Everytopic seminar
Everytopic Seminar  Fall 2014
Friday, 2:00pm3:20pm
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.

Friday September 19
Speaker: Ilya Vinogradov (University of Bristol)
Title: Effective Ratner Theorem for ASL(2, R) and the gaps of the sequence
\sqrt n modulo 1
Abstract: Let G=SL(2,\R)\ltimes R^2 and Gamma=SL(2,Z)\ltimes Z^2. Building on
recent work of Strombergsson we prove a rate of equidistribution for the
orbits of a certain 1dimensional unipotent flow of Gamma\G, which projects to
a closed horocycle in the unit tangent bundle to the modular surface. We use
this to answer a question of Elkies and McMullen by making effective the
convergence of the gap distribution of sqrt n mod 1.

Friday October 3
Speaker: Andreas Arvanitoyeorgos (University of Patras and Tufts University)
Title: Aspects of homogeneous geometry with application to invariant Einstein metrics
Abstract: According to F. Klein's Erlanger program geometry is the study of those properties of a space M, which are invariant under the action of a group G. When M is a smooth manifold and G is a Lie group which acts transitively on M, then M can be identified with the quotient space G/H, where H is the isotropy subgroup of a fixed point p of M. In this case, and under some assumptions, the geometry of the space M (e.g. curvature or geodesics) reduces to the study of the pair (g, h), where g, h are the Lie algebras of G and H respectively.
In the first part of the talk I will give some details on the theory of homogeneous spaces and Lie groups and then I will discuss some recent results on homogeneous Einstein metrics on two important class of homogeneous spaces, namely flag manifolds and Stiefel manifolds.

Friday October 10
Speaker: Stephen Hermes (Wellesley College)
Title: Maximal Green Sequences and SemiInvariant Pictures
Abstract: Quiver mutation is a combinatorial process with several striking connections to diverse areas of mathematics including (among others) representation theory, mathematical physics, and geometry. Maximal green sequences are particular sequences of quiver mutations originally introduced by Keller in order to give a combinatorial description of the (refined) DTinvariants of 3CalabiYau categories. Little is known about the enumeration of maximal green sequences in general. In particular, it is unknown if any quiver admits only finitely many maximal green sequences.
This talk is a report on work in progress with Brüstle, Igusa, and Todorov to interpret maximal green sequences in terms of the invariant theory of quiver varieties using ``semiinvariant pictures.'' Using the geometry of these pictures we give an alternative proof of a theorem of BrüstleDupontPérotin stating that socalled ``tame'' quivers admit only finitely many maximal green sequences.

Friday October 24
Speaker: Evgeniy Zorin (University of York)
Title: Mahler numbers and Diophantine approximation
Abstract: Mahler numbers originate in works of Kurt Mahler on transcendence. Their Diophantine properties have a surprising link with theoretical computer science. We will discuss the state of the art as well as the recent advancements in this area.

Friday October 31
Speaker: Jason Miller (MIT)
Title: Random Surfaces and Quantum Loewner Evolution
Abstract: What is the canonical way to choose a random, discrete, twodimensional manifold which is homeomorphic to the sphere? One procedure for doing so is to choose uniformly among the set of surfaces which can be generated by gluing together $n$ Euclidean squares along their boundary segments. This is an example of what is called a random planar map and is a model of what is known as pure discrete quantum gravity. Another canonical way to choose a random, twodimensional manifold is what is known as Liouville quantum gravity (LQG), which corresponds to the random surface whose metric when parameterized in isothermal coordinates is formally described by $e^{gamma h}(dx^2+dy^2)$ where $h$ is an instance of the Gaussian free field.
In this talk, I will describe various relationships between these discrete and continuum models for random surfaces (some conjectural, some now proved) as well as a related and new universal family of growth processes called quantum Loewner evolution (QLE). I will also explain how QLE is related to DLA, the dielectric breakdown model, and the SchrammLoewner evolution.
Based on joint works with Scott Sheffield.

Friday November 7
Speaker: Aaron Pixton (Harvard)
Title: The cohomology ring of the moduli space of curves
Abstract: Let $M_g$ be the moduli space of connected smooth genus g curves, a fundamental object in both algebraic geometry and topology. I will discuss what is known about its cohomology ring $H^*(M_g)$, focusing on a subring $RH^*(M_g)$ (known as the tautological ring) defined by Mumford in analogy with the cohomology of Grassmannians. There are two competing conjectural descriptions of the structure of this subring. I will also describe the analogous situation when $M_g$ is replaced by its compactification, the moduli space of DeligneMumford stable curves.

Friday November 14
Speaker: Jingyue Chen (Brandeis)
Title: Existence and rigidity of CalabiYau bundles
Abstract: Lian and Yau developed a global Poincare residue formula to study period integrals of families of complex manifolds, and used it to construct canonical PDE systems for period integrals. The notion of a CY bundle is a crucial ingredient in their construction. The idea is to lift line bundles from a manifold X to some larger space (a CY bundle) over X. A question was therefore raised as to whether there exists such a bundle that supports the lifting of all line bundles from X simultaneously. We give partial solutions to this problem. We also discuss the classification of CY bundles. This is a joint work with Bong H. Lian.

Friday November 21
Speaker: Vidya Venkateswaran (MIT)
Title: Signature characters of certain representations of the quantum group $U_{q}(gl_{n})$.
Abstract: The study of unitary representations is an important and active area in representation theory. We will discuss a related object, that of signature characters. We will see what information these invariants encode, as well as how they relate to unitarity. As an explicit example, we will consider the quantum group of type $A$, for $q=1$. We will discuss a technique for computing signature characters of highest weight representations, which relies on the combinatorics of the GelfandTsetlin bases. As an application, we recover some known results in the classical limit $q=1$ that were obtained by different means. We also obtain some information about unitarity for arbitrary $q$. We will also briefly discuss other situations in which one may compute signature characters by harnessing combinatorial structures of representations. We will define all the necessary terms, and provide enough background to be accessible to a general audience.

Friday December 5
Speaker: Lei Yang (Yale University)
Title: Equidistribution of expanding translates of curves in homogeneous spaces and its application to Diophantine approximation.
Abstract: We consider an analytic curve $\varphi: I \rightarrow \mathbb{M}(n\times m, \mathbb{R}) \hookrightarrow \mathrm{SL}(n+m, \mathbb{R})$, embed it into some homogeneous space $G/\Gamma$, and translate it via some diagonal flow
$\{a(t): t > 0 \} < \mathrm{SL}(n+m,\mathbb{R})$. Under some geometric conditions on $\varphi$, we prove the equidistribution
of the evolution of the translated curves $a(t)\varphi(I)$ in $G/\Gamma$, and as a result, we prove that for almost all points on the curve, the Dirichlet's theorem can not be improved. This is a joint work with Nimish Shah.
Map of Brandeis University  the Department of Mathematics (Goldsmith building) is U24 on this map.
For further information please contact:
Olivier Bernardi (bernardi at brandeis dot edu).
Previous semesters:
Spring 2014,
Fall 2013,
Spring 2013,
Fall 2012,
Spring 2012,
Fall 2011,
Spring 2011,
Fall 2010,
Spring 2010,
Fall 2009,
Spring 2009.