## Algebra, Combinatorics and Geometry Seminar

### This Weeks Lecture

April 18, 2013
12:00pm
427 Thackeray Hall
Prof. Greg Constantine, Univ. of Pittsburgh

" One error for the price of existence"

Abstract: Existence of optimal nonlinear codes, such as Hadamard codes, is difficult to establish. I show existence of codes that correct just one less error than the (possibly nonexistent) optimal codes, irrespective of dimension.

### Fall 2012/Spring 2013 Schedule

September 20 and 27, 2012
12:00pm

427 Thackeray Hall

Sevak Mkrtchyan, Carnegie Mellon University

"Asymptotic representation theory of symmetric groups"

Abstract: We will study local and global statistical properties of Young diagrams with respect to a Plancherel-type family of measures called Schur-Weyl measures and use the results to answer a question from asymptotic representation theory. More precisely, we will solve a variational problem to prove a limit-shape result for random Young diagrams with respect to the Schur-Weyl measures and apply the results to obtain logarithmic, order-sharp bounds for the dimensions of certain representations of finite symmetric groups. By studying the local fluctuations of the underlying point processes via the saddle point method we will prove that the Schur-Weyl measures have the asymptotic equipartition property. We will mention connections to combinatorics and random matrix theory.

October 18, 2012
12:00pm

427 Thackeray Hall

Prof. Armin Gholampour, Univ. of Maryland

"Donaldson-Thomas invariants of 2-dimensional sheaves and modular forms"

Abstract: We define the Donaldson-Thomas invariants associated to the moduli space of stable 2-dimensional sheaves on a smooth threefold X. If X is a smooth K3 fibration over a curve, we express the DT invariants of X in terms of the Euler characteristics of the moduli spaces of stable torsion free sheaves on a K3 surface and the Noether-Lefschetz numbers of the fibration. From this we conclude that the generating functions of the DT invariants of X are modular. We extend this to the case that the K3 fibration has finitely many fibers with nodal singularities. Finally, we sketch a method to compute the DT invariants of the Calabi-Yau complete intersections such as Fermat quintic in P^4.

October 25, 2012
12:00pm

427 Thackeray Hall

Prof. Howard Garland, Yale University

"Eisenstein series on loop groups"

Abstract: We will discuss the existence and meromorphic continuation of Eisenstein series on loop groups and the the possible application of this theory to automorphic L-functions associated to cusp forms on finite-dimensional groups.

November 1 and 8, 2012, 2012
12:00pm

427 Thackeray Hall

Dr. Rina Anno, University of Pittsburgh

"Fourier-Mukai transforms for enhanced triangulated categories"

Abstract: In algebraic geometry, the study of derived categories of sheaves is stumbled by the fact that the category of functors between triangulated categories is not triangulated. The usual way to treat this is to only consider Fourier-Mukai (integral) functors, replacing the category of functors by the category of their Fourier-Mukai kernels. The problem however is that this correspondence between kernels and transforms is neither full nor faithful. I will talk about a different idea: consider enhancement of derived categories of sheaves by DG categories of modules instead, and instead of Fourier-Mukai kernels get a DG category of bimodules that behaves much better.

November 15, 2012
12:00pm

427 Thackeray Hall

Dr. Chirs Manon, George Mason University

"The combinatorial commutative algebra of conformal blocks"

Abstract: Toric degenerations of schemes are a way to replace geometric or algebraic questions with questions about polyhedral geometry. In this talk we discuss how the combinatorics of objects from mathematical physics, the conformal blocks,  can be used to construct flat degenerations of the Cox ring of the moduli of quasi-parabolic principal bundles on an $n-$marked curve of genus $g.$  We will discuss when these degenerations are toric, and how the resulting combinatorial pictures can be used to prove structural theorems about this ring.

November 29, 2012
12:00pm

427 Thackeray Hall

Alexei Davydov, Ohio University

"Structure of braided tensor categories"

Abstract: A certain equivalence relation (Witt equivalence) allows one to organise braided tensor categories into a manageable set of classes. For example under this equivalence symmetric tensor categories form just two classes (Tannakian and super-Tannakian). In the fusion braided case the equivalence classes form a group (a generalisation of the Witt group). The structure of this group is related to a conjecture of Moore and Seiberg about chiral algebras of rational conformal field theories.

December 6, 2012
12:00pm

427 Thackeray Hall

Prof. Greg Constantine, University of Pittsburgh

"Design pairs"

Abstract: The general theme is that of constructing symmetric designs. It is often possible to accomplish this by working with two combinatorial structures, neither of which is a 2-design, but which act jointly to allow the construction of a 2-design.

Such design pairs offer an expression of certain integers that are equal to 3 mod 4 as a sum of two square moduli of cyclotomic integers. It is on occasion also possible to use the Seidel-Goethals' method to construct Hadamard matrices, with such design pairs playing a central role.

February 28, 2012
12:00pm

427 Thackeray Hall

Prof. Bogdan Ion, University of Pittsburgh

"BGG reciprocity for current algebras

Abstract: Current algebras are special maximal parabolic subalgebras of affine Lie algebras. In the case of untwisted affine Lie algebras they are isomorphic to the tensor product of a finite dimensional simple Lie algebra and the ring pf polynomials in one variable. The category of finite dimensional representations of current algebras is not semisimple. Recently Chari and collaborators have conjectured a version of the BGG reciprocity in this context, which connects simple finite dimensional representations, their projective covers, and standard modules. I will present a proof of this conjecture.

March 21, 2012
12:00pm

427 Thackeray Hall

Mr. Takuya Murata, University of Pittsburgh

"Asymptotic of multiplicities of the reductive group action: torus case"

Abstract: The Okounkov body of a projective variety is a compact convex set that encodes geometric invariants of the variety; e.g., the degree (i.e., the 1st Chern class of O(1)) of the variety. In the talk, we are interested in the case when there is an action of a complex connected reductive group on the variety. Okounkov studied the asymptotic of the multiplicity m_{k, k \lambda} as k goes infinity. We study that of m_{k, lambda}. The latter case goes back to Howe (finite group) and Brion (reductive group). The current result I and Kaveh have is when the group is a torus. The proof of the general case is also in preparation. In the talk, the symplectic approach (Riemann-Roch) may also be mentioned.

March 28, 2013
12:00pm
427 Thackeray Hall
Prof. Kuimars Kaveh, Univ. of Pittsburgh

"Geometric inequalities for multiplicities of ideals"

Abstract:

April 11, 2013
12:00pm
427 Thackeray Hall
Dr. Rina Anno, Univ. of Pittsburgh

"Braiding conditions for spherical twists"

Abstract:

A spherical functor between two triangulated categories is a functor with left and right adjoint for which cones of the four possible adjunction and coadjunction maps
are autoequivalences of categories. These autoequivalences are called spherical twists (or cotwists). In a number of situations, certain spherical twists generate a weak braid group or affine braid group action on a category. We are looking for a simplest possible condition on two spherical functors that ensures that their twists commute, or satisfy the braid relation. For the twists that are induced by a single object (or rather a functor from D^b(pt) to D^b(X)), these conditions are that Ext^*(E_1,E_2)=0 for commutation, and Ext^*(E_1,E_2) is 1-dimensional in degree one, for braiding. We seek to establish similar cohomologic criteria for spherical twists induced by certain fibrations, but first we need an abstract criterion in a verifiable form.

April 18, 2013
12:00pm
427 Thackeray Hall
Prof. Greg Constantine, Univ. of Pittsburgh

" One error for the price of existence"

Abstract: Existence of optimal nonlinear codes, such as Hadamard codes, is difficult to establish. I show existence of codes that correct just one less error than the (possibly nonexistent) optimal codes, irrespective of dimension.

### Fall 2011 Schedule

September 15, 2011
12:00pm
703 Thackeray Hall
Prof. Thomas Hales, Univ. of Pittsburgh
"Mathematics in the Age of the Turing Machine"

Abstract: Next year we celebrate the centennial of Alan Turing's birth.  This will be a talk for a general audience about some of the ways that computers shape mathematical research.  I will give examples both of "computer proofs" that make computation part of the proof and of "formal proofs" that use computers to check the logical reasoning behind proofs.

September 29, 2011
12:00pm
703 Thackeray Hall
Prof. Urs Schreiber, Utrecht University

"Differential characters from higher Lie integration"

Abstract: The process of integrating a Lie algebra to a Lie group can be generalized to give a canonical way of integrating an L-infinity algebra to a higher stack. This extends to L-infinity cocycles. I discuss how, in joint work with Fiorenza, Sati, and Stasheff, we used this to construct smooth differential cocycle-refinements of the first and second fractional Pontrjagin class, or of the second and fourth Chern classes in the complex case. These lead to differential and twisted refinements of higher notions of Spin structures, known  as string-structures and fivebrane-structures, and are motivated by an obstruction problem in the quantization of string theory.

October 13 and 20, 2011
12:00pm
703 Thackeray Hall
Prof. Hiham Sati, Univ. of Pittsburgh

"Topological modular forms"

Abstract: Topological modular forms (TMF) is a generalized cohomology theory characterized by the fact that its coefficient ring is essentially the graded ring of integral modular forms. I will explain what "essentially" means, why elliptic curves and their moduli stacks appear, and what they have to do with topology. I will also explain how this theory can be thought of as interpolating between number theory and homotopy theory. Generalizations and recent applications will be presented as time permits.

October 27 and November 3, 2011
12:00pm
703 Thackeray Hall
Prof. Alexander Borisov, Univ. of Pittsburgh

"Geometric approach to the two-dimensional Jacobian Conjecture"

Abstract: I will describe my approach to the two-dimensional Jacobian Conjecture using some ideas of birational algebraic geometry. The talk will include recent progress using determinants of weighted trees and applications of this approach to maps of small degree.

November 10 and 17, 2011
12:00pm
703 Thackeray Hall
Prof. Kiumars Kaveh, Univ. of Pittsburgh

"Toric degenerations, integrable systems and Okounkov bodies"

Abstract: A (completely) integrable system is a Hamiltonian system which admits a maximal number of "first integrals" (also called "conservation laws"). Integrable systems are abundant in physics and mathematics and are very well-studied. In this talk we make a connection between integrable systems and algebraic geometry, discussing a general method for constructing integrable systems on a large class of varieties. This relies on methods from algebra, namely degenerating a given variety to a "toric variety". Many well-known examples of integrable systems, e.g. Guillemin-Sternberg integrable system on the flag variety, fit into this picture.

December 1, 2011
12:00pm
703 Thackeray Hall
Prof. Jason DeBlois, Univ. of Pittsburgh

"Hyperbolic disk packings and the topology of moduli space"

Abstract: Each 2-cell of the Delaunay tessellation determined by a set of points in the hyperbolic plane is 'cyclic': its vertex set lies on a circle.  Call a 2-cell 'centered' if its interior contains the center of this circle.  There is a sense in which centered polygons behave better than those which are cyclic but not centered.  I will make this precise, then show that the set of non-centered 2-cells has a nice underlying structure that one can use to control their pathology.  As an application I will describe a sort of finite version of Mumford's compactness criterion for finding compact subsets of the genus-g moduli space.

January 26, 2012
12:00pm
427 Thackeray Hall
Chris Kapulkin, Univ. of Pittsburgh

"An introduction to polynomial functors"

Abstract: This talk is meant to be an introduction to the theory of polynomial functors and their applications. Building on set-theoretic intuitions I will introduce the notion of a polynomial functor on a (slice of) locally cartesian closed category and show some of its basic properties. Later, I will present several applications with a special emphasis on homotopy theory and higher category theory.

February 2, 2012
12:00pm
427 Thackeray Hall
Prof Alexander Borisov, Univ. of Pittsburgh

"Determinants of weighted trees and applications to plane compactifications"

Abstract: We derive "local" formulas for determinant matrices associated to weighted trees. Main motivation and applications come from the graphs of rational curves obtained by successive blowups "at infinity" of the projective plane. In particular, we define two integer invariants of these curves and interpret them as functions on the appropriate Zariski-Riemann space of valuations. We explain how these invariants naturally appear in the two-dimensional Jacobian conjecture and prove that if their values are fixed, the corresponding valuations form finitely many families, modulo polynomial automorphisms.

February 9, 2012
12:00pm
427 Thackeray Hall
Prof Gregory Constantine, Univ. of Pittsburgh

"Four Squares Of Sums Of Sets Of Cosines"

Abstract

March 29, 2012
12:00pm
427 Thackeray Hall
Prof. Kiumars Kaveh, Univ. of Pittsburgh

"Reciprocity Law on Algebraic Curves"

Abstract: I will talk about Galois theory of algebraic curves and Weil reciprocity.

The talk examines Galois theory and class field theory (nicely covered in Tom's course for number fields) for field of rational functions on an algebraic curve. I will give a simple proof of Weil reciprocity using Newton polygons. I will try to cover most of background material (no need to have attended Tom's course). Familiarity with Galois theory will be assumed.

April 5, 2012
12:00pm
427 Thackeray Hall

Prof. Ping Xu, Penn State University
"Geometry of Maurer-Cartan elements on complex manifolds"

Abstract: Maurer-Cartan elements on a complex manifold  are extensions of holomorphic Poisson structures. We study the  geometry of these structures, by investigating their cohomology and homology theory. In particular, we describe a duality on the homology groups, which generalizes the Serre duality of Dolbeault cohomology.

April 19, 2012
12:00pm
427 Thackeray Hall

Prof. Stephen DeBacker

"Unexpected twists"

Abstract:  The conjectural Local Langlands Correspondence (LLC) states that the set of irreducible smooth discrete series representations of a p-adic group may be partitioned into finite sets, called L-packets, such that many wonderful properties hold.  One of the expected properties states that an appropriate combination of characters of the representations in an L-packet will be stable (that is, as a function on the set of strongly regular semisimple rational elements, the combination should assume the same value at any two elements that are conjugate over the algebraic closure). We have found that, in a very natural setting, the “obvious" L-packet does not have this property. To overcome this difficulty, a certain twist must be added to the mix.

April 26, 2012
12:00pm

427 Thackeray Hall

Hoang Le Truong, Univ. of Pittsburgh and Hanoi Math Institute

"The equality (I^2 = q I) in sequentially cohen-macaulay rings"
Abstract

May 3, 2012
12:00pm

427 Thackeray Hall

Walter Freyn, University of Muenster

"From SL(2) and hyperbolic space to hyperbolic Kac-Moody algebras and their buildings"

Abstract: In this talk we start with with the finite dimensional Lie algebra sl(2) and hyperbolic space and describe how to construct from those elementary building blocks hyperbolic Kac-Moody algebras and their associated Kac-Moody groups. Associated to Kac-Moody algebras are twin buildings. We describe embeddings of hyperbolic twin buildings into the compact real forms of Kac-Moody algebras.

### Fall 2010/Spring 2011 Schedule

September 30, 2010
12:00pm
703 Thackeray Hall
Dr. Thomas Hales, Univ. of Pittsburgh
"The Fundamental Lemma for Beginners."

Abstract: At the International Congress of Mathematicians in India last month, Ngo Bao Chau was awarded a Fields medal for his proof of the "Fundamental Lemma."  This talk is particularly intended for students and mathematicians who are not specialists in the theory of Automorphic Representions.  I will describe the significance and some of the applications of the "Fundamental Lemma."  I will explain why this problem turned out to be so difficult to solve and will give some of the key ideas that go into the proof.

October 21, 2010
12:00pm
703 Thackeray Hall
Dr. Thomas Hales, Univ. of Pittsburgh
"The fundamental lemma and the Hitchin fibration"

Abstract: The fundamental lemma is a collection of identities of integrals that comes up in the study of a trace formula.  These identities have recently been proved by Ngo Bao Chau, for which he was awarded a Fields Medal earlier this year.  At the heart of his proof is a beautiful interpretation of these integrals in terms of the cohomology of the Hitchin fibration.  This talk will explain the geometry of the Hitchin fibration in relation to the fundamental lemma.

October 28, 2010
12:00pm
703 Thackeray Hall
Chris Kapulkin, Univ. of Pittsburgh
"From Atiyah to Lurie. 20 years of topological quantum field theory"

Abstract: I will define an n-dimensional TQFT as a symmetric monoidal functor from the category of n-cobordisms to the category of complex vector spaces and show some of its applications to algebraic topology and representation theory. This notion will be examine in case n=2---I will sketch the proof that the category of 2-dimensional TQFTs is equivalent to the category of commutative Frobenius algebras. In the last part of the talk, I will present some recent development: work of Jacob Lurie on Baez-Dolan Cobordism Hypothesis. Even though I will introduce such notions as monoidal structure or a symmetry, some familiarity with category theory will be assumed.

November 11, 2010
12:00pm
703 Thackeray Hall
Dr. Yimu Yin, Univ. of Pittsburgh
"Integration in algebraically closed valued fields with sections"

Abstract: I will describe how to construct Hrushovski-Kazhdan style motivic integration in certain expansions of ACVF(0, 0). Such an expansion is typically obtained by adding a full section from the RV-sort into the VF-sort and some (arbitrary) extra structure in the RV-sort. The construction of integration, that is, the inverse of the lifting map L, is rather straightforward. What is a bit surprising is that the kernel of L is still generated by one element, exactly as in the case of integration in ACVF(0, 0). I will also describe an application to zeta functions, showing that their rationality, shown by Denef and Pas in the 80s, is uniform.

November 18, December 2, 2010
12:00pm
Dr. Kiumars Kaveh, Univ. of Pittsburgh
"Convex polytopes, irreducible representations and flag varieties"

Abstract: I review some basic facts about the crystal bases for finite dimensional irreducible representations of a reductive group G. A remarkable property of a crystal basis (due to Littelmann and Bernstein-Zelevinsky) is that its elements can be naturally parametrized by the set of integral points in a convex polytope (a string polytope). I will then discuss a recent result that the Littelmann parametrization coincides with a geometric valuation on the field of rational functions of the flag variety of G. This valuation is constructed out of a sequence of Schubert varieties. This extends an earlier result of A. Okounkov for symplectic group and confirms the general philosophy that the string polytopes are analogues of Newton polytopes for toric varieties.

December 9, 2010
12:00pm
Dr. Bogdan Ion, Univ. of Pittsburgh
"Geometric Complexity Theory (after Mulmuley)"

Abstract: Geometric complexity theory (GCT) is an approach to the algebraic P vs NP problem laid out by Ketan Mulmuley and collaborators. I will give an overview of GCT and discuss the geometric and representation-theoretical conjectures on which everything ultimately relies.

February 3, 2011
12:00pm
703 Thackeray Hall

Chris Kapulkin

"Why do n-categories matter?"

Abstract: After its introduction in 1945 by Saunders MacLane and Samuel Eilenberg, the notion of category has been generalized in many different directions. One such direction is to consider so called higher dimensional categories which apart from the objects (0-cells) and morphisms (1-cells) also have morphisms between morphisms

(2-cells) and so on. The motivation to consider such structures comes from many different parts of mathematics, for example: topology, algebra, theoretical computer science, and mathematical physics. In this talk I will sketch those motivations and show how higher dimensional categories provide a universal framework to work with various algebraic, topological, and logical structures. Finally, I will try to sketch different approaches to the definition of higher dimensional category and discuss their advantages and disadvantages.

February 10 and 24, 2011
12:00pm
703 Thackeray Hall
Prof. Kiumars Kaveh

"Polytope algebra and cohomology rings of toric varieties"

Abstract: I will give a brief introduction to "toric varieties". Main example of a toric variety is the projective space. They are objects of much interest in algebraic geometry, combinatorics and topology. The geometry and topology of toric varieties are closely related to the geometry and combinatorics of convex polytopes. I will discuss a very nice description of cohomology ring of a smooth projective toric variety due to Khovanskii-Pukhlikov. This is related to the so-called polytope algebra associated to a convex polytope in R^n. If time permits I will mention a result of mine which describes the cohomology rings of flag variety and Grassmannian in a similar fashion. For the most part I just assume basic background in algebra, geometry and topology.

February 17, 2011
12:00pm
703 Thackeray Hall
Prof. Xander faber

"The Berkovich Ramification Locus for Rational Functions"

Abstract: Given a nonconstant holomorphic map f: X \to Y between compact Riemann surfaces, one of the first objects we learn to construct is its ramification divisor R_f, which describes the locus at which f fails to be locally injective. The divisor R_f is a finite formal linear combination of points of X that is combinatorially constrained by the Hurwitz formula.

Now let k be an algebraically closed field that is complete with respect to a nontrivial non-Archimedean absolute value. For example, k = C_p. Here the role of a Riemann surface is played by a projective Berkovich analytic curve. As these curves have many points that are not algebraic over k, some new (non-algebraic) ramification behavior appears for maps between them. For example, the ramification locus is no longer a divisor, but rather a closed analytic subspace. The goal of this talk is to introduce the Berkovich projective line and describe some of the interesting features of the ramification locus for self-maps f: P^1 \to P^1.

March 4, 2011
12:00pm
703 Thackeray Hall
Prof. Greg Constantine, Univ. of Pittsburgh

"Faithful characters of finite groups"

Abstract: Characters of the faithful representations of a finite group are examined, and a formula that their degrees verify is obtained. As a follow-up, a combinatorial construction of symmetric designs using generating functions and characters of Abelian groups is presented.

March 17, 2011
12:00pm
703 Thackeray Hall
Dr. Thomas Hales, Univ. of Pittsburgh
"The Use of the Fundamental Lemma"

Abstract:

In a series of lectures in Paris in 1980, Langlands conjectured the "fundamental lemma", a collection of identities of integrals associated with reductive groups over nonarchimedean local fields. These identities were proved by Ngo Bao Chau in a book published last year.  The fundamental lemma has already had many applications to the theory of automorphic forms and number theory.  This talk will give a survey of some theorems that rely on the fundamental lemma.

April 7, 2011
12:00pm
Prof. Greg Constantine, Univ. of Pittsburgh

"Constructions of symmetric designs"

Abstract: Methods of constructing symmetric designs, and outstanding open problems in this area are presented.

### Fall 2009/Spring 2010 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.

January 7, 2010
3:00pm

704 Thackeray Hall
APPLICANT COLLOQUIUM
Prof. Tonghai Yang, Univ. of Wisconsin
"The Gross-Zagier Formula Revisited"
More>

February 25, 2010
12:00pm
703 Thackeray Hall
Prof. Bogdan Ion, Univ. of Pittsburgh
"Generalized exponents and the combinatorics of minimal expressions"

Abstract: I will describe a general scheme for computing generalized exponents (and in fact all q-multiplicities). The fact on which ultimately everything rests is an explicit formula for Fourier coefficients of the Cherednik kernel (a non-symmetric partition function). I will explain some of the details required to prove this formula and the combinatorics necessary to express the answer.

March 3, 2010
3:00pm
704 Thackeray Hall
APPLICANT COLLOQUIUM
Prof. Kiumars Kaveh, McMasters Univ.
"Convex Bodies, Algebraic Equations & Group Actions"
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March 4, 2010
3:00pm
704 Thackeray Hall
APPLICANT COLLOQUIUM
Prof. Jeehoon Park, McGill University
More>

March 18, 2010
12:00pm
Hoang Le Truong
”Hilbert Coefficients and Sequentially Cohen-Macaulay Modules”
More>

March 25, 2010
12:00pm
Soon-Yi Kang, Univ. of Pittsburgh
"Mock modularity that appears in q-hypergeometric series and traces of singular moduli"

Abstract: The theory of mock modular forms, which is an extension of the classical modular forms, has been rapidly developed in recent years.  In this talk, we present two most famous examples of the mock modular forms along with their applications to partition and number theory. They are Ramanujan's mock theta functions and the generating series of the traces of singular moduli.

April 8, 2010
12:00pm
Tom Hales, Univ. of Pittsburgh

Abstract: This talk will survey some recent elementary results from the areas of packings and tilings.

April 15, 2010
12:00pm
Prof. Alexander Borisov, Univ. of Pittsburgh
"Lattice-free simplices"

Abstract: Consider an n-dimensional lattice Z^n inside the real space R^n.
A lattice-free simplex is a simplex with vertices in Z^n and no other points from Z^n inside or on the boundary. The ultimate goal is to classify these simplices up to the automorphisms of the lattice. In dimension two the answer is obvious, and in dimension three it is relatively easy and well-known. However even in dimension 4 the answer is not known. I will give a survey of old and recent results on this topic.

April 22, 2010
12:00pm
703 Thackeray Hall
Prof. Bogdan Ion
"Generalized exponents and the combinatorics of minimal expressions 3"

Abstract: I will describe a general scheme for computing generalized exponents (and in fact all q-multiplicities). The fact on which ultimately everything rests is an explicit formula for Fourier coefficients of the Cherednik kernel (a non-symmetric partition function). I will explain some of the details required to prove this formula and the combinatorics necessary to express the answer.

### 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

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.

2007-2008 Schedule