Algebra, Combinatorics and Geometry Seminar
Many of the talks this semester will be on the theme of Ngo's proof of the Fundamental Lemma of Endoscopy. Here are some suggested reading materials.
This Weeks Lecture
November 19, 2009
12:00pm
703 Thackeray Hall
Prof. Gregory Contantine, Univ. of Pittsburgh
"Combinatorics of the Bose-Mesner algebra"
Abstract: We describe the origins of the Bose-Mesner algebra, and its connections to optimal designs and codes through extreme spectral properties. The focus then shifts to the Johnson scheme by showing that extreme spectra lead to geometric symmetry. Within this context, a class of combinatorial problems, including flags of maximal length, shall be described.
Fall 2009 Schedule
September 3, 2009
12:00pm
703 Thackeray Hall
Prof. Julia Gordon, University of British Columbia
"On motivic-ness of some positive-depth characters"
Abstract: This talk will be about trying to use motivic integration to study Harish-Chandra characters of p-adic groups. I will talk about linear functionals in the context of motivic integration, and will prove that Harish-Chandra characters of some positive-depth supercuspidal representations of p-adic groups are "constructible motivic exponential functions" (I will define all these terms).
October 1, 2009
12:00pm
703 Thackeray Hall
Prof. Bogdan Ion, University of Pittsburgh
"On PBW bases"
Abstract: Virtually all the proofs of Poincare-Birkhoff-Witt type theorems are of combinatorial nature reducing one way or another to the knowledge of generators and relations for the algebras in question. I will explain how to obtain PBW theorems for reasonably large classes of algebras without requiring any explicit information about generators or relations.
October 5, 2009
12:00pm
Kyungyong Lee, Purdue Univeristy
Title : q,t-Catalan numbers
Abstract : The q,t-Catalan numbers naturally occur in the study of Macdonald polynomials, which are an important family of multivariable orthogonal polynomials introduced by Macdonald with applications to a wide variety of subjects including Hilbert schemes, harmonic analysis, representation theory, mathematical physics, and algebraic combinatorics. Haiman and Garsia-Haglund proved that they are polynomials of q and t with nonnegative coefficients. We give simple upper bounds on coefficients in terms of partition numbers, and find all coefficients which achieve the bounds. Our main idea is to develop a nontrivial morphism from the space of alternating polynomials to partitions. This is a joint work with Li Li.
October 15, 2009
12:00pm
703 Thackeray Hall
Prof. Alexander Borisov, University of Pittsburgh
"A geometric approach to the two-dimensional Jacobian Conjecture"
Abstract: The Jacobian Conjecture of Keller states that any unramified polynomial map from the affine complex space to itself must be invertible. We study such maps in dimension two by compactifying and blowing up points to get a map from some rational surface to the projective plane. Using ideas of the Minimal Model Program, we obtain strong restrictions on the combinatorial structure of this rational surface. We exhibit a surface satisfying these restrictions and explain how it might lead to a counterexample to the Jacobian Conjecture.
October 22, 2009
12:00pm
703 Thackeray Hall
Prof. Jeffrey Wheeler, University of Pittsburgh
"A Proof the Erdos-Heilbronn Problem Using the Polynomial Method of Alon, Nathanson, and Ruzsa"
Abstract: In the early 1960's, Paul Erdos and Hans Heilbronn conjectured that for any two nonempty subsets A and B of Z/pZ the number of restricted sums (restricted in the sense that we require the elements to be distinct) of an element from A with an element from B is at least the smaller of p and |A|+|B|-3. This problem is related to independent results of Cauchy and Harold Davenport which established that there are at least the minimum of p and |A|+|B|-1 sums of the form a+b (with the restriction removed). One thing that makes the problem interesting is that the results of Cauchy and Davenport were immediately established whereas the conjecture of Erdos and Heilbronn was open for more than 30 years.
We present the proof of the conjecture due to Noga Alon, Melvyn Nathanson, and Emre Rusza. This technique is known as the Polynomial Method and is regarded by many as a powerful tool in the area of Additive Combinatorics.
October 29, 2009
12:00pm
703 Thackeray Hall
Truong Nguyen, Univeristy of Pittsburgh
"Counting Points on Elliptic Curves over Finite Fields"
November 5, 2009
12:00pm
703 Thackeray Hall
Petr Pancoska, Center for Clinical Pharmacology, Department of Medicine, University of Pittsburgh
"Entromics as the theoretical foundation of individual genomics:
From gene sequencing to severity of cystic fibrosis using physics in graph theory."
Abstract: The goal of entromics is to derive a quantitative characterization of the energy cost for the assembly of the genome using the information about genome DNA sequence as the exclusive input. From this effort we derive a thermodynamic formula, which combines enthalpy term with a special (compensatory) entropy term that was not known before. We therefore benefit from the study of this novel entropy distribution along the genome, which leads us to the name "entromics".
Entromics uses Eulerian oriented multigraphs to describe DNA sequences. This enables recognizing sequences in the genome that are mutually homomorphic, while being dissimilar in all aspects considered by current biology. This opens a whole new dimension of genomics, discovering until now hidden, but biologically important relationships in the genome. The focus of the presentation will be to deriving physical and biological interpretation of the DNA homomorphism from selective combination of mathematical proposition results with basic physical principles. Examples of clinical applications of entromics will be also shown and related open mathematical problems will be presented for discussion.
November 12, 2009
12:00pm
703 Thackeray Hall
Tran Nam Trung, University of Pittsburgh
"Regularity index of Hilbert functions of powers of ideals"
Abstract: Let A be a Noetherian standard graded algebra over an Artinian ring A_0. For a finitely generated graded A-module M, there is a function H called the Hilbert function of M. It is well-known that there is a polynomial P with rational coefficients called the Hilbert polynomial of M such that agrees with the Hilbert function at all sufficiently large natural numbers m. The regularity index of the Hilbert function of M is defined by ri(M):= min {m_0 | H(m)=P(m) forall m >= m_0}. Let I be a homogeneous ideal of A. It is shown that the regularity index of the Hilbert function of I^n M is a linear function of n for all n large enough.
November 19, 2009
12:00pm
703 Thackeray Hall
Prof. Gregory Contantine, Univ. of Pittsburgh
"Combinatorics of the Bose-Mesner algebra"
Abstract: We describe the origins of the Bose-Mesner algebra, and its connections to optimal designs and codes through extreme spectral properties. The focus then shifts to the Johnson scheme by showing that extreme spectra lead to geometric symmetry. Within this context, a class of combinatorial problems, including flags of maximal length, shall be described.
Fall 2008/Spring 2009 Schedule
August 28, 2008
12:00pm
703 Thackeray Hall
Dr. Thomas Hales, Univ. of Pittsburgh
"The transfer principle for the fundamental lemma"
September 4, 2008
12:00pm
703 Thackeray Hall
Dr. Thomas Hales, Univ. of Pittsburgh
"What is the Fundamental Lemma?"
September 4, 2008
1:00pm
703 Thackeray Hall
Yimu Yin, Univ. of Pittsburgh
"Kazhdan Hrushovski Motivic Integration"
September 11, 2008
12:00pm
703 Thackaray Hall
Dr. Bogdan Ion, University of Pittsburgh
"The Fourier-Mukai transform"
Abstract: This is an expository talk on the definition and basic
properties of the Fourier-Mukai transform. Some applications (such as
Atiyah's classification of vector bundles over elliptic curves) will
also be discussed.
References:
1) Mukai. Duality between $D(X)$ and $D(\hat X)$ with its
application to Picard sheaves. Nagoya Math. J. (1981) vol. 81 pp.
153-175
2) Atiyah. Vector bundles over an elliptic curve. Proc. London Math.
Soc. (3) (1957) vol. 7 pp. 414-452
September 18, 2008
12:00pm
703 Thackeray Hall
Dr. Alexander Borisov, Univ. of Pittsburgh
"A geometric approach to the two-dimensional Jacobian Conjecture"
Abstract: We pursue the most natural (from the birational geometry viewpoint) approach to the classical two-dimensional Jacobian Conjecture. Starting with a possible counterexample, we resolve the singularities at infinity to get a map from a rational surface to the projective plane. A priori, one can say very little about the structure of the intersection graph of the blown-up curves. By careful analysis, we manage to put severe restrictions on this graph. As a corollary, we prove that all the images of these curves pass through a single point on the projective plane.
September 25, 2008
12:00pm
Jeffrey Wheeler
"The Erdos-Heilbronn Problem for Finite Groups"
Abstract: The Erdos-Heilbronn Conjecture states that for any two nonempty subsets A and B of Z/pZ we have |A \dot{+} B| \geq min { p, |A|+|B|-3 }, where A \dot{+} B is the set of sums a+b mod p with a \in A and b \in B and a \neq b. Dias da Silva and Hamidounne established the result for the case A = B in 1994 while Alon, Nathanson, and Ruzsa established the more general result in 1995. We further generalize this result and extend it from Z/pZ to arbitrary finite (including non-abelian) groups. This is a joint work with Paul Balister of the University of Memphis.
October 2, 2008
12:00pm
703 Thackeray Hall
Dr. Thomas Hales, Univ. of Pittsburgh
"Ngo's proof of the Fundamental Lemma (overview)"
Abstract: This talk will describe the general outline of Ngo's proof of the fundamental lemma. it will touch on some of the key structures in the proof: affine Springer fibers, the Hitchin fibration, the stabilization of the trace formula, and a key theorem on supports.
October 9, 2008
12:00pm
Dr. Greg Constanstine, Univ. of Pittsburgh
"New perspectives on Hadamard designs"
Abstract: Existence and and construction of Hadamard designs are examined from viewpoints of maximal cliques in association schemes, systems of distinct representatives, covering colored arborescences in complete graphs, and probability theory.
October 16, 2008
12:00pm
Dr. Bogdan Ion, Univ of Pittsburgh
"Affine Springer fibers (after Kazhdan, Lusztig, Bezrukavnikov)"
Abstract: This is an introduction to affine Springer fibers and their basic properties. We also give a proof of the dimension formula for fibers over regular semisimple elements, following Bezrukavnikov. References:[1] Kazhdan and Lusztig. Fixed point varieties on affine flag manifolds. Israel J. Math. (1988) vol. 62 (2) pp. 129-168 [2]
Bezrukavnikov. The dimension of the fixed point set on affine flag manifolds. Math. Res. Lett. (1996) vol. 3 (2) pp. 185-189
October 23, 2008
12:00pm
Tonghai Yang
"Arithmetic Intersection and the Non-abelian Chowla-Selberg formula"
Abstract: Let F=Q(sqrt D) be a real quadratic field. Let X be the Hilbert modular surface, viewed as an arithmetic 3-fold over integers. It has two families of naturally defined cycles, the arithmetic Hizebruch-Zagier divisors (dimension 2) T_m and arithmetic CM cycles CM(K) associated to a quartic CM number field K. They intersect properly when K is non-biquadratic. In this talk, we give an explicit formula for their intersections in terms of arithmetic on K. As an application, we explain how it implies the first non-abelian generalization of the celebrated Chowla-Selberg formula, a special case of the Colmez conjecture.
October 30, 2008
12:00pm
Dr. Bogdan Ion, Univ of Pittsburgh
"Affine Springer fibers II (after Kazhdan, Lusztig, Bezrukavnikov)"
Abstract: This is an introduction to affine Springer fibers and their basic properties. We also give a proof of the dimension formula for fibers over regular semisimple elements, following Bezrukavnikov. References:[1] Kazhdan and Lusztig. Fixed point varieties on affine flag manifolds. Israel J. Math. (1988) vol. 62 (2) pp. 129-168 [2]
Bezrukavnikov. The dimension of the fixed point set on affine flag manifolds. Math. Res. Lett. (1996) vol. 3 (2) pp. 185-189
November 6, 2008
12:00pm
Ruggero Gabbrielli, Centre for Orthopaedic Biomechanics,
Department of Mechanical Engineering,
University of Bath
"Periodic Space Partitions
from a Pattern Forming Equation"
Abstract:
A new counterexample to Kelvin’s Conjecture on minimal foams has
been found. The conjecture stated that the minimal surface area partition
of space into cells of equal volume was a tiling by truncated octahedra
with slightly curved faces. Weaire and Phelan found a counterexample
whose periodic unit includes two different tiles, a dodecahedron and a
polyhedron with 14 faces. Successively, Sullivan showed the existence
of a whole domain of partitions by polyhedra having only pentagonal and hexagonal faces that included the Phelan-Weaire structure..
Here is presented a new set of partitions with lower surface area than Kelvin's partition containing quadrilateral, pentagonal and hexagonal faces. These and other new partitions have been generated via the Voronoi diagram of spatially periodic sets of points obtained as local maxima of the stationary solution of the 3D Swift-Hohenberg partial differential equation in a triply periodic boundary, with pseudorandom initial conditions.
February 5, 2008
12:00pm
Peter Lumsdaine, Carnegie Mellon Univ.
"Higher Categories in Algebra"
Anstract: Higher categories have been studied since the 1970's in pure category theory, algebraic topology, and algebraic geometry. More recently, however, they have become of interest to a wider audience, as "categorification" techniques and results have emerged in a range of areas. I will give an introduction to higher categories, and a quick survey of some applications.
February 12, 2009
12:00pm
Dr. Greg Constantine, Univ. of Pittsburgh
"A construction of 2-designs of any block size with transitive automorphism groups"
Abstract: Large infinite families of nontrivial 2-designs are known to exist, yet a systematic listing by basic parameters, such as block size, was not known. I shall demonstrate how a nontrivial 2-design with automorphism group transitive on blocks can be constructed for any block size. These objects are, therefore, less sporadic than one may have thought.
February 19, 2008
12:00pm
Sophie Morel, Institute for Advanced Study
"On the cohomology of some non-compact Shimura varieties"
Abstract : In this talk, I will explain how the method originally developed by Ihara, Langlands and Kottwitz to compute the cohomology of a Shimura variety (use the Grothendieck-Lefschetz fixed point formula in positive characteristic to calculate the trace on the cohomology of a power of Frobenius at a good place times a Hecke operator trivial at that place, and then compare the result with Arthur's trace formula) applies to intersection cohomology of the Satake-Baily-Borel compactification of the Shimura varieties of unitary groups over Q and of the Siegel moduli varieties. I will also present applications to the calculation of the L-function of the intersection complex (for unitary groups and small-dimensional symplectic groups) and some applications involving base change from quasi-split unitary groups to general linear groups.
March 5 , 2009
1:00pm
703 Thackeray Hall
Dr. Yimu Yin, Univ of Pittsburgh
"Fourier Transform in Algebraically Closed Valued Fields"
February 26, March 19, 26 2008
1:00pm
Alexander Borisov, Univ of Pittsburgh
"An introduction to Higgs bundles"
Abstract: Higgs bundles on projective curves are in some sense natural generalizations of semistable holomorphic vector bundles. I will give some basic definitions and state some theorems regarding them.
Thurs April 2, 2009
1:00pm
703 Thackeray Hall
Dr. Thomas Hales, Univ. of Pittsburgh
"The Reinhardt Conjecture"
Abstract: In 1934, Reinhardt made a conjecture about the shape of a (centrally symmetric) disk in the plane with the property that its best possible packing in the plane is the worst. The conjecture is that an octagon with its edges clipped (called the "smoothed octagon") is the worst possible shape from the point of view of packings. This talk will describe some recent progress toward a solution to this conjecture.
April 9, 2009
1:00pm
703 Thackeray Hall
Dr. Thomas Hales, Univ. of Pittsburgh
"A progress report on the Flyspeck formal proof project"
Abstract: A few years ago the Flyspeck formal proof project was launched. The purpose of this project is to give a complete formal proof of the Kepler conjecture. The Kepler conjecture asserts that no packing of congruent balls in three dimensions can have density greater than the density of the familiar cannonball arrangement. A formal proof is a proof in which every logical step has been checked by a computer, based on the fundamental axioms of mathematics. Originally, this project was estimated to take 20 work years to complete. The project now appears to be over half-way complete. This talk will discuss some recent progress toward the completion of the Flyspeck project.
April 16, 2009
12:00pm
Thack 703
Florence Lecomte, University of Strasbourg
"Motives and Realizations"
Abstract: Without giving any construction, I will explain the main properties of Voevodsky's motives. With easy examples, I will show how they work and how you can realize them.