuitive, they are based on statistical techniques and they fit the data in a more
appropriate way.
We'll focus on reduced-form models, giving some ideas about how we can simulate a
higher correlation among defaults of different firms; we'll present the
doubly-stochastic approach, copula models and frailty models.
  There will be a very short history of mathematical finance, a quick presentation of
its main tools (so that one with no background in probability or finance can
follow), lots of pictures and some quotations from media.

November 5: David Offner

Title: An extremal problem on the hypercube

A typical problem in Extremal Combinatorics asks what can be learned about extremal
(e.g largest, smallest) examples of combinatorial objects (e.g. graphs, set systems)
subject to certain restrictions.

In this talk we introduce a classic problem:  How many edges may a graph on n
vertices have without containing a clique on r vertices as a subgraph?  The solution
to this problem is Turan's theorem.

Variations on this problem are known as Turan type problems, and there are many
interesting ones.  We will consider the one which asks how many edges a subgraph of
an n-dimensional hypercube may contain without containing a given graph as a
subgraph.  We will give some longstanding conjectures and results, and discuss
recent progress.


November 12: Alex Rand

Title: Towards Automatic 3D Mesh Generation For Finite Element Methods

Finite element methods for solving partial differential equations (PDE) require a
quality triangulation of the domain of the solution.  Over the last 20 years, 2D
Delaunay refinement algorithms have been developed to produce suitable
triangulations of arbitrary domains which are within a constant factor of optimal
size.  While many aspects of these algorithms can naturally be extended to 3D, some
issues in 3D mesh generation do not have 2D analogies.  (Specifically, there is no
2D concept similar to that of skew lines in 3D.)  We will discuss recent
improvements in 3D Delaunay refinement algorithms as well as the remaining issues
which must be overcome to produce an automatic 3D mesh generator for finite element
methods.  

November 19: Maxim Bichuch

Title: An Optimal Portfolio of Correlated Futures with Small Transaction Cost

December 3: Pall Melsted

Title: Random Walks for Cuckoo Hashing

A Dictionary is a fundamental data structure in computer science, that provides a
mapping from keys to elements. One implementation is the d-ary
Cuckoo Hashing Table, which uses d hash functions and guarantees that each key
present be stored in one of the locations it hashes to. Cuckoo hashing is
known to have good space efficiency and worst case access time, but insertion of
items into the table is harder. A depth first search tree algorithm is known to take
expected constant time, but its worst case performance can be polynomial. All
practical implementations use a random walk heuristic to insert items, which is
known to work well in practice but previous to this work no theoretical bounds were
known. We present an analysis of the random walk heuristic and show it runs in
expected and worst case polylogarithmic time. The talk will be accessible a general
math audience and no prior knowledge of data structures is necessary.