Monday, 12:15 PM, 511 LA

Organizers: Chris Beasley, Ana-Maria Castravet, Tony Iarrobino, Egon Schulte and Alex Suciu

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Previous Events

Monday, December 8, 2014

Speaker: Javier Bracho  (UNAM, visiting Northeastern)

Monday, December 1, 2014

Speaker: Nicholas Matteo  (Northeastern University)

Monday, November 24, 2014

Speaker: Barbara Bolognese  (Northeastern University)
Title: Generic strange duality on abelian surfaces
Abstract: Le Potier conjectured an unexpected duality between the complete linear series of certain natural divisors, called Theta divisors, on moduli spaces of sheaves on a surface. Such conjecture is widely known as Strange Duality conjecture. After the results of Marian and Oprea, who proved generic Strange Duality for sheaves on K3 surfaces, I will work in the setting of abelian surfaces. First, I will describe the geometric setup, including the moduli space, the Theta divisors and their relative version giving rise to the Verlinde sheaves. I will then present the duality in this setting and, time permitting, I will sketch its proof. This is joint work with A. Marian, D. Oprea and K. Yoshioka.

Monday, November 17, 2014

Speaker: Jarosław Buczyński (Polish Academy of Sciences)
Title: Apolarity and direct sum decompositions
Abstract: A polynomial is a direct sum if it can be written as a sum of two non-zero polynomials in some distinct sets of variables, up to a linear change of variables. We analyse criteria for a homogeneous polynomial to be decomposable as a direct sum, in terms of the apolar ideal of the polynomial. We prove that the apolar ideal of a polynomial of degree d strictly depending on all variables has a minimal generator of degree d if and only if it is a limit of direct sums. This is a joint work with Weronika Buczynska, Johannes Kleppe, and Zach Teitler.

Monday, November 10, 2014

Speaker: Alexandru Dimca (University of Nice and IAS Princeton)
Title: Jacobian relations and Hodge theory of singular hypersurfaces
Abstract: Let f in S=C[x0,…,xn] be a homogeneous polynomial, and denote by fj the partial derivative of f with respect to xj, for j=0,…,n and Jf the Jacobian ideal spanned by them in S. Let V(f) be the projective hypersurface given by f=0 and F(f) the associated Milnor fiber given by f=1 in Cn+1. When V(f) is smooth, the partial derivatives f0,…,fn form a regular sequence in S, the Hodge theory of the hypersurface V(f)(resp. of the Milnor fiber F(f)) was described by Ph. Griffiths (resp. by J. Steenbrink) in terms of the graded Jacobian (or Milnor) algebra S/Jf. In this talk we describe how these results change when V(f) acquires some isolated singularities, e.g. some nodes. The interplay between algebra and topology continues to function, but it becomes more involved.

Monday, November 3, 2014

Speaker: Idun Reiten (NTNU Trondheim)
Title: Support tau-tilting modules

Monday, October 27, 2014

Speaker: Ramis Movassagh (Northeastern University)
Title: Eigenvalue attraction
Abstract: We prove that the complex conjugate eigenvalues of a real matrix attract in response to additive real randomness. Consider the time discretization 0< t_1< t_2<… and define a stochastic process by M(t_i + dt)=M(t_i)+dt*P(t_i), where M(0) is a fixed real matrix, t_i < dt and each is a real random whose entries are independent with zero mean and bounded we prove that any complex conjugate pair of eigenvalues of to prove we first construct a smooth family of stochastic processes such that M'(t)=P(t), that in the limit recover the original discrete process. We then explicitly write down the differential equations governing the motion of any eigenvalue. We derive formulas for the expectation value of the force and find that the force is inversely proportional to the distance of any c.c. pair and directly proportional to the 2-norm squared of the corresponding left eigenvector. Therefore, c.c. pairs closest to the real axis, or those that are ill-conditioned, attract most strongly. We then prove that when the perturbation matrix is complex, there is no such force. A special limit of our results is applicable to the, often arising in application, small perturbations of a fixed matrix. We numerically illustrate the theory through various examples and discuss applications, including the Hatano-Nelson model. Time permitting we will discuss related results on the eigenpairs of Toeplitz matrices with singular Fisher-Hartwig symbols and the aggregation and low density of the eigenvalues of real random matrices on and near the real axis respectively.

Friday, October 24, 2014 (at 9:00am)

Speaker: Larry Smith (University of Göttingen)
Title: Poincaré duality algebras, Milnor’s diagonal elements, and Macaulay duality
Abstract: In his famous lectures on characteristic classes John Milnor makes use of a cohomology class in H^*(M X M, Δ(M)) of a closed manifold M, where Δ: M → M X M is the diagonal embedding, to describe Poincaré duality and to define Stiefel–Whitney classes. In this talk I will make use of one of Milnor’s lemmas to give a purely algebraic definition of an element u ∈ A ⊗A, where A is a Poincaré duality algebra, that plays the same role. I will use this Milnor diagonal element to describe a Macaulay dual for the kernel of the multiplication map A ⊗ A → A, relate u to the dimension of A as a vector space, and to define Stiefel–Whitney classes if A supports an unstable Steenrod algebra action. Using this definition of Stiefel–Whitney classes I will show that Wu’s formula for the action of the Steenrod algebra on them holds, recovering a new proof of an old result of J.F. Adams.

Monday, October 20, 2014

Speaker: Claus Ringel (University of Bielefeld)
Title: The Auslander varieties for a wild algebra
Abstract: Let k be an algebraically closed field and A a finite-dimensional (associative) k-algebra. Given an A-module M, the set of all submodules of M with fixed dimension vector is called a quiver Grassmannian in M. If C and Y are A-modules, then we consider Hom(C,Y) as an E(C)-module, where E(C) is the opposite of the endomorphism ring of C, and the Auslander varieties for A are the quiver Grassmannians in this E(C)-module Hom(C,Y). In his seminal Philadelphia Notes (published in 1978), M. Auslander exhibited his theory of morphisms determined by modules. It is an important frame for describing the poset structure of the category of A-modules and this setting allows to interpret the Auslander varieties as describing factorizations of morphisms of A-modules. If the algebra A is (controlled) wild, then one knows that any projective variety can be realized as an Auslander variety. The aim of the lecture is to analyse sets of factorizations which can be used to realize arbitrary projective varieties.

Monday, October 6, 2014

Speaker: Karim Adiprasito (Hebrew University and IHES)
Title: Some remarks on the geometry of simplicial polytopes
Abstract: We show an interesting relation between the shape of a convex simplicial polytope, graph chordality and the Hard Lefschetz Theorem for projective toric varieties of simplicial polytopes.

Friday, October 3, 2014 (at 10:30 am)

Speaker: Eric Friedlander (University of Southern California)
Title: Invariants of representations using Jordan types
Abstract: This is will be a very informal discussion of how Julia Pevtsova and I have used the data of Jordan types of p-nilpotent operators to give refined invariants (finer than support varieties) for modular representations.

Wednesday, October 1, 2014

Speaker: Justin Malestein (Hebrew University)
Title: Rigidity of Symmetric Frameworks
Abstract: Bar-joint frameworks are structures made of fixed-length bars connected by universal joints with full rotational freedom. The allowed motions preserve the length and connectivity of the bars, and a framework is rigid if all the allowed motions extend to rigid body motions. For finite frameworks with generic geometry, rigidity is known to depend only on the graph that has as its edges the bars, and, in dimension 2, the generically rigid graphs are known exactly. In recent years, the question of extending this kind of combinatorial theory to infinite frameworks or finite frameworks with special geometry such as symmetry has received a lot of attention, motivated, in part, by applications in crystallography.
In this talk, I will discuss both combinatorial and algebraic aspects of rigidity of bar-joint frameworks with symmetry. In particular, I will present “Laman-like” theorems for various kinds of symmetry in dimension 2; i.e. theorems which characterize generically rigid frameworks. The main focus of the talk will be on recent work of Theran and myself about rigidity and “ultrarigidity” of periodic frameworks (infinite frameworks in Euclidean space invariant under some lattice).

Monday, September 29, 2014

Speaker: Isabel Hubard (Instituto de Matematicas, UNAM, Mexico City)
Title: Chiral polytopes and alternating groups
Abstract: We review basic topics about abstract chiral polytopes and explain the difficulty of constructing such objects of high rank.  Then a construction of chiral 4-polytopes with automorphism groups isomorphic to alternating and symmetric groups is described. This is joint work with Marston Conder, Daniel Pellicer and Eugenia O’Reilly.

Monday, September 22, 2014

Speaker: Tony Iarrobino (Northeastern University)
Title: Jordan types of two commuting nilpotent matrices
Abstract: This work is joint with Leila Khatami, Bart Van Steirteghem and Rui Zhao.  The similarity class of an n by n nilpotent matrix B over a field k is given by its Jordan type, which is the partition P of n that specifies the sizes of the Jordan blocks.  The variety N(B) parametrizing nilpotent matrices that commute with B is irreducible, so there is a partition Q = Q(P) that is the generic Jordan type for matrices A in N(B).  The partition Q(P) has parts that differ pairwise by at least two (we call it stable). P. Oblak proposed a recursive conjecture about the map P to Q(P); we (briefly) describe progress on it by P. Oblak, T. Kosir, L. Khatami, L.Khatami-I., D.I. Panyushev, and R. Basili.

Our focus, rather, is on the set S(Q) of partitions P having a given stable partition Q as generic nilpotent commuting orbit; and also on their loci Z(P),  the set of A in N(B) having Jordan type P. We prove a Table Conjecture proposed by P. Oblak and R. Zhao: given Q = (u,u-r) with u > r > 1 then S(Q) can be arranged in an (r-1) by (u-r) table T(Q) such that the partition P_{i.j} in the i,j entry has i + j parts.  We conjecture with Mats Boij that the locus in N(B) of P_{i,j} is a complete intersection of i + j – 2 linear and quadratic equations. We propose a Box Conjecture for the set S(Q) for any stable Q.
Reference: ArXiv 1409.2192

Monday, September 8, 2014

Speaker: Bruno Benedetti (Freie University Berlin, visiting Northeastern University)
Title: Diameter of graphs of polytopes and beyond
Abstract: The graph of every convex polytope is connected. Balinski proved it is even d-connected, where d is its dimension. But how far away can two vertices be, in a d-polytope with n facets? The “distance” is here just the number of edges you need to walk along to go from one to the other. Hirsch conjectured that the answer should be nd, but the conjecture was disproved by Santos in 2010.  I will explain a couple of recent positive results:(1) the Hirsch conjecture holds for all flag polytopes. The proof uses methods from metric geometry. This is joint work with Karim Adiprasito, arxiv:1303.3598.

(2) (if time permits) The Stanley-Reisner correspondence suggests an elementary way to define the dual graph also for algebraic varieties, basically by looking at “connectedness in codimension one”. Using an algebraic proof, we show that Balinski’s theorem can be extended to the generality of Gorenstein subspace arrangements. This is joint work with Matteo Varbaro, arXiv:1403.3241.

Wednesday, April 16, 2014 at 10:30am

Speaker: Thomas Church (Stanford University, visiting MIT)
Title: Uniform generators for the Johnson filtration
It’s well-known that SLn(Z) is generated by elementary matrices. The elementary matrices Eij and Eji are contained in the subgroup <Eij,Eji> = SL2(Z), so SLn(Z) is generated by elements that are “supported” on some SL2(Z) subgroup. Similarly, the mapping class group Modg is generated by Dehn twists supported on a genus-1 subsurface.
The same question about subgroups is much harder! For congruence subgroups SLn(Z,p), asking whether SLn(Z,p) is generated by elementary matrices is essentially equivalent to the Congruence Subgroup Property. Johnson proved that the Torelli group Modg[1] is not generated by elements supported on genus-1 subsurfaces. However, Birman–Powell proved that the Torelli group is generated by elements supported on genus-2 subsurfaces.
I will give an overview of such “generated by elements of bounded support” results, and explain the ideas behind a new theorem: for every term Modg[k] of the Johnson filtration, there is a constant Gk so that Modg[k] is generated by elements supported on genus-Gk subsurfaces. Joint work with Andrew Putman.

Monday, April 14, 2014

Speaker: Jonathan Heckman (Harvard)
Title: What is a T-Brane?
In string theory, the notion of the position of a particle in spacetime naturally generalizes to non-commuting matrices.  Of particular interest are T-branes, corresponding to the special case where such matrices are nilpotent.  In this talk we focus on a particular manifestation of T-branes in string theory described by solutions to Hitchin’s system with gauge group G.  For G an ADE group, there is a well-established correspondence in the physics literature between non-singular points of the Hitchin system moduli space and the deformation theory of a curve of ADE singularities.  In singular limits such as those governed by T-branes, the spectral equation for the Higgs field degenerates to zn=0, and the classical correspondence with deformation theory breaks down.
Using the theory of limiting mixed Hodge structures, we explain how to track this correspondence in such limits in the case of A_n type gauge groups / singularities. Time permitting, we discuss ongoing work on the  extension of this work to situations other than A-type groups / singularities,  highlighting the special case of T-branes with no deformation moduli.
Based on joint work with L. Anderson, S. Katz, and L. Schaposnik.

Monday, April 7, 2014

Speaker: Mark Mixer (Northeastern University)
Title: Transitive Permutation Groups as String C-Groups of High Rank
Abstract: For any n ≥ 9, up to isomorphism and duality, there are exactly two string C-groups of rank at least n-2 that represent a transitive permutation group of degree n, in both cases the symmetric group Sym(n). In this talk I will extend this classification of high rank string C-groups to include rank n − 3. The result is that for each n ≥ 9 there are seven string C-groups of that rank that represent a transitive permutation group of degree n. Furthermore, the permutation group again is Sym(n) in each case. By completing this classification, I will construct all non-degenerate abstract regular polytopes of any rank d with automorphism group isomorphic to Sym(d+3).This is joint work with M.E. Fernandes and D. Leemans.

Monday, March 31, 2014

Speaker: Sarah Kitchen (University of Michigan)
Title: Generalized Harish-Chandra Modules
Abstract: Harish-Chandra modules, together with Beilinson-Bernstein localization, are well known to allow the study of representations of real Lie groups from a complex algebraic perspective. These modules are simultaneously a representation of the complexification of the Lie algebra and maximal compact subgroup of the real group. Generalized Harish-Chandra modules weaken these requirements on the pair by taking any semi-simple complex Lie algebra, and any reductive subalgebra. In this talk, I will explain new considerations that must be taken into account in order to localize these objects and a geometric approach to a conjecture of Penkov and Zuckerman, which categorifies a parameterization of some of the irreducible modules.

Monday, March 24, 2014

Speaker: Max Wakefield (US Naval Academy)
Title: Limits of rational models of k-equal arrangements
Abstract: Braid arrangements  arise through many different fields of mathematics as configurations spaces, as reflecting hyperplanes of type A Coxeter groups, as realizations of partition lattices, and as a method to compute chromatic polynomials of graphs. We define a monoidal structure on a set of generalizations (k-equal arrangements) of braid arrangements and study some limits of these arrangements and their rational models. Summing these limits together we present a kind of classifying algebra for the rational homotopy models of the complements of limits of k-equal arrangements. This algebra has a rich combinatorial structure. We will discuss some of its properties and applications.

Monday, March 17, 2014

Speaker: Ivan Losev (Northeastern University)
Title: Procesi bundles on Hilb^n(C^2)
Abstract: A Procesi bundle is a vector bundle on the Hilbert scheme of n points on the plane. It was first constructed by Haiman who used it to prove the Schur positivity for Macdonald polynomials. This bundle also provides a derived McKay equivalence for the Hilbert scheme. I will basically take the latter for an axiomatic description of a Procesi bundle. I will show that there are exactly two bundles with these properties: Haiman’s and its dual. Time permitting I will also discuss an extension of these results to other symplectic resolutions and a relation between the Procesi bundles and the tautological bundle conjectured by Haiman. The proofs are based on the study of symplectic reflection algebras.

Monday, March 10, 2014

Speaker: Julia Pevtsova (University of Washington)
Title: Modules of Constant Jordan Type and Their Generalizations
Abstract: Let G be a finite group (scheme) with the group algebra kG for a field k of positive characteristic p.  A module of constant Jordan type  over G is a module that exhibits the same behavior when restricted to various subalgebras of kG isomorphic to k[t]/t^p.  I’ll discuss properties of these modules and their generalizations, some open problems and recent progress towards them, and the connections between modules of constant Jordan type and the geometry of the projective variety Proj H^*(G,k).  For many of my examples, I’ll concentrate on the case of an elementary abelian p-group,  for which the geometric connections lead to a correspondence between modules of constant Jordan type and vector bundles on projective spaces (and between generalized modules of constant Jordan type and bundles on Grassmannians).This is joint work with J. Carlson and E. Friedlander.

Monday, February 24, 2014

Speaker: Anand Patel (Boston College)
Title: The Maroni Theory of Hurwitz Space
Abstract: In this talk I will discuss what I call the “Maroni theory” of the Hurwitz spaces parametrizing branched covers of a fixed curve. The goal is to understand the geometry of a decomposition of Hurwitz space into the union of certain special subvarieties, the so-called Maroni loci. This decomposition seems to play an important role in the solution to various problems. I will discuss some of these problems. No prior familiarity of Hurwitz spaces will be assumed.

Thursday, December 13, 2013

Speaker: Diane Maclagan  (University of Warwick)
Title: The Cox ring of wonderful compactifications
Abstract: The wonderful model Y~ of a hyperplane arrangement complement Y is a smooth compactification of Y with normal crossing boundary introduced by DeConcini and Procesi. Examples include the blow-up of P2 at any number of points, and the moduli space M0,n. I will introduce these varieties, and describe an invariant ring description of their Cox rings generalizing the Doran-Giansiracusa description for M0,n, plus an associated power ideal description of the graded pieces of the Cox rings that generalizes the Emsalem-Iarrobino description for blow-ups of P2.  This is joint work with Florian Block.