|
|
Spring 2005
Colloquium
Speaker: Tom Roby (California State University, Hayward)
Time: Tuesday, January 18, 2005 at 4:00 pm
Place: MSB 118 (UConn - Storrs)
Abstract:
- Graphs and partitions are both fundamental objects in combinatorics that have been well studied. Less well-known is the connection between these via the notion of “graphical degree sequence”, an integer partition which lists the vertex degrees of a graph in nonincreasing order. We call such a partition graphic. A threshold graph is one whose degree sequence is maximal (in the dominance order) among graphic partitions.
By exploiting the interplay between threshold graphs and partitions, we construct a lattice of threshold graphs which is isomorphic to the lattice of shifted shapes (i.e., partitions into distinct parts). We explore the graph theoretic structure of this lattice.
This is joint work with Russell Merris.
- Since the summer of 2000, the Alameda County Collaborative for Learning and Instruction in Mathematics (ACCLAIM) has run professional development (PD) institutes for public K-12 teachers in the East San Francisco Bay Area. Originally funded largely by the state, in ACCLAIM’s peak two years it served over 350 teachers in summer institutes lasting 8-14 days, with school year followup. This made it one of the largest math content PD programs in the state. The program survived the summer of 2004 entirely on funds available through school districts on a “fee for service” basis, serving 325 teachers in 1-week institutes.
In this talk we will explain how ACCLAIM worked, some of the challenges we faced, and what components went into making it sufficiently successful that it could survive after state funding dried up. Many difficulties arose in creating and continuing a collaboration between Cal State Hayward, the Alameda County Office of Education, and local districts; some were exacerbated by the politically-charged atmosphere of California educational politics.
The speaker has been the PI and co-Director of ACCLAIM since its inception and is Associate Professor of Mathematics and Computer Sciences at Cal State Hayward.
|
Colloquium
Speaker: Kyu-Hwan Lee (University of Toronto)
Time: Thursday, January 20, 2005 at 4:00 pm
Place: MSB 118 (UConn - Storrs)
Abstract: The Langlands reciprocity conjecture is one of the main problems in number theory. This talk will present what the conjecture is. We will see that the Satake isomorphism plays an important role in this conjecture. A two-dimensional generalization of the Satake isomorphism will be considered at the end of the talk.
|
Logic Seminar
Speaker: Antonio Montalban (Cornell University)
Time: Monday, January 24, 2005 at 4:45 pm
Place: Exley Science Center 618 (Wesleyan University)
Abstract: Clifford Spector proved in 1955 that every hyperarithmetic (or equivalently Delta^1_1) well ordering is isomorphic to a computable one. The direct generalization of Spector's Theorem to the class of linear orderings does not hold; It is not the case that every linear ordering with a hyperarithmetic presentation is isomorphic to a computable one. It is known that for every non-computable Turing degree, there is a linear ordering of that degree which has no computable copy. This was proved by Knight extending a result of Downey. But there are other ways in which we can generalize Spector's Theorem. We say that two linear orderings are equimorphic if each one can be embedded into the other one. Observe that if a linear ordering and a well ordering are equimorphic, then they are actually isomorphic. We prove the following generalization of Spector's theorem: every hyperarithmetic linear ordering is equimorphic to a computable one. Moreover, we prove that every linear ordering of Hausdorff rank less than omega^{CK}_1 is equimorphic to a computable one.
The equimorphism relation on linear orderings is very natural. Many properties of linear orderings are invariant under equimorphisms, as, for example, extendibility, indecomposability, well foundedness, being scattered, or having a certain Hausdorff rank. The structure of equimorphism types of countable linear orderings ordered by embeddablity is also an interesting object. For example, Fraisse's conjecture (which was proved by Laver) is the statement that says that this partial has no infinite descending sequences and no infinite antichains. This statement has interested logicians because of its proof theoretic complexity. We prove that for every ordinal alpha < omega_1^{CK}, the partial ordering of equimorphism types of linear orderings of Hausdorff rank less than alpha ordered by embeddablity, is computably presentable.
In this talk, I am planning describe all these results. Also, I will give and outline of the proofs and show some key ideas.
|
Colloquium
Speaker: Alfred Dolich (McMaster University)
Time: Tuesday, January 25, 2005 at 4:00 pm
Place: MSB 118 (UConn - Storrs)
Abstract: In model theory one generally studies classes of mathematical objects described by first order axioms in some fixed logical language, for example vector spaces over Q or algebraically closed fields of fixed characteristic. A crude though very basic question is whether for a fixed class K or equivalently for a fixed set of axioms T, we can find useful measures of the “complexity” of the class. Not surprisingly many different interpretations may be given to the term complexity, natural interpretations arise from considering the complexity of the axioms used to describe the class, the computational strength needed to describe certain members of the class, or notions of complexity arising from topological or algebraic considerations. In general it appears that these various notions can not be correlated, though if we sufficiently constrain the classes we consider we obtain some results. In particular I will discuss results showing that if we restrict ourselves to considering only sets of axioms T which are trivial and uncountably categorical (these are constraints on the algebraic or topological complexity alluded to above) we can bound the axiomatic or computational complexity when we bound the topological or algebraic complexity. As an upshot of this we obtain some new results on the computability-theoretic complexity of axiom systems T as above admitting a computable model. (Some of this joint work with M.C. Laskowski and A. Raichev.)
|
UConn Math Club
Speaker: Keith Conrad (University of Connecticut)
Time: Wednesday, January 26, 2005 at 5:30 pm
Place: MSB 118 (UConn - Storrs)
Abstract: Historically, the first method of assigning probabilities to the outcomes of a random event is due to Bernoulli. His recipe is this: when there is no reason to do otherwise, assign all outcomes equal probability. This is called the principle of insufficient reason, or principle of indifference. We will discuss an extension of the principle of insufficient reason, called the principle of maximum entropy. It applies in cases where one does not expect the outcomes to fall out uniformly.
We will describe the principle of maximum entropy and see how it works in some examples. In particular, we will see how this principle provides a unifying conceptual role for several standard probability distributions.
Comments: Free Refreshments
|
Colloquium
Speaker: Jamie Sutherland (University of Wisconsin, Madison)
Time: Thursday, January 27, 2005 at 4:00 pm
Place: MSB 118 (UConn - Storrs)
Abstract: In this talk, I develop a framework for analyzing how held values interact with policies concerning placement into undergraduate math classes. Of particular interest is how the values inherent in a particular university's placement practices may at times conflict with those the university has stated or which are implied by university practices, the literature concerning placement, and the students themselves. This conflict suggests that evaluation of placement practices should be sensitive to these inherent values and changes in practice may be necessary to ensure validity. This framework addresses the question of how placement policies are affected by these values and ultimately how they affect the practices of placement. I illustrate the use of this framework using an example of placement practices at a large Midwestern university.
|
Geometry Seminar
Colloquium
Speaker: Fabiana Cardetti (University of Connecticut)
Time: Tuesday, February 1, 2005 at 4:00 pm
Place: MSB 319 (UConn - Storrs)
Abstract:
- Linear Control Systems on Lie groups
It is well known that one of the most important classes of control systems is the classical linear system on Rn. They are fundamental both from the theoretical and practical points of view. But, in many cases, coming from applications the assumption that the state space is a vector space cannot be made. Many situations, however, can be formulated as control problems on Lie groups. Typical of these are certain problems which arise in the control of the attitude of a rigid body. In this talk we introduce the notion of linear control systems on Lie groups. We will show some properties of these type of systems. In addition, controllability results will be presented and illustrated on the special linear group SL(2,R).
- Improving K-12 Teachers Education
It is often assumed that, because the topics covered in school mathematics are so basic, they must also be easy to teach. However, research over the past decade of the mathematical knowledge of prospective and practicing U.S. teachers suggests that substantial mathematical understanding is needed even to teach whole number
arithmetics. Studies also show that U.S. teachers are outperformed by their colleges of some Asian countries. Prospective teachers need a solid understanding of mathematics so that they can teach it as a coherent, reasoned activity and communicate its elegance and power. In this part of the talk, the focus will be my personal experience improving teachers education.
Comments: Tea in the Math Lounge at 3:30pm
|
S.I.G.M.A. Seminar
Speaker: Jesse Ratzkin (University of Connecticut)
Time: Wednesday, February 2, 2005 at 4:30 pm
Place: MSB 117 (UConn - Storrs)
Abstract: Unfortunately, curvature is an object which (a) means many different things to different people and (b) is horribly complicated in all of these different interpretations. I will explain some integral concepts of how I think about curvature, particularly related to length, area, and volume. This talk will concentrate on pictures instead of rigorous definitions, and be almost devoid of proofs.
|
UConn Math Club
Speaker: Rachel Schwell (University of Connecticut)
Time: Wednesday, February 2, 2005 at 5:30 pm
Place: MSB 118 (UConn - Storrs)
Abstract: Knot theory is a relatively “new” field of mathematics; new in that it has only begun to be explored in the past one hundred or so years. We will examine knots from a more mathematical angle, including an accepted mathematical procedure of “untangling” a knot, if it can be so done, and determining whether two different-looking knots are actually the same. We will then consider a way to “add two knots together”, and compare this algebraic operation to addition and multiplication of non-negative integers. The only knowledge that is required is to know what a knot is and how to add and multiply!
Comments: Free Refreshments
|
Colloquium
Speaker: Joseph Miller (Indiana University at Bloomington)
Time: Thursday, February 3, 2005 at 4:00 pm
Place: MSB 118 (UConn - Storrs)
Abstract: Kolmogorov defined the complexity of a finite binary string to be the
length of the shortest program that generates it. Most strings have
Kolmogorov complexity close to their lengths; in other words, they
cannot be compressed. Such strings are considered to have high
information content. Martin-Lof, a student of Kolmogorov, considered
the complexity of infinite binary sequences. He defined a sequence to be
random if it passes certain effectively presented statistical tests. Levin
and Chaitin both modified Kolmogorov complexity to characterize
Martin-Lof randomness, while Schnorr gave a characterization in terms
of effective betting games. Thus we have three equivalent formulations
of randomness arising from three different fundamental intuitions:
random sequences are "unremarkable", "incompressible" and
"unpredictable".
This talk will focus on some of the main themes of my work on Kolmogov
complexity and Martin-Lof randomness. We will examine the complexity
of initial segments of random reals, giving new insight into the
behavior of this complexity and applying it to study the "degrees of
randomness" of infinite sequences. Along the way, we are able for the
first time to characterize Martin-Lof randomness in terms of
Kolmogorov's original complexity measure. We will also contrast the
notions of "information" found in computability theory and algorithmic
randomness. For example, we will see that an appropriate upper bound
on the computational power of a random sequence actually enforces a
lower bound on its degree of randomness. Similarly, there are
connections between high initial segment complexity and low
computational power. Finally, we look at recent progress on an
interesting question about how much unpredictability is really needed
to ensure Martin-Lof randomness.
Comments: Tea in the Math Lounge at 3:30pm
|
Geometry Seminar
Speaker: Hossein Namazi (Yale University)
Time: Tuesday, February 8, 2005 at 4:00 am
Place: 117 MSB (UConn - Storrs)
Abstract: It is well known that every compact orientable 3-manifold admits a Heegaard splitting. An important problem in studying 3-manifolds is to use "combinatorics" of the Heegaard splitting and obtain information about the topology of the 3-manifold. In particular, we expect a 3-manifold with a "complicated" splitting to have "rigid" structure. We state some results in this direction by studying the mapping class group of the Heegaard surface and relating it to the splitting.
|
S.I.G.M.A. Seminar
Speaker: Reed Solomon (University of Connecticut)
Time: Wednesday, February 9, 2005 at 4:30 pm
Place: MSB (room t.b.a.) (UConn - Storrs)
Abstract: The notation dx was first used by Leibniz in his development of calculus and he defined dx to be the difference in successive values of the variable x. Of course, since the real numbers form a continuum, this definition was not rigorous and dx was considered to be an infinitely small positive quantity. (Newton used a different system of notation but had a similar notion of infinitesimal quantity.) This intuition proved remarkably useful but was subject to various criticisms, most notably by Bishop Berkeley who published a scathing criticism of these methods in which he referred to infinitesimals as "the ghosts of departed quantities". By the end of the 19th century, the idea of an infinitely small quantity had been formalized using a rigorous construction of the reals and the modern definition of a limit. However, in 1966, Abraham Robinson published a remarkable book titled "Non-standard Analysis" in which he gave a formal treatment of analysis using infinitely small positive numbers. In effect, Robinson gave the first interpretation of analysis which followed the true spirit of Leibniz's original ideas.
In this talk, I will explain two methods of creating nonstandard number systems. First, we will use a construction from logic to build a nonstandard copy of the natural numbers and then we will use a more algebraic construction to create a nonstandard copy of the reals. We will conclude with a couple of examples from mathematics in which the perspective of nonstandard models has been particularly useful.
|
UConn Math Club
Speaker: Sonal Jain (Harvard)
Time: Wednesday, February 9, 2005 at 5:30 pm
Place: MSB 118 (UConn - Storrs)
Abstract: If asked to approximate a number, most people think about the initial piece of its decimal expansion, like the square root of 10 which
equals 3.162277660168....
This talk is about a different way to approximate numbers, using continued fractions. A continued fraction approximates a number in a more subtle (some would say better) kind of way than a decimal, with a few surprises. For instance, while the decimal digits above look random, the continued fraction for the square root of 10 is periodic!
Continued fractions provide a convenient explanation of the leap year rules (an extra day for each year divisible by 4, unless the year is a multiple of 100, except when the year is a multiple of 400...). They are also the main tool needed for exact rational number recognition on a computer. For instance, what is the “most likely” fraction whose decimal expansion begins with 2.34579439252? (Hint: the answer has a much smaller denominator than you would expect.)
Comments: Free Refreshments; USG funded
|
Logic Seminar
Speaker: Dan Mauldin (University of North Texas)
Time: Monday, February 14, 2005 at 4:45 pm
Place: Exley Science Center 618 (Wesleyan University)
Abstract: Let X and Y be uncountable Polish spaces. We discuss the problem of whether there is a Borel set B in XxY which is uniquely universal for various families. For example, is there a B such that each subset of Y which is the union of countably many compact sets occurs as a fiber of B once and only once and in addition that all fibers of B are such sets? We discuss the problem of the existence of a Borel set meeting each line in the plane in exactly two points and the problem of the existence of a Steinhaus set- a set meeting every isometric copy of Z^2 in exactly one point.
|
Colloquium
Speaker: Payman Kassaei (McGill University, Montreal)
Time: Tuesday, February 15, 2005 at 4:00 pm
Place: MSB 118 (UConn - Storrs)
Abstract: Modular forms are certain holomorphic functions on the complex upper half-plane that possess an infinity of symmetry. Number theorists are interested in the systems of eigenvalues obtained from the action of Hecke operators on modular forms. These (a priori complex) numbers are algebraic integers which are often of arithmetic significance. One is interested in studying congruences modulo (powers of) a prime p between these eigenvalues. This is most efficiently done through a systematic study of p-adic analytic variation of modular forms.
In my talk I will survey the progress in this area from the conception of the notion of a p-adic analytic family of modular forms to Coleman-Mazur’s construction of the eigencurve which is in some sense the “universal” such family.
|
S.I.G.M.A. Seminar
Speaker: Lance Miller (University of Connecticut)
Time: Wednesday, February 16, 2005 at 4:30 pm
Place: MSB (UConn - Storrs)
Abstract: Boolean Algebras arise naturally in a variety of areas of mathematics and computer science. It can be shown that finite Boolean Algebras arise only as powersets of a given set. However this is not the case in the infinite setting (i.e., there are Boolean Algebras which are not powerset algebras). M.H. Stone though gave a classification of (infinite) Boolean Algebras when he in 1936 proved that every Boolean Algebra is the algebra of clopen subsets of a compact hausdorff space. This talk will attempt to present this theorem by associating to a boolean algebra a commutative ring called a boolean ring and looking at the Zariski topology over its prime spectrum.
|
UConn Math Club
Speaker: Alexander Russell (University of Connecticut)
Time: Wednesday, February 16, 2005 at 5:30 pm
Place: MSB 118 (UConn - Storrs)
Abstract: We survey some elements of combinatorial game theory, including symmetry, strategy stealing, and decomposition. To illustrate these phenomena, we introduce some of the darlings of the discipline: the games Nim, Chomp, and Disjoint Discs.
Comments: Free Refreshments
|
Geometry Seminar
S.I.G.M.A. Seminar
Speaker: $Unique empty (University of Connecticut)
Time: Wednesday, February 23, 2005 at 4:30 pm
Place: MSB 117 (UConn - Storrs)
Abstract: I will begin by briefly recounting how I came to be in
mathematics, then analysis, finally Banach algebras and Banach function
spaces. Along the way I will define Banach algebra, show how once one has the basic theory of Banach algebras a famous theorem of Wiener becomes almost trivial, and describe how the revelation of this last fact had a formative influence on me. Next we will look at uniform algebras
(uniformly closed point-separating algebras of continuous complex-valued functions on a compact Hausdorff space), and notice that much of their study involves determining to what extent they resemble algebras of
analytic functions. We will move on to the study of the "real part" of a
uniform algebra, and see that a key role is played by a phenomenon known as "ultraseparation," which in turn involves the Stone-Cech
compactifications of certain non-compact spaces. The total effect is that
the areas of Banach algebras and Banach function spaces provide a fertile
meeting place for algebra, topology, complex analysis, real analysis, and
functional analysis.
|
UConn Math Club
Speaker: Gerald Dunne (University of Connecticut)
Time: Wednesday, February 23, 2005 at 5:30 pm
Place: MSB 118 (UConn - Storrs)
Abstract: In 2004, the Nobel Prize in Physics was awarded to David Gross, David Politzer and Frank Wilczek for their discovery of “asymptotic freedom”. Asymptotic freedom is a startling property of quantum chromodynamics (QCD), which is the theoretical framework which physicists use to describe the fundamental forces that act deep inside protons and neutrons. I will describe in very basic terms what asymptotic freedom is and why it is so important for physics. I will also describe in very simple terms why the related (but more difficult) problem of “confinement” in QCD is central to one of the Clay Institute's Millenium Prize Problems in Mathematics: the proof of the existence of a “mass gap” in Yang-Mills theory.
Not much (if any) prior physics knowledge will be needed.
Comments: Free Refreshments
|
Analysis and Probability Seminar
Speaker: Ron Blei (University of Connecticut)
Time: Friday, February 25, 2005 at 3:00 pm
Place: MSB 118 (UConn - Storrs)
Abstract: Fractional Cartesian products and a subsequent measurement
of combinatorial dimension appeared first in a harmonic-analytic
context in the course of filling "analytic" gaps between successive
(ordinary) Cartesian products of spectral sets. Succinctly put,
combinatorial dimension is an index of interdependence. Attached to a
subset of an ordinary Cartesian product, it gauges the interdependence
of restrictions to the set, of the canonical projections from the
Cartesian product onto its independent coordinates. We can analogously
gauge the interdependence of restrictions to the same set, of
projections from the Cartesian product onto interdependent coordinates
of a prescribed fractional Cartesian product. We thus obtain distinct
indices of interdependence associated, respectively, with distinct
fractional Cartesian products. In this talk I will describe some
results concerning relations between these indices.
This is joint work with Fuchang Gao.
|
Logic Seminar
Speaker: Dennis Hirschfeldt (University of Chicago)
Time: Monday, February 28, 2005 at 4:45 pm
Place: Exley Science Center 618 (Wesleyan University)
Abstract: There has been a recent upsurge in interest among computability
theorists in the study of effective randomness. One of the most
interesting consequences has been the emergence of a rich collection of
results relating to a natural class of ``far from random'' reals called
the K-trivial reals. This class had been investigated in the 1970's by
Solovay and Chaitin, but it is only in the last few years that we have
realized its importance, both in the theory of relative effective
randomness and in more classical parts of computability theory. In this
talk, I will introduce the concept of K-triviality and discuss recent work
which I believe indicates that it is both a natural notion and one of
considerable mathematical interest.
|
Geometry Seminar
Geometry Seminar
Speaker: Jesse Ratzkin (University of Connecticut)
Time: Tuesday, March 1, 2005 at 4:00 pm
Place: MSB 117 (UConn - Storrs)
Abstract: Consider n Brownian predators pursuing one Brownian prey, with all motion restricted to a line. For which n is the expected capture time finite? I will discuss joint work with Andrejs Treibergs which answers this question by finding effective lower bounds for the first eigenvalue of certain spherical domains.
|
S.I.G.M.A. Seminar
Speaker: Ray Mines
Time: Wednesday, March 2, 2005 at 4:30 pm
Place: MSB 118 (UConn - Storrs)
Abstract: I will look at some of the interesting situations that
arise when one wants to compute the generators of an ideal.
Prerequisite:
Understand the meaning of the sentence "The ring of integers is a PID."
|
Analysis and Probability Seminar
Speaker: Daniel Look (Boston University)
Time: Wednesday, March 2, 2005 at 3:00 pm
Place: MSB 118 (UConn - Storrs)
Abstract: For the family of quadratic maps Qc(z)=z2+c there exists
the "Fundamental Dichotomy" which states that the topological structure
of the Julia set for the map Qc depends greatly on the orbit of the
critical point. If the critical orbit escapes to infinity the Julia set
is a Cantor set whereas if the critical orbit is bounded the Julia set is
connected.
We will explore a similar occurrence in the family of maps given by
FL(z)=zn+L/zd. For this family we have an escape
trichotomy in that there are 3 distinct ways in which the critical orbit
can escape to infinity and we will show that the Julia set will be a
Cantor set, a Cantor set of circles or a Sierpinski curve, depending on
how the critical orbit escapes.
|
UConn Math Club
Speaker: James Bridgeman (University of Connecticut)
Time: Wednesday, March 2, 2005 at 5:30 pm
Place: MSB 118 (UConn - Storrs)
Abstract: Over the past 50 years or so, financial economists have developed a set of increasingly complex and sophisticated mathematical models for the behavior of financial markets and for the value of the financial instruments (stocks, bonds, currencies, and more exotic “derivatives” thereof) traded in financial markets. Over perhaps the past 35 years, with exponential growth in computing power, there has been an explosion in the application of these theories to practical management of investments and of the financial underpinnings of many businesses. New financial instruments created by using the theories have come to rival (some would say overwhelm) traditional financial markets in some sectors.
Perhaps surprisingly, behind the enormous complexities of these theories and of their applications, there often stands an extremely simple, even simple-minded, set of little assumptions about how markets ought to work, and how people ought to value financial instruments. The talk will try to uncover the simple-minded core assumptions behind two of the theories in constant use: the Capital Asset Pricing Model for the risk/reward equilibrium values of financial instruments in the market and the Risk-Neutral Pricing Model for a “derivative” financial instrument (one whose value “derives” in some way from the value of another financial instrument). Little more than some calculus, vector-space geometry, and probability theory will be used
Comments: Free Refreshments
|
Colloquium
Speaker: Alice Chang (Princeton University)
Time: Monday, March 14, 2005 at 4:00 pm
Place: MSB 118 (UConn - Storrs)
Abstract: Based on the work of C. Fefferman and R. Graham in 1985, there have been systematic study of the conformal invariants and conformally covariant operators on manifolds of any dimensions. A special case of such an operator is a fourth order linear elliptic operator with its leading symbol the bi-Laplace operator called the Paneitz operator. In this talk, I will discuss study of this operator on four manifolds, its associated curvature function called the Q-curvature, connection of Q-curvature to the study of eigenvalues of the Ricci tensor and applications to some problems in geometry. I will also discuss some open questions in the area.
|
Colloquium
Speaker: Paul Yang (Princeton University)
Time: Tuesday, March 15, 2005 at 4:00 pm
Place: MSB 118 (UConn - Storrs)
Abstract: In this talk, I will explain first the pseudo-hermitian geometry which bears strong analogy to conformal geometry in the setting of Cauchy-Riemann structures. I then explain the notion of minimal surface in such geometry. I will present some existence results as well as some classification results generalizing the classical Bernstein theorem to this setting.
|
UConn Math Club
Speaker: Rob Benedetto (Amherst College)
Time: Tuesday, March 15, 2005 at 5:30 pm
Place: MSB 118 (UConn - Storrs)
Abstract: If you take a polynomial like ƒ(z) = z2 - 1 and start to compose it with itself, you get higher and higher degree polynomials:
ƒ(ƒ(z)) = z4 - 2z2,
ƒ(ƒ(ƒ(z))) = z8 - 4z6 + 4z4 - 1, …
Discrete dynamics is the study of what happens when you take higher and higher compositions (or iterations) of ƒ; complex dynamics is discrete dynamics when the coefficients and the variable are all complex numbers. In general, there is no nice formula for what the n-th iterate of ƒ looks like. Fortunately, we can figure out a lot of useful properties of the iterates without actually finding formulas for them. In particular, we'll talk about periodic points (and especially attracting periodic points) and how they can be used to understand a lot about the dynamics of ƒ. That will lead us into a look at Fatou and Julia sets, and then to a brief introduction to the famous Mandelbrot set, which is more than just a pretty picture.
No knowledge of complex analysis will be assumed, but it will help to have at least seen complex numbers used before, even in passing.
Comments: Free Refreshments; USG funded
|
S.I.G.M.A. Seminar
Speaker: Keith Conrad (University of Connecticut)
Time: Wednesday, March 16, 2005 at 4:30 pm
Place: MSB 118 (UConn - Storrs)
Abstract: Any complete normed vector space has plenty of linear
functionals, in fact enough to distinguish different points. A particular
class of such examples is the Lp-spaces for p at least 1. What happens in
Lp-spaces if 0 < p < 1? We will define these spaces in a reasonable way
and show their dual space is 0. Then we will try to put the ideas from the
proof into a more general context.
|
Colloquium
Speaker: Mariel Vasquez (University of California at Berkeley)
Time: Thursday, March 17, 2005 at 1:00 pm
Place: ITEB Conference Room 336 (UConn - Storrs)
Abstract: DNA topology is the study of geometrical (supercoiling) and topological
(knotting) properties of DNA loops and circular DNA molecules. Virtually
every reaction involving DNA is influenced by DNA topology, or has
topological effects. Site-specific recombinases and topoisomerases are
enzymes able to change the topology of circular DNA by breaking the DNA and introducing one or more crossing changes. Mathematical analysis of such changes may provide relevant information about the possible enzymatic pathways, and about DNA conformation at the moment of double-stranded break induction. In this talk I will discuss some of the problems that I am currently interested in, and the topological tools used in their analyses.
First I will talk about Xer recombination and how we applied, and
extended, the tangle model for site-specific recombination to propose a
unique enzymatic mechanism. I will then present the Java applet
TangleSolve that makes the tangle model easily accessible to the
interested molecular biologist.
Finally I will talk about our recent work on DNA unknotting by type II topoisomerase.
Comments: Discussion and tea afterwards, 2-3pm, in ITEB 201A
|
Colloquium
Speaker: Chengyang Xu (Siemens Corporate Research)
Time: Thursday, March 17, 2005 at 3:00 pm
Place: ITE 336 (UConn - Storrs)
Abstract: In the past four decades, computerized image segmentation has
played an increasingly important role in medical imaging. Segmented
images are now used routinely in a multitude of different
applications, such as the quantification of tissue volumes,
diagnosis, localization of pathology, study of anatomical
structure, treatment planning, partial volume correction of
functional imaging data, and computer-assisted surgery. Image
segmentation remains a difficult task, however, due to both the
tremendous variability of object shapes and the variation in image
quality. In particular, medical images are often corrupted by noise
and sampling artifacts, which can cause considerable difficulties
when applying classical segmentation techniques such as edge
detection and thresholding. As a result, these techniques either
fail completely or require some kind of postprocessing step to
remove invalid object boundaries in the segmentation results.
To address these difficulties, deformable models have been
extensively studied and widely used in medical image segmentation,
with promising results. Deformable models are curves or surfaces
defined within an image domain that can move under the influence of
internal forces, which are defined within the curve or surface
itself, and external forces, which are computed from the image
data. By constraining extracted boundaries to be smooth and
incorporating other prior information about the object shape,
deformable models offer robustness to both image noise and boundary
gaps and allow integrating boundary elements into a coherent and
consistent mathematical description. Such a boundary description
can then be readily used by subsequent applications. The popularity
of deformable models is largely due to the seminal paper "Snakes:
Active Contours" by Kass, Witkin, and Terzopoulos in 1989. Since
its publication, deformable models have grown to be one of the most
active and successful research areas in image segmentation.
There are basically two types of deformable models: parametric
deformable models and geometric deformable models. Parametric
deformable models represent curves and surfaces explicitly in their
parametric forms during deformation. This representation allows
direct interaction with the model and can lead to a compact
representation for fast real-time implementation. Adaptation of
the model topology, however, such as splitting or merging parts
during the deformation, can be difficult using parametric models.
Geometric deformable models, on the other hand, can handle
topological changes naturally. These models, based on the theory of
curve evolution and the level set method, represent curves and
surfaces implicitly as a level set of a higher-dimensional scalar
function. Their parameterizations are computed only after complete
deformation, thereby allowing topological adaptivity to be easily
accommodated. Despite this fundamental difference, the underlying
principles of both methods are very similar.
In this talk, I will present an overall description of the
development in deformable models research and their applications in
medical imaging. I will first introduce parametric deformable
models, and then describe geometric deformable models. Next, I
will present an explicit mathematical relationship between
parametric deformable models and geometric deformable models.
Finally, I will present several extensions to these deformable
models by various researchers and point out future research
directions.
Comments: This is a Joint Seminar with the department of Electrical and Computer Engineering
|
Analysis and Probability Seminar
Speaker: Vladimir Koltchinskii (University of New Mexico)
Time: Friday, March 18, 2005 at 3:00 pm
Place: MSB 118 (UConn - Storrs)
Abstract: The abstract can be viewed more accurately as a PDF file using the link at the end.
Let (X,Y), (X1,Y1), ... ,(Xn,Yn) be a sequence of i.i.d. random
couples in S × R, S being a compact, for simplicity.
Let ? denote the distribution of X.
Based on (X1,Y1), ... ,(Xn,Yn), we want to approximate the
function gast:S -> R that minimizes
L(g):=E(Y-g(X))2
(in fact, gast= E(Y|X=x) and it is called the
regression function).
An approach that has become very popular in the recent years is based
on trying to find a good approximation of gast in a reproducing
kernel Hilbert space (RKHS) HK with some kernel K. It is done,
for instance, by minimizing
Ln,?(g):=n-1?j=1n (Yj-g(Xj))2+?
|g|HK2
over the whole space HK
where ?>0 is a regularization parameter. Let hat{g}n denote
the solution of the above optimization problem. The question is what
one can say about the size of |hat{g}n-gast|2L2(?)
in terms of the kernel K (say, its spectral properties).
This question has been looked at by a number of authors (Smale and
Cucker;
Mendelson; Blanchard, Bousquet and Massart among others).
The goal of the talk will be to explain how this and other questions
of this nature can be answered based on very general excess risk
bounds
that are related to famous concentration inequalities by Talagrand.
<Extra Information>
|
PDE and Image Analysis Seminar
Speaker: Congming Li (University of Colorado)
Time: Monday, March 21, 2005 at 3:00 pm
Place: MSB 118 (UConn - Storrs)
Abstract: I will present the joint work with T. Hou about the global
well-posedness of the Lagrangian averaged Euler equations in three
dimensions. We show that a necessary and sufficient condition for the
global existence is that the BMO norm of the stream function is
integrable in time. We also derive a sufficient condition in terms of
the total variation of certain level set functions, which guarantees the
global existence. Furthermore, we obtain the global existence of the
Lagrangian averaged 2D Boussinesq equations and the Lagrangian averaged
2D quasi-geostrophic equations in finite Sobolev space in the absence of
viscosity or dissipation.
|
Algebra Seminar
Speaker: Keith Conrad (University of Connecticut)
Time: Tuesday, March 22, 2005 at 4:00 pm
Place: MSB 118 (UConn - Storrs)
Abstract: This expository talk will survey work on the proportion of number fields with a given discriminant and
Galois group, from very classical results to recent work of Bhargava.
|
Geometry Seminar
Speaker: John Sullivan (T. U. Berlin)
Time: Tuesday, March 22, 2005 at 4:30 pm
Place: MSB 411 (UConn - Storrs)
Abstract: Following Kuperberg, we define which subarcs of a knotted
curve are (topologically) essential, creating the knottedness.
Using a result of Denne - that knots have essential alternating
quadrisecants - we can show the ropelength of any knot is
at least 15.66, within 5 percent of the known upper bound for the trefoil.
By considering the shortest essential arc of a knot, we can
show that the Gromov distortion is at least 3.99, more than
twice the value for an unknotted circle. This is largely joint
work with Elizabeth Denne and Yuanan Diao.
|
UConn Math Club
Speaker: Peter Garrity (Columbia)
Time: Wednesday, March 23, 2005 at 5:30 pm
Place: MSB 118 (UConn - Storrs)
Abstract: Math for America has put forth a great opportunity
for any math, engineering, computer science or applied math
major who has an interest in teaching. It may even be
a newfound interest, particularly as you approach graduation
and go through the "What am I going to do now?" angst. Consider
applying for a Newton Fellowship.
Each fellowship includes a series of substantial stipends,
a Master's in Mathematics Education and the chance to
teach for four years in a New York City high school.
Teachers College/Columbia University has been selected
as the premier institution to facilitate the
Newton Fellowship Program and work with you as you begin
your teaching career in New York City. There are 20 fellowships
for the next academic year.
The benefits to you include:
-
A tuition-free masters in mathematics education.
-
A first-year stipend of $28,000 as you get your masters.
-
An exciting four years of teaching in a New York City high school.
-
Additional salary compensation of 11, 13, 17, and 20 thousand
dollars above your New York City teaching salary and benefits.
-
A great education in life and the opportunity to make a difference.
Dr. Peter Garrity of Teachers College is responsible for the
development and facilitation of this program at Columbia.
He will explain the details of the program, and answer
any questions you may have.
Check out the Math for America website!
Comments: Free Refreshments
|
Colloquium
Speaker: Steve Miller (Brown University)
Time: Thursday, March 24, 2005 at 4:00 pm
Place: MSB 118 (UConn - Storrs)
Abstract: A fundamental problem in science is to understand spacings between adjacent events, where the events range from prime numbers to energy levels of Uranium to eigenvalues of matrices to zeros of L-functions. Surprisingly, very diverse systems exhibit similar behavior. In particular, the techniques developed to understand Uranium can be used to study the primes and elliptic curves! We'll explore some of these systems, and show the number theory analogues of the nuclear physics tools.
|
S.I.G.M.A. Seminar
Speaker: Tom Peters (University of Connecticut)
Time: Thursday, March 24, 2005 at 4:30 pm
Place: ITEB 336 (UConn - Storrs)
Abstract: This talk will introduce an emerging view of computational
topology in geometric modeling when spline surfaces are used for
approximate representations of manifolds. Applications in engineering and
molecular modeling will be presented. Intersection algorithms are
fundamental in these constructions and work on our NSF I-TANGO project
(Intersections -- Topology, Accuracy and Numerics for Geometric Objects)
will be reported, both completed results and ongoing activity. Supporting
animations will be shown to aid intuition, particularly regarding our use
of a strong form of topological equivalence from knot theory.
|
Logic Seminar
Algebra Seminar
Speaker: Keith Conrad (University of Connecticut)
Time: Tuesday, March 29, 2005 at 4:00 pm
Place: MSB 118 (UConn - Storrs)
Abstract: This talk is meant as preparatory material for next week's talk. We will
discuss the Artin isomorphism in algebraic number theory. After defining and
illustrating a few general notions from algebraic number theory (ideal
class groups, ramification of primes, and Frobenius elements), we will see how
the Artin isomorphism lets us view ideal class groups as Galois groups and look
at some examples.
|
S.I.G.M.A. Seminar
Speaker: Jonathan Axtell (University of Connecticut)
Time: Wednesday, March 30, 2005 at 4:30 pm
Place: MSB 117 (UConn - Storrs)
Abstract: In 1941, S. M. Ulam and P. J. Kelly made the conjecture that any
two graphs with the same set of one-vertex deleted subgraphs
must be isomorphic. While Kelly showed that all trees are
reconstructible, very few classes of reconstructible graphs are
known. The Reconstruction Conjecture is much like Goldbach's
Conjecture in that it is so easy to state, yet so difficult to prove.
In 1964, Frank Harary gave the problem a colorful twist by
imagining each unlabeled one-vertex deleted subgraph as being
drawn on a 3x5 index card. The collection of such cards is called
a deck. Then, the conjecture states that, two graphs are isomorphic
if and only if they have isomorphic decks.
Besides giving the basic definitions and history, I will try to give
another point of view for approaching this problem.
|
UConn Math Club
Speaker: Robert Pollack (Boston University)
Time: Wednesday, March 30, 2005 at 5:30 pm
Place: MSB 118 (UConn - Storrs)
Abstract: We will look carefully at a few polynomial equations and attempt to find all
of their integer solutions. For instance, what are all the integer
solutions to y2 = x3 - 2 .
In general, finding all the integer solutions to equations of this kind can
be extremely difficult, but we will see that by using
“prime factorization”
in a very clever way, a great deal of information can be teased out of these
equations.
Comments: Free Refreshments; USG funded
|
Special Talks
Speaker: Jonathan Bilmes
Time: Thursday, March 31, 2005 at 11:00 am
Place: MSB 403 (UConn - Storrs)
Abstract: In the 1980s, Connecticut made a major philosophical and financial commitment to municipal solid waste to
energy technology. This lecture will discuss the fundamental technical and political issues related to municipal
solid waste to energy. In addition, the lecture will demonstrate that this technology is entirely compatible
with recycling efforts. The lecture will consist of a short video, a presentation and Q & A.
Comments: Guest Lecture at Mathematical Modeling in the Environment Course
<Extra Information>
|
Colloquium
Speaker: Fuchang Gao (University of Idaho)
Time: Thursday, March 31, 2005 at 4:00 pm
Place: MSB 118 (UConn - Storrs)
Abstract: Biofilms are communities of microorganisms attached to a surface. According to a public announcement of NIH, "biofilms are medically important, accounting for over 80 percent of microbial infections in the body." Yet, the effective treatment of biofilm infection remains underdeveloped. One of the main challenges is that the bacteria within biofilms tend to be far more resistant to biocides and antibiotics. Despite the increasing interest, little is known about how such resistance develops in biofilms. Various mechanisms have been postulated or demonstrated. The conclusions of these studies converge to a need of a better understanding of the biofilm spatial heterogeneities. Because the spatial heterogeneities in biofilms involve multiple factors, to study them requires a non-destructive method, which the current experimental techniques alone fail to achieve.
By mathematical modeling and computer simulations, we investigate how the spatial structure of the biofilm affects the physicochemical and physiological heterogeneities, and how the latter in turn influences the development of the spatial structure. Through simulation, we show how the non-Fickian diffusion process in biofilms, together with the variation of metabolic activities, creates heterogeneous local environments in biofilms, which may pose selective pressure. As a consequence, we found that there are regions in a biofilm where the energy level of the cells is much lower than that in other regions. This may relate to the rpoS genes expression pattern in some E.coli biofilms.
The main mathematical content of this work is the modeling of non-Fickian diffusion process, which in some ideal case, is related to Fractional Brownian Motion.
Part of this talk is from a joint work with Zhong, La, Krone and Forney
|
Geometry Seminar
Speaker: Saul Schleimer (Rutgers University)
Time: Friday, April 1, 2005 at 3:30 pm
Place: 117 MSB (UConn - Storrs)
Abstract: Rubinstein and Thompson gave an algorithm to decide if a triangulated three-manifold is homeomorphic to the three-sphere. We will discuss why the problem lies in NP. As the ideas behind their algorithm and my improvements are complicated, I'll begin by surveying a related simpler problem, solved by Hass et al.: unknot recognition lies in NP.
|
Algebra Seminar
Speaker: Michael Bush (University of Massachusetts)
Time: Tuesday, April 5, 2005 at 4:00 pm
Place: MSB 118 (UConn - Storrs)
Abstract: This talk will begin with an introduction to some basic
objects in algebraic number theory including class groups and
(Hilbert) class towers. A computational method for obtaining the Galois
group associated to the Hilbert p-class tower of a number field K
will then be described together with results for various imaginary
quadratic fields K when p = 2 or 3. If time permits we will discuss
some exciting new developments regarding the structure of Galois
groups of maximal p-extensions where ramification at a finite set
of primes (excluding p) is allowed. The talk should be accessible to graduate students.
|
Geometry Seminar
Speaker: Fred Gardiner (CUNY)
Time: Tuesday, April 5, 2005 at 4:00 pm
Place: MSB 117 (UConn - Storrs)
Abstract: We discuss how the Teichmuller model for the full horseshoe map leads to consideration of the baker map, a hyperelliptic cover of infinite genus, and a pseudo-Anosov axis in Teichmuller space.
|
S.I.G.M.A. Seminar
Speaker: Gorjan Alagic (University of Connecticut)
Time: Wednesday, April 6, 2005 at 4:30 pm
Place: MSB 117 (UConn - Storrs)
Abstract: In classical computation, random walks are an important algorithmic and theoretical concept. Recent research in quantum computation has led to the development of some quantum analogues of the classical random walk.
The talk will begin with an introduction to the basic ideas of random walks on graphs and groups. We will then attempt to sketch out the basic ideas behind a version of quantum walks prevalent in current research: the continuous-time quantum walk. Some recent results (joint work with Alexander Russell) will be mentioned.
|
UConn Math Club
Speaker: Sarah Glaz (University of Connecticut)
Time: Wednesday, April 6, 2005 at 5:30 pm
Place: MSB 118 (UConn - Storrs)
Abstract: A flexagon is a polygon folded from a strip of paper, which
has the fascinating property of changing faces when it is flexed.
Flexagons where discovered by accident in 1939, when Arthur Stone,
a 23 year old graduate student at Princeton, had to trim off
a strip of paper from his British notebook page to be able
to fit it into his American binder. By folding the trimmed-off
strip of paper in various ways, he soon constructed the
first flexagon: the three-faced trihexaflexagon.
I will elaborate on some interesting aspects of the history
and properties of flexagons. I will also show you how to
construct and flex your own trihexaflexagon, which you
may take with you to dazzle your family members and friends.
We will also use the trihexaflexagon to sneak up on a group.
Comments: Free Refreshments
|
Colloquium
Speaker: Matt Szczesny (University of Pennsylvania)
Time: Thursday, April 7, 2005 at 4:00 pm
Place: MSB 118 (UConn - Storrs)
Abstract: Mathematicians have devoted much effort towards rigorously
describing various aspects of quantum field theory (QFT). One
mathematical object that captures the algebraic content of (some)
QFT's is a vertex algebra.
Given a complex manifold $X$, one can attach to it a QFT known as the
"supersymmetric sigma-model with target X", denoted $Sigma (X)$. One
would of course like to have a geometric construction of the vertex
algebra underlying $Sigma (X)$. A candidate for such a construction
is the chiral de Rham complex introduced by Malikov, Schechtman, and
Vaintrob in 1998. It is a sheaf of vertex algebras on $X$, denoted
$Omega^{ch}_X$, whose cohomology captures interesting "stringy"
invariants related to $Sigma(X)$.
This talk will introduce vertex algebras, the sheaf $Omega^{ch}_X$,
and its relation with algebraic loop-spaces. I will discuss joint work
with Edward Frenkel extending the construction of $Omega^{ch}_X$ to
orbifolds. I will describe how it captures stringy geometric
invariants of orbifolds such as Chen-Ruan orbifold cohomology and
orbifold elliptic genera. Finally, I will discuss joint work with
Anatoly Libgober on discrete torsion, and how beautiful generating
functions that arise in this context seem to have a deep relation with
the theory of automorphic lifts.
|
Algebra Seminar
Speaker: Sarah Glaz (University of Connecticut)
Time: Tuesday, April 12, 2005 at 4:00 pm
Place: MSB 118 (UConn - Storrs)
Abstract: Let R be a commutative ring, and let f be a polynomial with coefficients in R. The content of f, c(f), denotes the ideal of R generated by the coefficients of f. A ring R is called a Gaussian ring if c(fg) = c(f)c(g) for all polynomials f and g with coefficients in R. Gaussian rings were defined by Tsang (1965). Gilmer (1967) had shown that for an integral domain R the Gaussian property and the Prufer domain property are equivalent. This talk revolves around some recent results obtained by the speaker jointly with Silvana Bazzoni, regarding the extension of Gilmer’s result to rings with zero-divisors. In particular, we consider two very closely related extensions of the notion of a Prufer domain to rings with zero-divisors, arithmetical rings and Prufer rings. The class of arithmetical rings is contained in the class of Gaussian rings, which is in turn contained in the class of Prufer rings. We explore the extent to which these three classes of rings are different, and the conditions under which they coincide.
|
Geometry Seminar
S.I.G.M.A. Seminar
Speaker: Oscar Levin (University of Connecticut)
Time: Wednesday, April 13, 2005 at 4:30 pm
Place: MSB 117 (UConn - Storrs)
Abstract: Ramsey Theory is a branch of combinatorics that studies the
conditions under which seemingly random structures must contain very
specific and "nice" substructures. For instance, when must a
multi-dimensional tic-tac-toe game always have a winner? And there are
applications to mathematics as well. When must a sequence of natural
numbers contain an arbitrarily long arithmetic progression? In this talk we
shall answer both of those and other questions, as well as pose some open
questions for consideration.
|
UConn Math Club
Speaker: Max Lieblich (Brown University)
Time: Wednesday, April 13, 2005 at 5:30 pm
Place: MSB 118 (UConn - Storrs)
Abstract: How many prime numbers are there? OK, most of us know the answer to this one: there are infinitely many. A related question is: how many ways are there to prove this? The answer: there are also infinitely many! We will discuss all of them (if time permits).
Comments: Free Refreshments; USG funded
|
Logic Seminar
Algebra Seminar
Speaker: Roger Wiegand (University of Nebraska)
Time: Tuesday, April 19, 2005 at 4:00 pm
Place: MSB 118 (UConn - Storrs)
|
Geometry Seminar
Speaker: Craig Westerland (Institute for Advanced Study)
Time: Tuesday, April 19, 2005 at 4:00 pm
Place: MSB 117 (UConn - Storrs)
Abstract: I will discuss Koszul duality for operads in a slightly broader context than described by Ginzburg and Kapranov. This will alow us to compute the Koszul dual of the framed disk operads. I'll then make some applications to and raise a few conjectures about string topology and how it interacts with the Koszul duality picture.
|
S.I.G.M.A. Seminar
Speaker: Brian Conrad (University of Michigan)
Time: Wednesday, April 20, 2005 at 4:30 pm
Place: MSB 117 (UConn - Storrs)
Abstract: Matrix groups show up quite a lot in mathematics, and sometimes it is important to know that certain such groups are connected. For example, the group of invertible n by n real matrices is not connected because the determinant on this group is continuous and jumps from positive to negative values without going through 0. One is then led to wonder if the determinant is the only obstruction to connectivity: is the group of invertible real matrices with positive determinant connected? We show how the Gram-Schmidt algorithm, viewed in a suitably dynamic manner, enables us to "see" that such connectivity does hold, and that it even provides explicit paths linking any matrix to the identity. If time permits, we show how this geometric visualization also enables one to give geometric proofs of the well-definedness of the signature of quadratic forms over the reals and the fact that the "space" of quadratic forms with a fixed signature is itself path-connected (i.e., any two such forms can be continuously deformed into each other). An important principle in all of these arguments is to avoid the explicit use of coordinates (which tends to mask the underlying structures at hand) and to let geometry show us what to do.
|
UConn Math Club
Speaker: Brian Conrad (University of Michigan)
Time: Wednesday, April 20, 2005 at 5:30 pm
Place: MSB 118 (UConn - Storrs)
Abstract: It has been said that integration is an art, whereas
differentiation is merely a skill. Even beginning calculus students
quickly recognize that integration is a fundamentally much harder problem
than differentiation.
In fact, in the 19-th century Liouville proved that certain simple kinds of
functions
do not have anti-derivatives that can be written down
“in elementary terms.”
(Of course, one has to give a reasonable and precise definition of what
“elementary” means.) This is analogous to the fact that
the roots to many polynomials of degree above four can't be written down
exactly using standard arithmetic operations and root extractions.
There is a powerful analogy between the theories of solving for roots of
polynomials and solving certain kinds of
differential equations (integrating).
The main new idea that we need to understand for Liouville's theorem
is a differential field. Assuming just knowledge of calculus and an
interest in
understanding how on earth one could really prove that a function doesn't
have an “elementary”
anti-derivative, we will introduce differential fields and apply them to
the solvability of some concrete differential equations. In particular,
we will see why
e–x2
cannot be integrated in elementary terms.
Comments: Free Refreshments; USG funded
|
Special Talks
Speaker: Allison Pacelli (Williams College)
Time: Thursday, April 21, 2005 at 4:00 pm
Place: IMS 20 (UConn - Storrs)
Abstract: Have you ever wondered if your vote really matters? Recent
Presidential Elections may have convinced you that it does. Hanging chads,
butterfly ballots, recount upon recount... What happened in 2000, and how
can we be sure it doesn't happen again? Rather than blame the Electoral
College, Florida, or Ralph Nader, perhaps we should reconsider the very
notion of voting. Is there a better method to a better democracy? The answer,
revealed in this presentation, may surprise you. Is democracy doomed, or
will mathematics save the day?
NO mathematical background is assumed.
Comments: Awards Day
|
Analysis and Probability Seminar
Speaker: Ron Blei (University of Connecticut)
Time: Friday, April 22, 2005 at 3:00 pm
Place: MSB 118 (UConn - Storrs)
Abstract: Fractional Cartesian products and a subsequent measurement
of combinatorial dimension appeared first in a harmonic-analytic
context in the course of filling "analytic" gaps between successive
(ordinary) Cartesian products of spectral sets. Succinctly put,
combinatorial dimension is an index of interdependence. Attached to a
subset of an ordinary Cartesian product, it gauges the interdependence
of restrictions to the set, of the canonical projections from the
Cartesian product onto its independent coordinates. We can analogously
gauge the interdependence of restrictions to the same set, of
projections from the Cartesian product onto interdependent coordinates
of a prescribed fractional Cartesian product. We thus obtain distinct
indices of interdependence associated, respectively, with distinct
fractional Cartesian products. In this talk I will describe some
results concerning relations between these indices.
This is joint work with Fuchang Gao.
|
Algebra Seminar
Geometry Seminar
Speaker: Andy Haas (University of Connecticut)
Time: Tuesday, April 26, 2005 at 4:00 pm
Place: MSB 117 (UConn - Storrs)
Abstract: A generic geodesic on a finite area, non-compact surface with a hyperbolic orbifold structure travels densely about the surface, making excursions out the non-compact end and then returning to a compact piece of the surface. We showed that for almost all geodesics on such a surface there is a limiting distribution for the sequence of depths of the excursions out a non-compact end. The distribution is independent of the surface and closely resembles the distribution arising from a sequence of continued fraction approximations in the metrical theory of diophantine approximation. This metrical theory can then be reproduced in the setting of approximation by parabolic fixed points in a Fuchsian group.
|
S.I.G.M.A. Seminar
Speaker: Matthew Jura (University of Connecticut)
Time: Wednesday, April 27, 2005 at 4:30 pm
Place: MSB 117 (UConn - Storrs)
Abstract: Finite fields are used extensively in the areas of applied
algebra and number theory. In this talk we will investigate properties
of finite fields – first a brief background in the area of finite fields
and field extensions will be supplied, culminating with an example of the
construction of a finite field. We will then deduce a nice equation
relating the parity of the number of irreducible factors of a monic
square free polynomial over a perfect (finite) field of odd prime
characteristic to the degree of the polynomial. We will show how this
result is a direct consequence of the fact that the Galois group of any
finite field is cyclic, and therefore the result holds for any field with
cyclic Galois group.
|
UConn Math Club
Speaker: Bill Wickless (University of Connecticut)
Time: Wednesday, April 27, 2005 at 5:30 pm
Place: MSB 118 (UConn - Storrs)
Abstract: We will present some facts about the well-known Rubik's Cube, a popular puzzle introduced some years ago. In doing so, we'll illustrate the concept of a group, a basic mathematical notion that arises in a number of areas. No previous knowledge of either groups or Rubik's Cube will be required.
Comments: Free Refreshments
|
Analysis and Probability Seminar
Speaker: Evarist Gine (University of Connecticut)
Time: Friday, April 29, 2005 at 3:00 pm
Place: MSB 118 (UConn - Storrs)
Abstract: A notion of local U-statistic process is introduced and central limit
theorems in various norms are obtained for it. This involves the
development
of several inequalities for U-processes that may be useful in other
contexts. This local U-statistic process is based on an estimator of the
density of a function of several sample variables proposed by Frees
(1994)
and, as a consequence, uniform in bandwidth central limit theorems in
the
sup and in the $L_p$ norms are obtained for these estimators.
LIL's are also obtained. This is joint work with David Mason.
|
Colloquium
Speaker: Alexander Zelikovsky (Georgia State University)
Time: Thursday, May 19, 2005 at 11:00 am
Place: ITEB 336 (UConn - Storrs)
Abstract: Arguably, the most intriguing questions in population genetics are
concerned with haplotype association of complex genetic diseases. In
order to establish such associations it is necessary to resolve numerous
ambiguities in real genetic data. The main obstacle is that the methods
of inferring two haplotypes from genotype individual data are too
expensive. Although computational haplotype inferring is well-explored,
still too high error rate deteriorates association abilities. Another
important problem is to computationally determine SNPs that should be
taken in account while dropping others thus reducing noisy and linked
data. Finally, even if clean data is available, it is unclear if common
statistical methods can determine susceptibility of complex diseases. We
discuss mostly discrete combinatorial approaches to allowing prediction
of Crohn's disease with high accuracy.
________________________________________________________________________
Short bio
Alexander Zelikovsky received the Ph.D. degree in Computer Science from
the Institute of Mathematics of the Belorussian Academy of Sciences in
Minsk (Belarus) in 1989. Alexander is an Associate Professor at Computer
Science Department of Georgia State University. He authored over 90
refereed publications. Dr. Zelikovsky's research interests include VLSI
physical design, computational biology, ad-hoc wireless networks, and
combinatorial optimization. He served as a Chair or PC member of SLIP
2000, ISMP 2000, APPROX 2002, IWBRA 2005, SAWN 2005.
|
|
|
University of Connecticut | 
