Dr. Maia Martcheva
University of Florida

will present

Vaccine Induced Pathogen Type Replacement

Thursday, April 24, 2008, 5:00pm
Ungar Room 506

Refreshments at 4:30pm in Ungar Room 521
All interested persons are welcome to attend.


Abstract: Many pathogens in nature exist in multiple variants, called strains. Vaccines, designed to protect the population from disease-causing microparasites, protect well against several of the strains included in the vaccine, but only partially or not at all against the rest of the strains. This property of the vaccines is called differential effectiveness. Vaccination campaigns with differentially effective vaccines lead to reduction in the number of cases caused by the strains in the vaccine but often also to rise in the number of cases caused by the non-vaccine strains -- a property called strain replacement. Differential effectiveness of the vaccines causes strains replacement but would strain replacement occur if the vaccines protect 100% against all pathogen strains involved ("perfect" vaccines)?

In this talk I would address this question and show that the answer is "yes". Strain replacement with perfect vaccination can occur if some coexistence mechanism, such as super-infection, mediates the coexistence of the strains. Do all coexistence mechanisms lead to strains replacement with "perfect" vaccination? What is common for the mechanisms that do, and those that do not?

---

Dr. J. Davidov
Bulgarian Academy of Sciences
and
Florida International University

will present

Twistor Spaces of Generalized Complex Structures

Friday, April 18, 2008, 5:00pm
Ungar Room 506

Refreshments at 4:30pm in Ungar Room 521
All interested persons are welcome to attend.


Abstract: The notion of a generalized complex structure has been introduced by N. Hitchin. It generalizes both the concept of a complex structure and that of a symplectic one and can be considered as a complex analog of the notion of a Dirac structure introduced by T. Courant and A. Weinstein to unify Poisson and presymplectic geometries. The generalized complex geometry has been further developed by M. Gualtieri and has recently attracted the interest of many mathematicians and physicists.

A generalized complex structure on a smooth manifold $M$ is an endomorphism $J$ of the bundle $TM\oplus T^{\ast}M$ satisfying the following conditions:\\ $(a)$ $J^2=-Id$,~\\ $(b)$ $J$ preserves the natural metric $=\frac{1}{2}(\xi(Y)+\eta(X))$, $X,Y\in TM$, $\xi,\eta\in T^{\ast}M$,~ \\$(c)$ the $+i$-eigensubbundle of $J$ in $(TM\oplus T^{\ast}M)\otimes {\Bbb C}$ is involutive with respect to the bracket introduced by T. Courant. \\If $J$ satisfies only the conditions $(a)$ and $(b)$, it is called generalized almost complex structure. The integrability condition $(c)$ is equivalent to vanishing of the Nijenhuis tensor of $J$ defined by means of the Courant bracket instead of the Lie one.

Every complex or symplectic structure induces a generalized complex structure in a natural way. Examples of generalized complex structures that cannot be obtained from a complex or a symplectic structure have been given first by M. Gualtieri and G. Cavalcanti.

The main purpose of the talk is to provide other examples of this type by means of the twistor construction.

---

Dr. Pierre Magal
University of Le Havre, France

will present

P-gp Transfer and Acquired Multi-drug Resistance in Tumors Cells

Friday, April 18, 2008, 4:00pm
Ungar Room 402

Refreshments at 3:30pm in Ungar Room 521
All interested persons are welcome to attend.


Abstract: Multi-Drug resistance for cancer cells as been a serious issue since several decades. In the past, many models have been proposed to describe this problem. These models use a discrete structured for the cancer cell population, and they may include some class of resistant, non resistant, and acquired resistant cells. Recently, this problem has received a more detailed biological description, and it turns out that the resistance to treatments is due in 40% of cancers to a protein called P-glycoprotein (P-gp). Moreover it has been proved that P-gp can be transferred from cell to cell by an osmotic phenomenon. This transfers turn to be responsible for the acquired resistance of sensitive cells. The goal of this talk is to introduce this problem, and to present a cell population dynamic model with continuous P-gp structure.

---

Dr. Pavel Etingof
MIT

will present

Orbifold Hecke Algebras

Tuesday, April 8, 2008, 5:00pm
Ungar Room 506

Refreshments at 4:30pm in Ungar Room 521
All interested persons are welcome to attend.


Abstract: To a group G acting discretely on a simply connected complex manifold X, I will attach a Hecke algebra H_q(G,X), which is a deformation of the group algebra of G. We will see that if H^2(X,C)=0 then this deformation is flat. We will also see that this setting unifies many known types of Hecke algebras - usual (finite), affine, double affine (Cherednik), Hecke algebras of complex reflection groups (Broue-Malle-Rouquier), and many others. In particular, there are orbifold Hecke algebras which provide quantization of Del Pezzo surfaces and their Hilbert schemes.

---

Professor Xiuxiong Chen
University of Wisconsin

will present

On the Space of Kaehler Metrics

Thursday, April 3, 2008, 5:00pm
Ungar Room 506

Refreshments at 4:30pm in Ungar Room 521
All interested persons are welcome to attend.


Abstract: On the infinite dimensional space of Kaehler metrics, Mabuchi, Semmes and Donaldson introduce a Weil-Peterson type metric. Under this metric, this space becomes an infinite dimensional symmetric space of non-compact type with semi-negative curvature. Donaldson made several important conjectures concerning the geometric structure of this space; and the resolution of these conjectures of Donaldson has important consequences on Kaehler geometry. For instance, the well known problem of uniqueness of "best metric" in each Kaehler class is settled in 2005 through this program. In this lecture, I will give an expository account of this program as well as some recent updates on Kaehler geometry.

---

Dr. John Shareshian
Department of Mathematics
Washington University

will present

Inversion Arrangements and Lower Intervals in the Bruhat Order

Tuesday, March 25, 2008, 5:00pm
Ungar Room 506

Refreshments at 4:30pm in Ungar Room 521
All interested persons are welcome to attend.


Abstract: I will present results from joint work with Axel Hultman, Savante Linusson and Jonas Sjöstrand of KTH in Stockholm. The Bruhat order on a Weyl group is of interest in the theory of algebraic groups. Let w be an element of the symmetric group S_n. A. Postnikov conjectured, and we proved, that

(1) The number of elements below w in the Bruhat order is at least the number of connected components of the complement in Rn of the union of the hyperplanes determined by the equations x_i=x_j for all inversions (i,j) of w, and
(2) equality holds in (1) if an only if w avoids certain patterns.

In fact, we proved a version of (1) for a general finite reflection group. I will explain our proof, which employs Zaslavsky's formula for the number of regions in the complement of a real hyperpalne arrangement and the theory of EL-shellability.

---

Dr. Eric Sharpe
Virginia Tech

will present

Recent Developments in Landau-Ginzburg Models

Monday, March 24, 2008, 4:00pm
Ungar Room 411

Refreshments at 3:30pm in Ungar Room 521
All interested persons are welcome to attend.


Abstract: In this talk, we shall give an overview of 'Landau-Ginzburg models,' certain physical theories, defined over Riemann surfaces, which have many ties to mathematics. In the first part of the talk, we will describe Landau-Ginzburg models over Riemann surfaces without boundary, and two corresponding topological field theories one can build from Landau-Ginzburg models. One of those, the "A twist," is a recent development. We will also describe how Landau-Ginzburg models are related to other physical theories and use those relations to give some checks of the computations we will outline. In the second part of the talk, we will describe Landau-Ginzburg models over Riemann surfaces with boundary, and the notion of 'matrix factorization' that is required to make sense of Landau-Ginzburg models in the presence of boundaries. After outlining some standard results in matrix factorizations, we will present a few physically-motivated mathematics conjectures regarding matrix factorizations and some unanswered questions.

---

Dr. Mikhail Kapranov
Yale University

will present

Hodge Structures and Equivariant Connections

Thursday, March 20, 2008, 5:00pm
Ungar Room 506

Refreshments at 4:30pm in Ungar Room 521
All interested persons are welcome to attend.


Abstract: We give a 'gauge-theoretic' diescription of the category of mixed Hodge structures. This category is identified with the category of vector bundles with connections on some toric stack. This description identifies the "Hodge-Galois group" with the group of unparametrized loops in the plane with the operation being.

---

Dr. Fedor Bogolomov
NYU

will present

Isogeny and Divisbility

Wednesday, March 19, 2008, 5:00pm
Ungar Room 402

Refreshments at 4:30pm in Ungar Room 521
All interested persons are welcome to attend.


Abstract: The main topic is a generalization of Tate's theorem for abalian varieties over finite fields.

Theorem. Let $A,B$ be two ableian varieties defined over (may be different) finite fields $k_0,l_)$. Assume that there is an infinite (in fact sufficiently long) sequence of finite extension $k_i:k_0$ and $l_i:l_0$ with $deg k_i:k_0= deg l_i:l_0$ and the number of points $A(k_i)$ is divisble by $B(l_i)$ Then there is a surjective map $A\to B$ In particular $char k_0 = char l_0$.

---

Dr. Yuri Safarov
Kings College, London

will present

Comparison of Dirichlet and Neumann Eigenvalues

Tuesday, March 18, 2008, 5:00pm
Ungar Room 506

Refreshments at 4:30pm in Ungar Room 521
All interested persons are welcome to attend.

---

Jonathan Jannarone
Vice President and Appointed Actuary
Assurant Solutions, Miami

will present

Actuaries in the Workplace: Solving Math Puzzles for a Living

Monday, March 3, 2008, 5:00pm
Ungar Room 402

Refreshments at 4:30pm in Ungar Room 521
All interested persons are welcome to attend.


Abstract: Actuaries help insurance companies in many areas, such as accounting, claims, lawsuits, investments, mergers & acquisitions, and premium calculations. Join Jonathan Jannarone, UM math graduate and Vice President of Life Actuarial at Assurant, as he explains how actuaries apply mathematics, statistics, and computer science to their daily jobs. Learn about how to become an actuary, consistently picked as one of the top 5 occupations in the country.

---

Dr. Ernest Shult
Kansas State University

will present

A Tale of Two Theorems

Thursday, February 28, 2008, 5:00pm
Ungar Room 506

Refreshments at 4:30pm in Ungar Room 521
All interested persons are welcome to attend.


Abstract: A light-weight discussion of a purely combinatorial phenomenon.

---

Dr. Nicholas Eriksson
University of Chicago

will present

Combinatorial Methods in Evolutionary Biology

Friday, February 15, 2008, 5:00pm
Ungar Room 402

Refreshments at 4:30pm in Ungar Room 521
All interested persons are welcome to attend.


Abstract: I'll talk about three areas of evolutionary biology using a combination of statistics and discrete math: viral population diversity, the evolution of drug resistance, and phylogenetics. Knowledge of the diversity of viral populations is important for understanding disease progression, vaccine design, and drug resistance, yet it is poorly understood. New technologies (pyrosequencing) allow us to read short, error-prone DNA sequences from an entire population at once. I will show how to assemble the reads into genomes using graph theory, allowing us to determine the population structure.

Next, I will describe a new class of graphical models inspired by poset theory that describe the accumulation of (genetic) events with constraints on the order of occurrence. Applications of these models include calculating the risk of drug resistance in HIV and understanding cancer progression.

Finally, I'll describe a polyhedral method for determining the sensitivity of phylogenetic algorithms to changes in the parameters. We will analyze several datasets where small changes in parameters lead to completely different trees and see how discrete geometry can be used to average out the uncertainty in parameter choice.

---

Dr. Alexander Goncharov
Brown University

will present

Hodge Correlators

Thursday, February 14, 2008, 5:00pm
Ungar Room 506

Refreshments at 4:30pm in Ungar Room 521
All interested persons are welcome to attend.


Abstract: I describe the periods of the mixed Hodge structure on the fundamental group of a complex curve as correlators for a Feynman integral.

---

Dr. Kenneth Baker
Georgia Institute of Technology

will present

Genera of Knots in 3-manifolds

Wednesday, February 13, 2008, 5:00pm
Ungar Room 402

Refreshments at 4:30pm in Ungar Room 521
All interested persons are welcome to attend.


Abstract: A knot is an embedding of a circle into a 3-manifold. Knots in the 3-sphere bound compact, orientable surfaces commonly known as Seifert surfaces. The genus of a knot is defined to be the minimum genus among all its Seifert surfaces. What enables a knot to have a Seifert surface is that, when oriented, it represents the trivial element of the 1st homology of the ambient 3-manifold. We extend this notion of genus to non-null homologous knots in (closed, orientable) 3-manifolds and investigate the set of genera of knots within a given 1st homology class. These sets for non-trivial homology classes have curious structures and are related to outstanding problems in Dehn surgery.

---

Dr. Aobing Li
University of Wisconsin-Madison

will present

A Liouville Type Theorem for Some Conformally Invariant Equations

Friday, February 11, 2008, 5:00pm
Ungar Room 402

Refreshments at 4:30pm in Ungar Room 521
All interested persons are welcome to attend.


Abstract: Obata and Gidas-Ni-Nirenberg classified all the positive C2 solutions of the Yamabe equation -Δ u=c(n)u(n+2)/(n-2) on Rn (n>2) under the assumption of finite volume. Caffarelli-Gidas-Spruck removed the assumption. In our talk, we will consider a fully nonlinear Yamabe equation σk (Ag) on Rn and classify all the positive solutions, where σk is the kth fundamental symmetric function and Ag is the Schouten tensor of the metric u4/(n-2)(dx 12+...+dxn2). The classification of solutions is related to obtain the compactness results of the corresponding geometric problem.

---

Dr. Larry Shepp
Rutgers University
Statistics Department

will present

Closed-loop Control of Diabetes

Friday, February 8, 2008, 4:30pm
Ungar Room 402

Refreshments at 4:00pm in Ungar Room 521
All interested persons are welcome to attend.


Abstract: Diabetics require tedious and constant monitoring of their blood sugar level to avoid serious consequences. If two recent technologies, the insulin pump and the continuous glucose monitor, can be unified, it gives the possibility to allow programmed control. Various algorithms have been proposed for the control problem involved in the unification. I will discuss how to compare algorithms using a method based loosely on ergodic theory. This is work in progress.

---

Dr. Herbert Wilf
University of Pennsylvania
Recipient of the 1998 AMS Steele Prize for Seminal Contribution to Research

will present

The Outstanding Elements of Permutations and Words

Thursday, February 7, 2008, 5:00pm
Ungar Room 506

Refreshments at 4:30pm in Ungar Room 521
All interested persons are welcome to attend.

---

Dr. Dan Lee
Duke University

will present

The Notion of Mass in General Relativity and Riemannian Geometry

Monday, February 4, 2008, 5:00pm
Ungar Room 402

Refreshments at 4:30pm in Ungar Room 521
All interested persons are welcome to attend.


Abstract: The Penrose inequality asserts that the mass of a gravitational system is bounded below in terms of the area of a black hole. The goal of this talk is to provide enough background and context to appreciate this result. I will begin with a review of some facts from Newtonian gravity, and then I will introduce the corresponding ideas from general relativity.

---

Professor Herbert Freedman
Department of Mathematical Sciences
University of Alberta

will present

Persistence and Extinction in a Mathematical Model of Cell Population Affected by Radiation

Friday, February 1, 2008, 4:30pm
Ungar Room 402

Refreshments at 4:00pm in Ungar Room 521
All interested persons are welcome to attend.


Abstract: A mathematical model consisting of ordinary differential equations is formulated to represent the interrelationship between healthy and radiated cells at a given site. Three different modes of radiation delivery are considered: constant, decaying and periodic. For the constant case, precise criteria for the persistence or extinction are obtained. In the decaying case it is shown that the radiated cells always become extinct. In the periodic case, criteria for a perturbed positive periodic solution are obtained.

---

Dr. Frank Kutzschebauch
University of Bern

will present

Embeddings of Stein Manifolds into Euclidean Spaces

Friday, January 25, 2008, 4:30pm
Ungar Room 402

Refreshments at 4:00pm in Ungar Room 521
All interested persons are welcome to attend.


Abstract: We start the talk with introductory material about Stein manifolds and the general embedding results of Remmert, Gromov- Eliashberg and Schürmann. Then we motivate and make precise the following two questions: Suppose that a Stein manifold X can be embedded into C_n for some dimension n,
1.) what additional properties of the embedding can one prescribe?
2.) in how many different ways can X be embedded into C_n?
During an overview about known results in this direction we indicate the main technique for producing interesting holomorphic embeddings. Finally we present recent joint results with our student Borell: The existence of uncountably many embeddings of C_k into C_n (for 0 < k < n) and results concerning the "ball embedding property", a property of a Stein manifold introduced by Borell.

---

Dr. Simon Levin
George M. Moffett Professor of Biology, Princeton University
Director, The Center for Biocomplexity
Member, National Academy of Sciences
Member, Academy of Arts & Sciences

will present

Crossing Scales: Evolutionary Approaches to Ecological Interactions

Friday, January 18, 2008, 4:30pm
Ungar Room 402

Refreshments at 4:00pm in Ungar Room 521
All interested persons are welcome to attend.


Abstract: There is a long history of mathematical research into the dynamics of populations, their ecological relationships, and their patterns of movement. There is a similarly rich literature in the dynamics of infectious diseases. In both cases, the bulk of the literature assumes particular relationships and asks what the consequences will be. But these relationships and the parameters that govern them have been shaped and continue to be shaped by evolution; this is, for example, why influenza A continues to be a scourge despite the fact that it confers lifetime immunity, why bacteria acquire resistance to antibiotics, and why collective behavior exists in societies from bacteria to humans. In this lecture, I will elucidate some of these issues, and suggest mathematical approaches to addressing them.

---

Dr. Mark Skandera
Lehigh University, Bethlehem, PA

will present

On the Cluster Basis of Z[x_1,1,...,x_3,3]

Wednesday, December 19, 2007, 4:00pm
Ungar Room 402

Refreshments at 3:30pm in Ungar Room 521
All interested persons are welcome to attend.


Abstract: We show that the set of cluster monomials for the cluster algebra of type D4 forms a basis of the Z-module Z[x_1,1,...,x_3,3]. We also show that the transition matrices relating the cluster basis of this module to the natural and the dual canonical bases are unitriangular and nonnegative. These results support a conjecture of Fomin and Zelevinsky on the equality of the cluster and dual canonical bases of Z[x_1,1,...,x_3,3]. In the event that this conjectured equality is true, our results also imply an explicit factorization of each dual canonical basis element of the module as a product of cluster variables.

---

Dr. Thomas Bruestle
Universite de Sherbrooke and Bishops University

will present

Algebraic Structures Given by Surface Triangulations

Thursday, December 13, 2007, 4:00pm
Ungar Room 402

Refreshments at 3:30pm in Ungar Room 521
All interested persons are welcome to attend.


Abstract: The number of dissections of a polygon, the Catalan number, appears in many areas of mathematics. But only recently, the triangulations of an arbitrary oriented surface with boundary were found to be closely related to algebraic objects: Fomin, Shapiro and Thurston showed in 2006 that every surface triangulation gives rise to a cluster algebra (a class of algebras introduced earlier by Fomin and Zelevinsky to study coordinate rings of classical groups), the case of a polygon corresponding to type A_n.

We explain their construction in this talk and illustrate how nicely properties of the triangulation correspond to algebraic concepts.

---

Dr. Oleg Viro
SUNY at Stony Brook

will present

Compliments to Bad Spaces

Wednesday, December 5, 2007, 5:00pm
Ungar Room 402

Refreshments at 4:30pm in Ungar Room 521
All interested persons are welcome to attend.


Abstract: The ways that mathematical theories get into the core of mainstream mathematical curriculum sometimes are strongly influenced by accidental circumstances. Often basic definitions could be made more convenient than the present ones. In the talk we will consider a few examples. Speaking on differentiable manifolds, we usually pretend that they have no singular siblings. This causes lots of inconveniences. Another example: most mathematicians (besides, probably, specialists in combinatorics) believe that all finite topological spaces are either trivial or nasty. Topology appears to be the only mathematical field that feels ashamed of its finite objects.

---

Dr. Shripad Tuljapurkar
Stanford University

will present

Dynamic Heterogeneity in Life Histories

Friday, November 9, 2007, 4:30pm
Ungar Room 402

Refreshments at 4:00pm in Ungar Room 521
All interested persons are welcome to attend.


Abstract: Biologists view the life histories of vertebrates as types: long or short life, early or late reproduction. Mathematical theory accordingly views a life history as static, and life history evolution as taking place on a static, if complex, fitness surface. Recent work on longitudinal data on natural populations shows that life histories are dynamic -- individuals move between "states" of reproduction and mortality in a surprisingly stochastic way. There is a large amount of "dynamic" heterogeneity both along an individual life course and between individuals. I discuss models of this process, the analogy between plants and animals, and the consequences for population and evolutionary theory.

---

Dr. Thomas Mark
University of Virginia

will present

Knotted Surfaces in 4-manifolds

Thursday, October 25, 2007, 5:00pm
Ungar Room 402

Refreshments at 4:30pm in Ungar Room 521
All interested persons are welcome to attend.


Abstract: One of the reasons that 4-dimensional manifold theory is "special" is the distinction between the topological and smooth categories: there are many examples of infinite families of smoothly distinct closed 4- manifolds that are all topologically equivalent, a phenomenon that does not occur any other dimension. Several years ago Fintushel and Stern showed that this happens also in a relative context: they constructed infinite families of smooth surfaces embedded in a given closed 4-manifold and showed that they are all topologically equivalent. Under certain circumstances they were able to distinguish the smooth isotopy classes of these surfaces, exhibiting a knotting phenomenon in the smooth category that does not appear in the topological category. I will describe their construction and a new proof of their result that applies to a broader range of cases.

---

Professor Pierre Magal
University of Le Havre, France

will present

A Model of Antibiotic Resistant Bacterial Epidemics in Hospitals

Friday, August 24, 2007, 4:00pm
Ungar Room 402

Refreshments at 3:30pm in Ungar Room 521
All interested persons are welcome to attend.


Abstract: The development of drug-resistant strains of bacteria is an increasing threat to society, especially in hospital settings. Many antibiotics that were formerly effective in combating bacterial infections in hospital patients are no longer effective due to the evolution of resistant strains. The evolution of these resistant strains compromises medical care worldwide. In this article, we formulate a two-level population model to quantify key elements in nosocomial infections. At the bacteria level patients infected with these strains generate both nonresistant and resistant bacteria. At the patient level susceptible patients are infected by infected patients at rates proportional to the total bacteria load of each strain present in the hospital. The objectives are to analyze the dynamic elements of non-resistant and resistant bacteria strains in epidemic populations in hospital environments and to provide understanding of measures to avoid the endemicity of resistant antibiotic strains.

---

Dr. Jibin Li
Zhejiang Normal University, China

will present

On the Study for a Class of Singular Nonlinear Travelling Wave Equations:
Dynamical System Appproach

Thursday, August 16, 2007, 4:00pm
Ungar Room 402

Refreshments at 3:30pm in Ungar Room 521
All interested persons are welcome to attend.


Abstract: The existence of solitary wave solutions, kink wave solutions and periodic wave solutions of a class of singular reaction-diffusion equations is obtained using some effective methods from the dynamical systems theory. Specially, for a class of nonlinear wave equations, fundamental properties of profiles of travelling wave solutions determined by some bounded orbits of the travelling wave systems are rigorously proved. Parametric conditions that guarantee the existence of the aforementioned solutions are derived and given explicitly.

---

Professor Piotr Chrusciel
Oxford University
and
Universite de Tours

will present

Black Holes

Wednesday, August 1, 2007, 4:00pm
Ungar Room 506

Refreshments at 3:30pm in Ungar Room 521
All interested persons are welcome to attend.


Abstract: I will briefly review the notion of black hole, in general relativity but also in the context of other non-linear field equations. I will point out the existing experimental evidence, and turn to a discussion of old and new results about stationary vacuum black holes in 3+1, but also in higher, dimensions.

---

Dr. Bertrand Clarke
Department of Statistics at British Columbia

will present

TBA

Tuesday, May 1, 2007, 4:00pm
Ungar Room 402

Refreshments at 3:30pm in Ungar Room 521
All interested persons are welcome to attend.

---

Professor Chengzhi Li
Peking University, P.R.China

will present

The Weak Hilbert's 16th Problem and Its Solution in Quadratic Case

Monday, April 30, 2007, 4:00pm
Ungar Room 402

Refreshments at 3:30pm in Ungar Room 521
All interested persons are welcome to attend.


Abstract: The second part of Hilbert's 16th problem, asking for the maximum H(n) of the numbers of limit cycles and their relative positions for all planar polynomial differential systems of degree n, is still open even for the quadratic case (n=2).

A weak form of this problem, proposed by V. I. Arnold, asking for the maximum Z(m,n) of the numbers of isolated zeros of Abelian integrals of all polynomial 1-forms of degree n over algebraic ovals of degree m, is also extremely hard to grasp. The number Z(n)=Z(n+1,n) can be chosen as a lower bound of H(n), so far only Z(2)=2 was proved, and this was done by several authors over a period of about 10 years.

In my lecture I will first briefly introduce the Hilbert's 16th Problem and its weak form, then talk about the solution of Weak Hilbert's 16th Problem for n=2.

---

Professor Jianguo Cao
Department of Mathematics
University of Notre Dame

will present

Kaehler Version of Cheeger-Gromoll-Perelman Soul Theory with Positive Bi-sectional Curvature

Thursday, April 5, 2007, 5:00pm
Ungar Room 506

Refreshments at 4:30pm in Ungar Room 521
All interested persons are welcome to attend.


Abstract: In 1980's, S. T. Yau conjectured that any open complete Kaehler manifold with positive bi-sectional holomorphic curvature is bi-holomorphic to the Euclidean space. We will use Morse theory for distance function to show that, under some mild assumptions, any open complete Kaehler manifold with positive bi-sectional holomorphic curvature is diffeomorphic to the Euclidean space C^n.

This result can be viewed as a part of Kaehler version of Cheeger-Gromoll-Perelman soul theory.

---

Professor Jie Qing
Department of Mathematics
University of California, Santa Cruz

will present

On Asymptotically Hyperbolic Manifolds

Thursday, March 29, 2007, 5:00pm
Ungar Room 506

Refreshments at 4:30pm in Ungar Room 521
All interested persons are welcome to attend.


Abstract: In this talk I will talk on some recent developments on the study of asymptotically hyperbolic manifolds. An asymptotically hyperbolic manifold is a conformally compact Riemannian manifold whose curvature tends to -1 at its infinity. I will discuss three related questions. The first is about the rigidity of conformally compact Einstein manifolds. The rigidity was first obtained by Andersson and Dahl for spin manifolds. We will introduce a recent proof that shows the rigidity without spin assumption. The second is about the study of the topology of conformal Einstein manifolds based on the size of the so-called renormalized volume. This is inspired by the so-called holography principle in the AdS/CFT correspondence in physics. The third is on the positive mass of asymptotically hyperbolic manifolds. We will present a proof to a positive mass theorem to allow to have corners along a hypersurface in an asymptotically hyperbolic manifold.

---

Professor Herb Wilf
University of Pennsylvania
Recipient of the 1998 Steele Prize for Seminal Contribution to Research

will present

Partition Congruences and Geographical Clusters of Cases of a Disease

Friday, March 23, 2007, 4:00pm
Ungar Room 506

Refreshments at 3:30pm in Ungar Room 521
All interested persons are welcome to attend.


Abstract: [The two subjects in the title are unrelated except that I'll talk about both of them] Ramanujan conjectured and proved a number of results about congruence properties of the integer partition function, in particular that p(5n+4) is divisible by 5 for all n. This was refined by Berkovich and Garvan, and we will give a short proof of a similar refinement. We show that not only is p(5n+4) divisible by 5, but actually the number of partitions of 5n+4 whose "BG rank" is j, is divisible by 5, for all n and j.

In the second sub-talk I'll show how the distribution of the maximum cell occupancy in balls-in-boxes problems can be rapidly computed using combinatorial methods, thus allowing determination of whether an apparent cluster of cases of acute leukemia which occurred in a small locality was really unusual or was to be expected. Older methods for this problem required exponential time.

---

Professor John Morgan
Columbia University

will present

The Poincare Conjecture and Classification of 3-manifolds

Thursday, March 22, 2007, 5:00pm
Ungar Room 402

Refreshments at 4:30pm in Ungar Room 521
All interested persons are welcome to attend.


Abstract: For over 100 years the Poincare Conjecture has been the central problem in topology. It conjectures a characterization of the 3-sphere as the only simply connected, closed 3-manifold. It has been generalized to a conjectured characterization of the sphere in all dimensions, and by 1986 had been solved in all dimensions except dimension 3. In the early 1980's Richard Hamilton proposed an attack on this conjecture, and a more general conjecture, due to Thurston, about the classification of all compact 3-manifolds. The method of attack proposed by Hamilton was analytic and differential geometric in nature. His idea was to use an evolution equation for a Riemannian metric on a manifold, deforming the metric by it curvature, which is an analogue of the heat equation. The intuition is that just as the heat equation distributes the temperate evenly around the manifold, the analogous tensor equation should distribute the curvature equally around the manifold. Hamilton showed that under certain geometric assumptions this program works perfectly, and the evolution equation converges to a round metric, from which it is easy to reach topological conclusions. In spite of this positive evidence, there was a serious obstacle in Hamilton's program. The equation is non-linear and, as is usual in non-linear evolution equations, singularities develop in finite time. In order to establish topological conclusions, one must continue the evolution process through these singularities and define an evolution for all positive time. Perelman provided the insights that allowed the extension of the equation through the singularities and showed how to complete Hamilton's program and prove the Poincare Conjecture and the more general classification of all closed 3-manifolds.

In this talk we will describe some of the history surrounding the Poincare Conjecture, indicating its central importance in much of 20th century topology. Then we will describe Hamilton's program and briefly indicate how Perelman overcame the difficulties that Hamilton had encountered in carrying out the program.

BREAKTHROUGH OF THE YEAR: The Poincare Conjecture-Proved
On August 22, 2006, the ICM awarded Perelman the Fields Medal for his work on the conjecture, but Perelman refused the medal. John Morgan spoke at the ICM on the Poincare conjecture on August 24, 2006, declaring that "in 2003, Perelman solved the Poincare Conjecture."

---

Professor Fedor Bogomolov
New York University

will present

On Special Construction of Projective Surfaces with Infinite Fundamental Groups

Wednesday, March 14, 2007, 4:00pm
Ungar Room 402

Refreshments at 3:30pm in Ungar Room 521
All interested persons are welcome to attend.


Abstract: I will discuss a special construction of nonsimply connected surfaces as families of curves and related problems concerning the quotients of mapping class groups.

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Professor Maxim Kontsevitch
Institute des Hautes Etudes Scientific
Recipient of a Fields Medal in 1998

will present

Derived Non-commutative Algebraic Geometry III

Friday, March 9, 2007, 2007, 4:00pm
Ungar Room 402

Refreshments at 3:30pm in Ungar Room 521
All interested persons are welcome to attend.

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Professor Maxim Kontsevitch
Institute des Hautes Etudes Scientific
Recipient of a Fields Medal in 1998

will present

Derived Non-commutative Algebraic Geometry II

Wednesday, March 7, 2007, 2007, 4:00pm
Ungar Room 402

Refreshments at 3:30pm in Ungar Room 521
All interested persons are welcome to attend.

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Tony Pantev
University of Pennsylvania

will present

Langlands Duality for the Hitchin System

Tuesday, March 6, 2007, 4:00pm
Ungar Room 402

Refreshments at 3:30pm in Ungar Room 521
All interested persons are welcome to attend.

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Professor Maxim Kontsevitch
Institute des Hautes Etudes Scientific
Recipient of a Fields Medal in 1998

will present

Derived Non-commutative Algebraic Geometry I

Monday, March 5, 2007, 4:00pm
Ungar Room 402

Refreshments at 3:30pm in Ungar Room 521
All interested persons are welcome to attend.

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D. Orlov
RAS

will present

Motives and Derived Categories

Friday, March 2, 2007, 2:00pm
Cox Building 217

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P. Bressler
IAS

will present

Riemman Roch Theorems for Real Algebraic Varieties

Friday, March 2, 2007, 11:15pm
Memorial Building 203

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N. Turin
RAS

will present

Geometric Quantization and Sympletic Algebraic Geometry

Friday, March 2, 2007, 10:00am
Memorial Building 203

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Professor Maxim Kontsevitch
Institute des Hautes Etudes Scientific
Recipient of a Fields Medal in 1998

will present

Quantum Integrable Systems over Finite and Local Fields

Friday, March 2, 2007, 4:30pm
Ungar Room 402

Refreshments at 4:00pm in Ungar Room 521
All interested persons are welcome to attend.