Title: Moduli space of G2-instantons on 7-dimensional product manifolds

Date: 11/01/2018

Time: 11:00 AM - 12:00 PM

Place: C304 Wells Hall

$G_2$-instantons are 7-dimensional analogues of flat
connections in dimension 3. It is part of Donaldson-Thomas’ program to
generalize the fruitful gauge theory in dimensions 2,3,4 to dimensions 6,7,8.
The moduli space of $G_2$-instantons, with virtual dimension 0, is
expected to have interesting geometric structure and yield enumerative
invariant for the underlying 7-dimensional manifold.
In this talk, in some reasonable special cases and a fairly complete manner,
we will describe the relation between the moduli space of $G_2$-instantons
and an algebraic geometry moduli on a Calabi-Yau 3-fold.

Title: Cluster Monomials and Theta Bases via Scattering Diagrams

Date: 11/01/2018

Time: 3:00 PM - 4:00 PM

Place: C117 Wells Hall

In this talk I will add to Nick’s presentation from last time by describing a portion of the scattering diagram using c-vectors and g-vectors. Then I will present some examples of computing cluster monomials using broken lines. If there is time I will compute an element of the theta basis which is not a cluster monomial.

Title: Nonlocal Geometric Variational Problems: Isotropic and Anisotropic Extensions of Gamow's Liquid Drop Problem and Beyond

Date: 11/01/2018

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

The liquid drop (LD) model, an old problem of Gamow for the shape of atomic nuclei, has recently resurfaced within the framework of the modern calculus of variations. The problem takes the form of a nonlocal isoperimetric problem on all 3-space with nonlocal interactions of Coulombic type.
In this talk, we first state and motivate the LD problem, and then summarize the state of the art for global minimizers.
We then address certain recent anisotropic variants of the LD problem in the small mass regime, with a particular focus on the minimality of the Wulff shape.
In the second half of the talk, we address a related nonlocal geometric problem based solely on competing interaction potentials of algebraic type. This problem is directly related to a wide class of self-assembly/aggregation models for interacting particle systems (eg. swarming).
This talk includes joint work with Almut Burchard (Toronto), Robin Neumayer (IAS and Northwestern), and Ihsan Topaloglu (Virginia Commonwealth).

Title: Data-driven computational modeling of 3D structures of genomes

Date: 11/02/2018

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

The three-dimensional (3D) structure of a genome is important for genome folding, genome function, genome methylation, spatial gene regulation, and cell development, but has not been well studied due to lack of experimental techniques for genome structure determination. In this talk, I will present our large-scale computational optimization methods for effectively reconstructing the 3D structure (shape) of the human genome from chromosomal conformation capturing data. The highly scalable algorithms were able to build the highest-resolution structures of human chromosomes to date and one of the first structures of the entire human genome. The computational modeling enables the visualization of the previously unknown 3D shape of the genome consisting of millions of units and opens a new avenue to conduct genomics research in the 3D perspective.

Title: Around Waring problem for homogeneous polynomials

Date: 11/07/2018

Time: 2:00 PM - 3:00 PM

Place: C304 Wells Hall

Waring problem for homogeneous polynomials (forms) asks to represent a given form of degree k*d as a sum of d-th powers of forms of degree k. The main objective is to find a presentation with a small number of summands. The classical case going back to J.J. Sylvester deals with k=1 and binary forms. We will survey some of the results in this area and pose some elementary looking open problems. No preliminary knowledge of the topic is required

Speaker: Guozhen Lu, University of Connecticut; Nobody Else

Title: Fourier analysis on hyperbolic spaces and sharp higher order Hardy-Sobolev-Maz'ya inequalities

Date: 11/07/2018

Time: 4:10 PM - 5:00 PM

Place: C517 Wells Hall

In this talk, we will describe some recent works on
the sharp higher order Hardy-Sobolev-Maz'ya and Hardy-Adams inequalities on hyperbolic balls and half spaces. The relationship between the classical Sobolev inequalities and the Hardy-Sobolev-Maz'ya inequalities for higher order derivatives will be established. Our main approach is to use the Fourier analysis on hyperbolic spaces and Green's function estimates.

Title: Topological Hochschild Homology and Higher Characters

Date: 11/08/2018

Time: 2:00 PM - 3:00 PM

Place: C304 Wells Hall

In this talk I'll explain how Hochschild homology and duality theory in bicategories can be used to obtain interesting Euler characteristic-type invariants in a number of mathematical contexts (all of the terms in the previous sentence will be explained). A topological refinement, using THH, of this reasoning very easily yields interesting fixed point invariants, such as the Lefschetz trace and Reidemeister trace. Using this, one can show that the cyclotomic trace from algebraic K-theory is computing fixed point invariants. Time permitting, I'll explain how zeta functions relate to the above. Prerequisites: an appetite for category theory, and a belief in, but not knowledge of, the stable homotopy category.

Title: Singularity Formation in General Relativity

Date: 11/08/2018

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

The celebrated Hawking–Penrose theorems are breakdown results for solutions to the Einstein equations of general relativity, which are a system of highly nonlinear wave-like PDEs. These theorems show that, under appropriate assumptions on the matter model, a large, open set of initial data lead to geodesically incomplete solutions. However, these theorems are “soft” in that they do not yield any information about the nature of the incompleteness, leaving open the possibilities that i) it is tied to the blowup of some invariant quantity (such as curvature) or ii) it is due to a more sinister phenomenon, such as incompleteness stemming from lack of information for how to uniquely continue the solution (this is roughly known as the formation of a Cauchy horizon). In various works, some joint with I. Rodnianski, we have obtained the first results in more than one spatial dimension that eliminate the ambiguity for an open set of initial data: for the solutions that we studied, the incompleteness is tied to the blowup of various spacetime curvature scalars along a spacelike hypersurface. Physically, this phenomenon corresponds to the stability of the Big Bang and/or Big Crunch singularities. From an analytic perspective, the main theorems are stable blowup results for quasilinear systems of elliptic-hyperbolic PDEs. In this talk, I will provide an overview of these results and explain how they are tied to some of the main themes of investigation by the mathematical general relativity community. I will also discuss the role of geometric and gauge considerations in the proofs, as well as intriguing connections to other problems concerning stable singularity formation.

Title: Optimal transport or: How I learned to stop worrying and grew my own shipping empire

Date: 11/09/2018

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

Optimal transport, a.k.a. the Monge-Kantorovich problem has been an active area of mathematics recently. It is an optimization problem, but has connections to PDEs, geometry, economics, image processing, kinetics, and probability, among other areas. I will start with the discrete optimal transport problem and explain some results about existence, duality, and computation. If time permits, I will discuss the continuous framework and some other variations of the problem.

Speaker: Wenrui Hao, Pennsylvania State University

Title: Computational modeling for cardiovascular risk evaluation

Date: 11/09/2018

Time: 4:10 PM - 5:00 PM

Place: 1502 Engineering Building

Atherosclerosis, the leading cause of death in the United State, is a disease in which a plaque builds up inside the arteries. The LDL and HDL concentrations in the blood are commonly used to predict the risk factor for plaque growth. In this talk, I will describe a recent mathematical model that predicts the plaque formation by using the combined levels of (LDL, HDL) in the blood. The model is given by a system of partial differential equations within the plaque with a free boundary. This model is used to explore some drugs of regression of a plaque in mice, and suggest that such drugs as used for mice may also slow plaque growth in humans. Some mathematical questions, inspired by this model, will also be discussed. I will also mention briefly some related projects about abdominal aortic aneurysm (AAA) and red blood cell aggregation, which would have some potential blood biomarkers for diagnosis of AAA.

I'll define universal central extensions of groups, and prove a criterion for a central extension to be universal. Time permitting, I'll connect this to the 2nd algebraic K-group of a ring, and a certain group cohomology.

I will give a brief description of main ingredients of abstract topological recursion, which is a method for constructing correlation functions in a number of models admitting "genus", or 1/N filtration. I will show how the condition of total symmetricity of the correlation functions naturally leads to algebras of second-order differential operators annihilating the partition function of a model. I will also discuss the transition between global and local models on a spectral curve.

Title: W-algebra constraints and higher Airy structures

Date: 11/13/2018

Time: 3:00 PM - 4:00 PM

Place: C304 Wells Hall

Virasoro constraints are omnipresent in enumerative geometry. Recently, Kontsevich and Soibelman introduced a generalization of Virasoro constraints in the form of Airy structures. It can also be understood as an abstract framework underlying the topological recursion of Chekhov, Eynard and Orantin. In this talk I will explain how the triumvirate of Virasoro constraints, Airy structures and topological recursion can be generalized to W-algebra constraints, higher Airy structures and higher topological recursion. The resulting formalism includes as a special case the W-algebra constraints satisfied by generating functions for intersection numbers on the moduli space of r-spin curves, but is much more general.
This is joint work with Gaetan Borot, Nitin Chidambaram and Dmitry Noschenko.

Title: Dimension estimates for the set of points with non-dense orbit in homogeneous spaces

Date: 11/13/2018

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

Abstract: In this talk we study the set of points in a homogeneous space whose orbit escapes the complement of a fixed compact subset. We find an upper bound for the Hausdorff dimension of this set. This extends the work of Kadyrov, where he found an upper bound for the Hausdorff dimension of the set of points whose orbit misses a fixed ball of sufficiently small radius in a compact homogeneous space. We can also use our main result to produce new applications to Diophantine approximation. This is joint work with Dmitry Kleinbock.

Title: On Bergman type spaces of functions of nonstandard growth and some related questions.

Date: 11/14/2018

Time: 4:10 PM - 5:00 PM

Place: C517 Wells Hall

Abstracts: We study various Banach spaces of holomorphic functions on the unit disc and half plane. As a main question we investigate the boundedness of the corresponding holomorphic projection. We exploit the idea of V.P.Zaharyuta, V.I.Yudovich (1962) where the boundedness of the Bergman projection in Lebesgue spaces was proved using Calderon-Zygmund operators. We treat the cases of variable exponent Lebesgue space, Orlicz space, Grand Lebesgue space and variable exponent generalized Morrey space. The major idea is to show that the approach can be applied to a wide range of function spaces. This opens a door in a sense for introducing and studying new function spaces of Bergman type in complex analysis. We also study the rate of growth of functions near the boundary in spaces under consideration and their approximation by mollifying dilations.

The classical Künneth formula provides a relationship between the homology of a product space and that of its factors. In this talk, I will briefly review persistent homology and show Künneth-type theorems for it. That is, for two different notions of products, we show how the persistent homology of a filtered product space relates to that of the factor filtered spaces.

Title: The stabilization distance and knot Floer homology

Date: 11/15/2018

Time: 2:00 PM - 3:00 PM

Place: C304 Wells Hall

Given a knot K in S^3, we consider the set of oriented surfaces in B^4 which bound K. A natural question is how many stabilizations and destabilizations one must perform to move from one surface to another. Similarly, one may wonder how many double point birth/deaths must occur in a regular homotopy. In this talk, we consider several metrics on the set of surfaces bounding K, based on the number of stabilizations which must occur in a stabilization sequence connecting the two surfaces, or in the minimal number of double points which appear in a generic regular homotopy. We will describe how the link Floer TQFT can be used to construct lower bounds. This is joint work with Andras Juhasz.

A 2-SLE$_\kappa$, $\kappa\in(0,8)$, is a pair of random curves $(\eta_1,\eta_2)$ in a simply connected domain $D$ connecting two pairs of boundary points such that conditioning on any curve, the other is a chordal SLE$_\kappa$ curve in a complement domain. We prove that, for the exponent $\alpha=\frac{(12-\kappa)(\kappa+4)}{8\kappa}$, for any $z_0\in D$, the limit $\lim_{r\to 0^+}r^{-\alpha}\mathbb{P}[\mbox{dist}(\eta_j,z_0)<r,j=1,2]$ converges to a positive number, called the two-curve Green’s function. To prove the convergence, we transform the original problem into the study of a two-dimensional diffusion process, and use orthogonal polynomials to derive its transition density and invariant density.

Title: More on Scattering Diagram and Theta Functions

Date: 11/15/2018

Time: 3:00 PM - 4:00 PM

Place: C117 Wells Hall

Abstract: I will continue the discussion on scattering diagram and theta functions and relate them to the classical cluster theories. I will sketch Gross-Hacking-Keel-Kontsevich’s proofs of positive Laurent phenomenon, sign coherence, and a weak version of the cluster duality conjecture.

Speaker: Jun Song, University of Illinois at Urbana-Champaign

Title: Spectral and Statistical Analyses of Nucleosome Positioning: New Answers to Old Questions

Date: 11/16/2018

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

Nucleosomes form the fundamental building blocks of eukaryotic
chromatin, and previous attempts to understand the principles
governing their genome-wide distribution have spurred much interest
and debate in biology. In particular, the precise role of DNA sequence
in shaping local chromatin structure has been controversial.
In this talk, I will described categorical spectral analysis methods
and statistical physics approaches for rigorously quantifying the
contribution of hitherto-debated sequence features to three
distinct aspects of genome-wide nucleosome landscape: occupancy,
translational positioning, and rotational positioning.

Title: The Brauer Class Associated to the Binary Cubic Generic Clifford Algebra

Date: 11/19/2018

Time: 1:00 PM - 2:00 PM

Place: C517 Wells Hall

The Clifford algebra associated to a quadratic form is a classical algebraic object with many applications. Its definition can be generalized to forms of higher degree. In the case of a binary cubic form, the center of the associated Clifford algebra is isomorphic to the affine coordinate ring of an elliptic curve. Furthermore, it is an Azumaya algebra over its center and thus it defines an element in the Brauer group. We study this phenomenon in families and show that the universal class (the class of the binary cubic generic Clifford algebra) is never trivial. This is joint work with Rajesh Kulkarni.

After a brief introduction of Seiberg-Witten equations on closed smooth four manifolds, we will see how moduli space of solutions leads to an oriented compact manifold and a topological invariant (Seiberg-Witten Invariant) for the four manifold. Then for the purpose of computation of this invariant on Kähler manifolds, we will rewrite the equation in terms of complex geometry and see for most of the Kähler Surfaces the answer will be in terms of algebraic geometric criterion of the surface. Most of the technical details will be omitted but some brief sketches will be there. I will follow John Morgan's Book on Seiberg Witten equations.

Title: Active Learning 2.0: Being Intentionally Inclusive

Date: 11/27/2018

Time: 1:30 PM - 3:00 PM

Place: 252 EH

Active learning has many documented benefits both for students and instructors. Moreover, there is increasing evidence that it disproportionately benefits women, students of color, and students who were previously denied the same learning opportunities as others. However, the empirical evidence for this disproportionate benefit doesn't explain why it happens, nor does it guarantee that all students will benefit from active learning. In fact, my own experience with active learning is that it is difficult to do well and sometimes it can have detrimental effects on students if we're not careful. So, we should aim not just for active learning, but learning that is both active and inclusive. We'll discuss some principles and practical strategies for making active learning more inclusive.

Title: Improving L^1 Poincar\'e inequality on Hamming cube

Date: 11/28/2018

Time: 4:10 PM - 5:00 PM

Place: C517 Wells Hall

L^1 Poincar\'e inequality on hypercube is related to many interesting questions in random graph theory (like Margoulis graph connectivity theorem, e.g.). The sharp constant is unknown, but I will show how to improve the previously known constant \pi/2 obtained by Ben Efraim and Lust--Piquard by using non-commutative harmonic analysis. The approach will be probabilistic and luckily commutative. For Gaussian space the constant is known, it is \sqrt{\pi/2}, and the short proof belong to Maurey--Pisier.

Reproducing kernel Hilbert spaces (RKHSs) are Hilbert spaces of functions on which point evaluation functionals are continuous. Thanks to the existence of an inner product, RKHSs are well-understood in functional analysis. Successful and important machine learning methods based on RKHSs include support vector machines, regularization networks and kernel-based approximation.
In the past decade, there has been emerging interest in constructing reproducing kernel Banach spaces (RKBSs) for applied and theoretical purposes for instance sparse approximation. Recently, we propose a generic definition of RKBS and a framework of constructing RKBSs that unifies existing constructions in the literature, and leads further to new RKBSs. As a by-product, the space C([0,1]) of all continuous functions on the interval [0,1] is an RKBS.
Motivated by sparse multi-task learning, we constructed a class of vector-valued RKBSs with the l1 norm based on multi-task admissible kernels. The relaxed linear representer theorem holds for regularization networks in the obtained spaces if and only if the Lebesgue constant of kernels is uniformly bounded. A class of translation-invariant kernels of limited smoothness admissible for construction are given. Numerical experiments demonstrate the advantages of the proposed construction and regularization models.
This talk is based on two joint papers with Prof. Guohui Song (Clarkson University), Haizhang Zhang (Sun Yat-sen University), and Jun Zhang (University of Michigan).