Everytopic Seminar - Spring 2015
Room 226, Goldsmith building
Organizer: Olivier Bernardi
The Everytopic Seminar is a seminar aimed at a broad audience of mathematicians.
The talks are 80 minutes long. The first half the talk should be given in a
colloquium style, and should be accessible to faculty and graduate students
from any field of mathematics. It should allow everyone to grasp the content and
significance of the results being discussed. The second half of the talk can be in
the style of a regular research seminar, providing the audience with a deeper understanding
of the results as well as some details of the proofs.
Tuesday February 3
Speaker: Siu-Cheong Lau (Harvard)
Title: A constructive and functorial approach to mirror symmetry
Abstract: Homological mirror symmetry conjecture asserts an equivalence between the category of Lagrangian submanifolds and the category of coherent sheaves of the mirror. The conjecture has been verified in several interesting cases by computing generators and relations of the categories. However such a computational approach hardly explains why homological mirror symmetry exists in general. In this talk I will introduce a mirror construction which always comes with a functor between the two categories. It produces mirrors which are not reachable by the SYZ construction, and is naturally connected with non-commutative geometry.
Tuesday February 24
Speaker: Dylan Rupel (Northeastern)
Title: Quantum groups, quantum cluster algebras, and their categorifications
Abstract: Quantum groups, quantum cluster algebras, and their categorifications
Abstract: At the heart of the definition of a quantum cluster algebra is a deep conjecture that the recursively, combinatorially defined cluster monomials of a conjectural quantum cluster algebra structure are elements of the (dual) canonical basis of a quantum group. In this talk I will describe a proof of this “dual canonical basis conjecture" in the special case of acyclic, skew-symmetric quantum cluster algebras by relating the categorifications on either side of this picture.
Tuesday March 3
Speaker: Arunima Ray (Brandeis)
Title: Shake slice and shake concordant knots.
Abstract: Consider the set of knots in the 3-sphere. There are a number of four-dimensional equivalence relations on knots, which have interesting implications towards the study of 4-manifolds. The most well-studied of these is the relation called concordance, under which knots form an abelian group; knots concordant to the trivial knot are said to be slice. We study a generalization of concordance, called shake concordance; in this realm, the analogue for slice knots are called shake slice knots. Any slice knot is shake slice, but the converse is unknown. We construct infinite families of knots that are pairwise shake concordant but not concordant. We also give a complete characterization of shake concordant and shake slice knots in terms of concordance, which allows us to construct new infinite families of shake slice knots. (This is a joint project with Tim Cochran.)
Tuesday March 10
Speaker: Paul Monsky (Brandeis)
Title: Some Characteristic 2 Hecke algebras. A theme of Nicolas and Serre, with variations
Abstract: For each odd prime p, there is a formal Hecke operator Tp: Z/2[[x]] -> Z/2[[x]]. The mod 2 reductions of modular forms give rise to certain Tp stable subspsaces of Z/2[[x]]. Nicolas and Serre have analyzed the action of the Tp on a space attached to level 1 forms. Their space is spanned by the odd powers of x+x^9+x^25+x^49+... . They show that the Hecke algebra arising is a power series ring in T3 and T5.
I'll sketch an alternative proof and present related results on some spaces attached to level 3 forms. One of the Hecke algebras I get is a power series ring in T7 and T13 with an element of square 0 adjoined. My arguments are not technical, and the talk should be accessible.
Tuesday March 17
Speaker: Anna Medvedovsky (Brandeis)
Title: Nilpotent recursion operators + applications to mod-p Hecke algebras
A recursion operator on a polynomial algebra over a field is a linear
operator T: F[y] -> F[y] so that the images of powers of y satisfy a
linear recurrence over F[y]. Suppose every polynomial f in F[y] is
killed by some power of T, and write N(f) for the least such power. We
will analyze the growth of this nilpotence index N(y^n) in the case
that F has characteristic p, and describe applications to determining
the structure of Hecke algebras acting on modular forms modulo p. The
main result uses elementary methods only, and the talk should be
accessible to a wide audience.
Tuesday March 24
Speaker: Jonathan Novak (MIT)
Title: Random lozenge tilings and Hurwitz numbers
Abstract: This talk will be about random lozenge tilings of a class of planar domains which I like to call ``sawtooth domains.'' The basic question is: what does a random lozenge tiling of a large sawtooth domain look like? At the first order of randomness, a remarkable form of the law of large numbers emerges: the tiling separates into "frozen" and "liquid" phases separated by an "arctic curve." I will discuss the next order of randomness, in which one wants to analyze fluctuations of tiles near a turning point of the arctic curve. Quite remarkably, this analytic problem can be solved in an essentially combinatorial way, using a desymmetrized version of the double Hurwitz numbers from enumerative algebraic geometry.
Tuesday March 31
Speaker: Charles Smart (MIT)
Title: The Abelian sandpile on periodic Euclidean graphs
Abstract: I will discuss joint work with Lionel Levine and Wesley Pegden. The Abelian sandpile is a model for "self-organized criticality" that generates approximations of striking fractal images on periodic graphs. Our work explains the appearance of these images: the scaling limit of the sandpile is a nonlinear partial differential equation with unexpected algebraic structure. I will discuss some older work on the square lattice as well as recent work on more general graphs.
Tuesday April 14
Speaker: Alison Miller (Harvard)
Title: Classical knot invariants, arithmetic invariant theory, and counting simple (4q+1)-knots
Certain knot invariants coming from the Seifert matrix and Alexander polynomial of a knot have interesting number-theoretic structure. I'll explain how a classical construction of knot invariants that are ideal classes of rings fits into the context of Bhargava and Gross's arithmetic invariant theory.
I'll then discuss the following related asymptotic counting question: on the set of knots with squarefree Alexander polynomial of fixed degree and bounded height, our ideal class invariant can take on only finitely many values -- how many? This is a quantitative version of the question "how much additional information is contained in the Seifert matrix that is not captured by the Alexander polynomial?" It also has consequences in higher-dimensional knot theory, where Kearton, Levine, and Trotter identified an important class of knots, the "simple knots", which are entirely classified by higher-dimensional analogues of our invariants.
Here are some information about how to reach Brandeis Mathematic department.
For further information please contact:
Olivier Bernardi (bernardi at brandeis dot edu).