Colloquia, AY 2009-2010

Colloquium, AY 2009-2010



Thursdays 4:10 pm in 5211 Stevenson Center, unless otherwise noted
Tea at 3:30 pm in 1425 Stevenson Center

September 3, 2009  

Welcome Event




 

October 8, 2009  

Faculty meeting — No Colloquium.

Tuesday,
October 20, 2009
 

Cornel Pasnicu, University of Puerto Rico



Noncommutative zero dimensional topological spaces

Abstract:
A C*-algebra can be thought as a noncommutative topological space or as a collection of infinite matrices of complex numbers endowed with an interesting algebraic and topological structure. The C*-algebras have significant applications in different areas of mathematics (geometry, topology, ergodic theory), parts of physics (quantum mechanics and statistical mechanics) and other sciences. Understanding the structure and classification of C*-algebras was and continues to be one of the most important researh directions in Operator Algebras (Elliott and Kirchberg, I.C.M. 1994, Rordam, I.C.M. 2006). In this talk I will present, in a natural context and giving basic definitions and examples, a joint work with Mikael Rordam (J. Reine Angew. Math. 2007) in which we characterize, in the separable case, for a large and important class of C*-algebras that are “infinite” in some specific sense (introduced 10 years ago by Kirchberg and Rordam) a certain condition of noncommutative zero dimensionality (introduced by Brown and Pedersen) that proved to be very successful in Elliott’s well known Classification Program for C*-algebras (I.C.M. 1994). (It is perhaps worth to mention also that many C*-algebras of interest happen-sometimes surprisingly-to satisfy this condition). Some interesting consequences of this result that concern the structure of C*-algebras will be also discussed. Our theorem strongly generalizes a result of Perera and Rordam (J. Funct. Anal. 2004) and, in the separable case, a result of Zhang.

Contact person: Guoliang Yu

October 22, 2009  

Fall Break — No Colloquium.

October 29, 2009  

Alexander Razborov, University of Chicago



Flag Algebras


Abstract:
A substantial part of extremal combinatorics studies relations existing between densities with which given (fixed size) combinatorial structures may appear in unknown (and presumably very large) structures of the same type. Using basic tools and concepts from algebra, analysis and measure theory, we develop a general framework that allows to treat all problems of this sort in an uniform way and reveal mathematical structure that is common for most known arguments in the area. The backbone of this structure is made by commutative algebras defined in terms of finite models of the associated first-order theory.

In this talk I will give a general impression of how things work in this framework, and we will pay a special attention to concrete applications of our methods.

Contact person: Mark Sapir

November 5, 2009  

Viktor Ginzburg, University of California, Santa Cruz



The Conley conjecture: infinitely many periodic points of Hamiltonian diffeomorphisms


Abstract:
One distinguishing feature of Hamiltonian dynamical systems is that such systems, with very few exceptions, tend to have numerous fixed and periodic points. In 1984 Conley conjectured that a Hamiltonian diffeomorphism (i.e., the time-one map of a Hamiltonian flow) of a torus has infinitely many periodic points or, more precisely, such a diffeomorphism with finitely many fixed points has simple periodic points of arbitrarily large period. This fact was proved by Hingston some twenty years later, in 2004. Similar results for Hamiltonian diffeomorphisms of surfaces of positive genus were also established by Franks and Handel. Of course, one can expect the Conley conjecture to to hold for a much broader class of closed manifolds and this is indeed the case. For instance, by now, the conjecture has been proved for the so-called closed, symplectically aspherical manifolds (including tori and surfaces of positive genus) and the Calabi-Yau manifolds using symplectic topological techniques.

In this talk we will discuss the underlying reasons for the existence of periodic orbits for Hamiltonian flows and maps and outline a proof of the Conley conjecture.

Contact person: Ba┼čak Gürel

November 12, 2009  

Doron Lubinsky, Georgia Tech



de Branges spaces, universality limits, and orthogonal polynomials


Abstract: Universality limits for random matrices describe the spacings between successive eigenvalues of random matrices, and their distribution. In the unitary case, one way to establish such universality limits is via the the theory of entire functions of exponential type, and de Branges spaces. We shall discuss the method and some recent results. No background on de Branges spaces, or universality is assumed.

Contact person: Ed Saff

November 19, 2009  

Hanfeng Li, SUNY Buffalo and Vanderbilt University



Combinatorial independence in dynamics


Abstract: Combinatorial independence originated from the work of
Rosenthal on characterization of Banach spaces
containing l_1 isomorphically. Based on joint work with Wen Huang,
David Kerr, and Xiangdong Ye, I will discuss
how it leads to unified combinatorial and functional-analytic
approaches to the study of various mixing properties in
dynamics.

Contact person: Guoliang Yu

November 26, 2009  

Thanksgiving Break — No Colloquium.

December 3, 2009  

Faculty meeting — No Colloquium.

December 10, 2009  

Miklos Maroti, University of Szeged, Hungary



The constraint satisfaction problem for algebras of bounded
width and few subpowers



Abstract: The constraint satisfaction problem CSP(G) for a directed
graph G is the problem of deciding of an input directed graph H
whether there exists a homomorphism from H to G. For the two-element
complete directed graph G with no loops this problem is solvable in
polynomial time (corresponds to the class of bipartite directed
graphs), while for the three-element complete directed graph G with no
loops it is NP-complete (corresponds to the class of 3-colorable
directed graphs). The dichotomy conjecture formulated by Feder and
Vardi in 1993 asserts that for any directed graph (or finite
relational structure) G the problem CSP(G) is in P or NP-complete,
therefore the intermediate complexity classes, which exist by Ladner’s
result if P does not equal NP, cannot be realized with constraint
satisfaction problems. This conjecture has been verified in numerous
special cases but it is still open in general. The latest results have
been achieved with the help of universal algebra. First, we can fully
characterize in algebraic terms those relational structures G for
which the so called “local consistency” algorithm works. Second, we
can fully characterize in algebraic terms those relational structures
G for which the set of all homomorphisms from H to G can be
represented in polynomial space (this is a generalization of solving a
system of linear equations over a finite field). We review these
results, and show how these two algorithms can be combined to solve an
even larger class of problems.

Contact person: Ralph McKenzie

January 14, 2010  

Faculty meeting — No Colloquium.

January 21, 2010  

James McKernan, MIT



Boundedness in Algebraic Geometry


Abstract: It is a classical result that if C is a compact Riemann
surface of genus g at least two then C has at most 84(g-1) automorphisms. On the other hand, suppose that f(z_1,z_2,…,z_n) is a holomorphic
function. If f(0)=0 then 1/|f|^2 is not an integrable function. The
largest value of t for which 1/|f|^{2t} is integrable is a real number
between zero and one which is a measure of how much f vanishes at the
origin. The set of all such t, as the holomorphic function f varies,
exhibits some unusual boundedness properties. We show that these two apparently unrelated phenomena are in fact
closely related.

Contact person: Dietmar Bisch

January 28, 2010  

Faculty meeting — No Colloquium.

February 4, 2010  

Friedrich Wehrung, University of Caen, France



Approximating the finite by the infinite: Larders and CLL


Abstract: We are given categories A, B, and S together with functors F:A–>S and
G:B–>S such that for each object a of A, there exists an object b of
B such that F(a) is isomorphic to G(b). We are asking whether the
assignment from a to b can be made functorial. CLL is a theorem of
pure category theory, whose statement is complex but that can be
summarized as follows: Theorem (P. Gillibert and F. Wehrung, 2008). If A, B, S, F, G form a
“larder”, then such a thing can be done. This result can be used either to deduce properties of “large”
objects (how large being determined by infinite combinatorics) from
properties of “small” objects, but also conversely, for example by
establishing a lifting property of a finite diagram of finite Boolean
semilattices by way of a representation result on uncountable
algebras. We illustrate this result by applications to various open
problems in universal algebra, lattice theory, and ring theory.

Contact person: Ralph McKenzie

February 11, 2010  

Vaughan Jones, UC Berkeley



Associative algebras from planar algebras


Abstract: We will explain how planar pictures give rise to a host of
associative algebras and subalgebras including some that
allow reconstruction of the origingal planar structure. Connections
with random matrices, algebraic geometry and low dimensional
topology will be explored.

Contact person: Dietmar Bisch

February 18, 2010  

Boris Zilber, Oxford University



On Model Theory, noncommutative geometry and physics


Abstract: Studying possible relations between a mathematical structure and its
description in a formal language Model Theory developed a hierarchy
of ‘logical perfection’. On the very top of this hierarchy we discovered
a new class of structures called Zariski geometries. A joint theorem by
Hrushovski and the speaker (1993) indicated that the general Zariski
geometry looks very much like an algebraic variety over an algebraically
closed field, but in general is not reducible to an algebro-geometric
object. Later the present speaker established that a typical Zariski
geometry can be explained in terms of a possibly noncommutative ‘co-ordinate’
algebra. Moreover, conversely, many quantum algebras give rise to
Zariski geometries and the correspondence ‘Co-ordinate algebra – Zariski
geometry’ for a wide class of algebras is of the same type as that
between commutative affine algebras and affine varieties. General quantum Zariski geometries can be approximated (in a certain
model-theoretic sense)
by quantum Zariski geometries at roots of unity. The latter are of a finitary
type, where Dirac calculus has a well-defined meaning. We use this to give a
mathematically rigorous calculation of a
Feynman propagator in a few simple cases.

Contact person: Mark Sapir

February 25, 2010  

Mauro Maggioni, Duke University



Geometry and Analysis of point sets in high dimensions


Abstract: The analysis of high dimensional data sets is useful in a large variety of applications, from machine learning to dynamical systems: data sets are often modeled as low-dimensional, noisy data sets embedded in high-dimensional spaces; dynamical systems often have very high-dimensional state spaces but sometimes interesting dynamics occurs on low-dimensional sets. We discuss several problems associated with the analysis of the geometry of such sets, and with the approximation of functions on such sets, together with some solutions: in particular we discuss how to construct random walks on such data sets and perform multiscale analysis of them and their applications (especially to machine learning); how to construct robust coordinate systems for data sets; how to estimate reliably the intrinsic dimensionality of the data when only few noisy samples are available.

Contact person: Akram Aldroubi

March 4, 2010  

Yuan Lou, Ohio State University



Non-random Dispersal of Interacting Species in Heterogeneous Landscapes


Abstract: From habitat degradation and climate change to spatial spread of invasive species, dispersals play a central role in determining how organisms cope with a changing environment. The dispersals of many organisms depend upon local biotic and abiotic factors, and as such are often non-random. In this talk we will discuss some recent progress on the effects of non-random dispersal on two competing species in heterogeneous environments via reaction-diffusion-advection models.

Contact persons: Phil Crooke, Zhian Wang, and Glen Webb

March 11, 2010  

Spring Break — No Colloquium.

March 18, 2010  

Peter Palfy, Mathematics Institute, Budapest, Hungary



When does the size tell us the structure of a group?


Abstract: Everybody learns in the introductory group theory course that every
group of prime order is cyclic. In fact, there are other numbers with
this property. Namely, every group of order n is cyclic if and only
if n and φ(n) are relatively prime, where φ is
Euler’s totient function. This was observed by George A. Miller in 1899.
In 1948 Paul Erdos proved that
the number of positive integers n < x with this property is asymptotically e-γx/logloglog x, where γ denotes
Euler’s constant. In this survey talk I will discuss similar results characterizing
those numbers n for which every group of order n is abelian,
nilpotent, solvable, etc. and giving an asymptotic for the number of
positive integers n < x having these properties. In some cases the relevant question is whether a group with some
property (for example, a simple group) of a given order exists.
Paul Erdos has a forgotten paper on this question. I will
speculate why this paper was omitted from his list of publications.
Now the classification of finite simple groups yields that
there are asymptotically 3.21/3 x1/3/log x simple group
orders less than x. I will also tell how I got Erdos number 1.

Contact person: Ralph McKenzie

Friday,
March 26, 2010
3:10 pm
 

Peter Polacik, University of Minnesota



Liouville-type theorems in partial differential equations and their applications


Abstract: Just like in complex analysis, Liouville theorems in partial differential equations assert that if u is an entire solution of a specific equation and it is contained in an admissible class of functions, then it is the trivial solution. We will start the talk with an overview of Liouville theorems for nonlinear elliptic and parabolic equations. Then we will show some typical applications of Liouville theorems in qualitative studies of solutions.

Contact person: Gieri Simonett

March 30, 2010  

Special Colloquium – Aldroubi-Azhari Prize Winner



Romain Tessera, CNRS



The large-scale geometry of Lie groups


Abstract: Lie groups have been studied from almost all possible points of view. However, most of the attention has been focused on semi-simple Lie groups. The large scale geometry of a general connected Lie group is a challenging field, where many natural questions are still wide open. In this talk, we will consider a continuous analogue of the “word problem” in Lie groups. We will see that as a corollary of our results, we obtain an algebraic characterization of Gromov-hyperbolic homogeneous Riemannian manifolds. Similar results are obtained in the p-adic setting.

An award ceremony will be held before the colloquium at 3:00 p.m. in SC 1425.

April 1, 2010  

Jaroslav Nesetril, Charles University, Prague



Existence vs Counting


Abstract: We consider the dichotomy in the title in the context of properties of large sparse finite structures. This in turn leads to the nowhere dense vs somewhere dense dichotomy which can be defined in a surprising variety of different ways. But basically most of these characterizations are static – using properties of the homomorphism order. Recently the nowhere dense classes were characterized by properties of counting function of subobjects. This then relates to Lovasz and al. (for dense graphs) and Benjamini-Schramm (for ultrasparse graphs) statistics.

Contact person: Ralph McKenzie

April 8, 2010  

Carl de Boor, University of Wisconsin-Madison



Issues in multivariate polynomial interpolation


Abstract: While univariate polynomial interpolation has been a basic tool of scientific computing for hundreds of years, multivariate polynomial interpolation is much less understood. Already the question from which polynomial space to choose an interpolant to given data has no obvious answer.

The talk presents, in some detail, one answer to this basic question, namely the “least interpolant” of Amos Ron and the speaker which, among other nice properties, is degree-reducing, then seeks some remedy for the resulting discontinuity of the interpolant as a function of the interpolation sites, then addresses the problem of a suitable representation of the interpolation error and the nature of possible limits of interpolants as some of the interpolation sites coalesce.

The last part of the talk is devoted to a more traditional setting, the complementary problem of finding correct interpolation sites for a given
polynomial space, chiefly the space of polynomials of degree <= k for some k, and ends with a particular recipe for good interpolation sites in the square, the Padua points.

References: http://pages.cs.wisc.edu/~deboor/multiint/

Contact person: Larry Schumaker

 
Colloquium Chair (2009-2010): Mark Sapir

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