Among the classical geometric evolution equations, YM flow is the least nonlinear and best behaved. Nevertheless, curvature concentration is a subtle problem when the base manifold has dimension four. I'll discuss my proof that finite-time singularities do not occur, and briefly describe the infinite-time picture.

Title: Cluster Donaldson-Thomas Transformation of Grassmannian

Date: 09/06/2018

Time: 3:00 PM - 4:00 PM

Place: C117 Wells Hall

Abstract: On the one hand, there is a 3d Calabi Yau category with stability conditions associated to a quiver without loops or 2-cycles with generic potential, and one can study its Donaldson-Thomas invariants. On the other hand, such a quiver also defines a cluster Poisson variety, which is constructed by gluing a collection of algebraic tori in a certain way governed by combinatorics. In certain cases, the Donaldson-Thomas invariants of the former category can be captured by an automorphism on the latter space. In this talk, I will recall the cluster Poisson structure on the moduli space of configurations of points in a projective space, and state my result on constructing the corresponding cluster Donaldson-Thomas transformation, and give a new proof of Zamolodchikov’s periodicity conjecture in the $A_m\boxtimes A_n$ cases as an application. If time permits, I will also talk about the generalization of this result to double Bruhat cells.

Title: Gromov-Monge Quasimetrics and Distance Distributions

Date: 09/07/2018

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

In applications in computer graphics and computational anatomy, one seeks measure-preserving maps between shapes which preserve geometry as much as possible. Inspired by this, we define a distance between arbitrary compact metric measure spaces by blending the Monge formulation of optimal transport with the Gromov-Hausdorff construction. We show that the resulting distance is an extended quasi-metric on the space of compact mm-spaces, which has convenient lower bounds defined in terms of distance distributions. We provide rigorous results on the effectiveness of these lower bounds when restricted to simple classes of mm-spaces such as metric graphs or plane curves.This is joint work with Facundo Mémoli.

I will give the basic definitions of Hochschild cohomology, and discuss interpretations of the first few cohomology groups H^0, H^1, and H^2. Time permitting, I will discuss how the groups H^2 and H^3 are related to formal deformations of algebras and quantization of Poisson brackets.

Among the classical geometric evolution equations, YM flow is the least nonlinear and best behaved. Nevertheless, curvature concentration is a subtle problem when the base manifold has dimension four. I'll discuss my proof that finite-time singularities do not occur, and briefly describe the infinite-time picture.
This talk will be more analytic and contains <50% overlap with my talk last Thursday.

Title: On Tame Subgroups of Finitely Presented Groups

Date: 09/13/2018

Time: 2:00 PM - 2:50 PM

Place: C304 Wells Hall

We describe several examples of tame subgroups of finitely presented groups and prove that the fundamental groups of certain finite graphs of groups are locally tame.

Abstract: Compatible subsets of edges in a maximal Dyck path were introduced by Lee, Li, and Zelevinsky as a tool for constructing nice bases for rank two cluster algebras. In this talk, I will present a generalization of this combinatorics and give two applications. The first application is a combinatorial construction of non-commutative rank two generalized cluster variables which proves a conjecture of Kontsevich. The second application gives a combinatorial description of the cells in an affine paving of rank two quiver Grassmannians, this part is joint work with Thorsten Weist.

Speaker: Dongwook Lee, University of California, Santa Cruz

Title: New Polynomial-free, Variable High-order Methods using Gaussian Process Modeling for CFD

Date: 09/14/2018

Time: 4:10 PM - 5:00 PM

Place: 1502 Engineering Building

In this talk, an entirely new class of high-order numerical algorithms for computational fluid dynamics is introduced. The new method is based on the Gaussian Processes (GP) modeling that generalizes the Gaussian probability distribution. The new approach is to adopt the idea of the GP prediction technique which utilizes the covariance kernel functions and use it to interpolate and/or reconstruct high-order approximations for computational fluid dynamics simulations. The new GP high-order method is proposed as a new numerical high-order formulation in finite difference and finite volume frameworks, alternative to the conventional polynomial-based approaches.

The notion of an absolute value function can be generalized to arbitrary fields; one example is the p-adic absolute value on the rational. Ostrowski's theorem classifies all absolute values on the rationals. An absolute value induces a metric in a natural way, so we can "complete" a field with respect to a given absolute value. We'll also discuss the close relationship between absolute values and discrete valuation rings.

Title: Where do Seiberg-Witten equations come from?

Date: 09/17/2018

Time: 4:10 PM - 5:30 PM

Place: C304 Wells Hall

The question in title is akin to asking where the equation of motion of a free falling object a + bt + 1/2 gt^2 in 3-space come from? then discovering that the "objects fall with constant acceleration" rule. Similarly, we derive Seiberg-Witten equations (which also have a linear part and a quadratic part) from the deformation equations of an "isotropic associative submanifold" of a complex G_2 Manifold. For this, we will define the notion of complex G_2 manifold and notion of complexification of a G_2 manifold (this is a joint work with Ustun Yildirim).

Title: Elliptic Regularity of J-Holomorphic Curves

Date: 09/19/2018

Time: 4:00 PM - 4:50 PM

Place: A202 Wells Hall

One of the fundamental estimates for the L^p theory of elliptic operators is Calderon-Zygmund inequality. I’ll follow Mcduff & Salamon’s book for the proof of regularity theorem, raising the order of nonlinear Cauchy Riemann equation and making use of mean value property.

Speaker: Richard Kollar, Comenius University, Bratislava, Slovakia

Title: Krein signature - three unexpected lessons

Date: 09/19/2018

Time: 4:10 PM - 5:00 PM

Place: C517 Wells Hall

Krein signature is an algebraic quantity characterizing purely imaginary eigenvalues of linearized Hamiltonian systems. Instabilities growing from a stable state in these systems are caused by Hamiltonian-Hopf bifurcations, i.e. events when two purely imaginary eigenvalues collide and split off the imaginary axis. The necessary condition for such an event is that the colliding eigenvalues must have mixed signature. In the talk we present three elegant results related to Krein signature - graphical Krein signature and its use to simplify proofs, a connection to stability in general extended systems, and ability to characterize the nature of the eigenvalue collisions directly from the reduced dispersion relation.

In 2014 it was conjectured that the equality of the cluster algebra and upper cluster algebra is equivalent to the existence of a maximal green sequence. In this talk we will discuss a stronger result for cluster algebras from mutation-finite quivers with an emphasis on surface cluster algebras. Specifically we show that for all quivers from surface cluster algebras there exists a maximal green sequence if and only if the cluster algebra is equal to the cluster algebra if and only if the cluster algebra is locally-acyclic. We will also provide a counterexample to show that the result does not hold in general.

Speaker: Brendon Rhoades, University of California, San Diego

Title: The combinatorics, algebra, and geometry of ordered set partitions

Date: 09/20/2018

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

An {\em ordered set partition} of size $n$ is a set partition of $\{1, 2, \dots, n \}$ with a specified order on its blocks. When the number of blocks equals the number of letters $n$, an ordered set partition is just a permutation in the symmetric group $S_n$. We will discuss some combinatorial, algebraic, and geometric aspects of permutations (due to MacMahon, Carlitz, Chevalley, Steinberg, Artin, Lusztig-Stanley, Ehresmann, Borel, and Lascoux-Sch\"utzenberger). We will then describe how these results generalize to ordered set partitions and discuss a connection with the Haglund-Remmel-Wilson {\em Delta Conjecture} in the field of Macdonald polynomials. Joint with Jim Haglund, Brendan Pawlowski, and Mark Shimozono.

The SPORT program at the National Security Agency (NSA) offers graduate students the opportunity to apply their academic knowledge in a stimulating professional environment. As a SPORT intern you will work closely with full time Operations Research analysts applying academic and technical skills to challenging, real-world problems. Internships are paid and are 12 weeks in duration (May-August). Applications accepted September 1st through October 31st

Title: Noncommutative Geometry and Character Varieties

Date: 09/26/2018

Time: 4:00 PM - 5:00 PM

Place: A202 Wells Hall

Roughly speaking, noncommutative geometry studies noncommutative rings and algebras from a "geometric" perspective. I will discuss some philosophies and approaches to the subject, which leads to the study of character varieties, which I will define and discuss.

Title: Hitchin representations and positive configurations of apartments

Date: 09/27/2018

Time: 2:00 PM - 3:00 PM

Place: C304 Wells Hall

Hitchin singled out a preferred component in the character variety of representations from the fundamental group of a surface to PSL(d,R). When d=2, this Hitchin component coincides with the Teichm\"uller space consisting of all hyperbolic metrics on the surface. Later Labourie showed that Hitchin representations share many important differential geometric and dynamical properties.
Parreau extended previous work of Thurston and Morgan-Shalen to a compactification of the Hitchin component whose boundary points are described by actions of the fundamental group of the surface on a building.
In this talk, we offer a new point of view for the Parreau compactification, which is based on certain positivity properties discovered by Fock and Goncharov. Specifically, we use the Fock-Goncharov construction to describe the intersection patterns of apartments in invariant subsets of the building that arises in the boundary of the Hitchin component.

Title: The reflection group of a regular tetrahedron

Date: 09/27/2018

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

I will explain how the group of affine orthogonal transformations generated by the reflections into the four facets of a regular tetrahedron and its symmetries appears as a discrete group of motions of the 9-dimensional hyperbolic space, as the full group of automorphisms of some algebraic surfaces and as a lattice in a projective linear group over the 3-adic numbers.