Seminars and Colloquia by Series

Elliptic integrands in geometric variational problems

Series
Job Candidate Talk
Time
Thursday, January 9, 2020 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Antonio De Rosa – NYU –

Elliptic integrands are used to model anisotropic energies in variational problems. These energies are employed in a variety of applications, such as crystal structures, capillarity problems and gravitational fields, to account for preferred inhomogeneous and directionally dependent configurations. After a brief introduction to variational problems involving elliptic integrands, I will present an overview of the techniques I have developed to prove existence, regularity and uniqueness properties of the critical points of anisotropic energies. In particular, I will present the anisotropic extension of Allard's rectifiability theorem and its applications to the Plateau problem. Furthermore, I will describe the anisotropic counterpart of Alexandrov's characterization of volume-constrained critical points. Finally, I will mention some of my ongoing and future research projects.

Lorentzian polynomials

Series
School of Mathematics Colloquium
Time
Thursday, January 9, 2020 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
June Huh – Princeton University –

Lorentzian polynomials link continuous convex analysis and discrete convex analysis via tropical geometry. The tropical connection is used to produce Lorentzian polynomials from discrete convex functions. Although no specific background beyond linear algebra and multivariable calculus will be needed to enjoy the presentation, I advertise the talk to people with interests in at least one of the following topics: graphs, convex bodies, stable polynomials, projective varieties, Potts model partition functions, tropicalizations, Schur polynomials, highest weight representations. Based on joint works with Petter Brändén, Christopher Eur, Jacob Matherne, Karola Mészáros, and Avery St. Dizier.

Analysis and computation of nonlocal models

Series
Job Candidate Talk
Time
Tuesday, January 7, 2020 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Xiaochuan Tian – University of Texas at Austin –
Nonlocal models are experiencing a firm upswing recently as more realistic alternatives to the conventional local models for studying various phenomena from physics and biology to materials and social sciences. In this talk, I will describe our recent effort in taming the computational challenges for nonlocal models. I will first highlight a family of numerical schemes -- the asymptotically compatible schemes -- for nonlocal models that are robust with the modeling parameter approaching an asymptotic limit. Second, I will discuss nonlocal-to-local coupling techniques so as to improve the computational efficiency of using nonlocal models. This also motivates the development of new mathematical results -- for instance, a new trace theorem that extends the classical results. 
 
Although new nonlocal models have been gaining popularity in various applications, they often appear as phenomenological models, such as the peridynamics model in fracture mechanics. Here I will illustrate how to characterize the origin of nonlocality through homogenization of wave propagation in periodic media. 

Localization in Khovanov homology

Series
Geometry Topology Seminar
Time
Monday, January 6, 2020 - 14:00 for 1 hour (actually 50 minutes)
Location
Skile 006
Speaker
Melissa Zhang – UGA

When a topological object admits a group action, we expect that our invariants reflect this symmetry in their structure. This talk will explore how link symmetries are reflected in three generations of related invariants: the Jones polynomial; its categorification, Khovanov homology; and the youngest invariant in the family, the Khovanov stable homotopy type, introduced by Lipshitz and Sarkar. In joint work with Matthew Stoffregen, we use Lawson-Lipshitz-Sarkar's construction of the Lipshitz-Sarkar Khovanov homotopy type to produce localization theorems and Smith-type inequalities for the Khovanov homology of periodic links.

Random matrix theory and supersymmetry techniques

Series
Job Candidate Talk
Time
Monday, January 6, 2020 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Tatyana Shcherbyna – Princeton University –
Starting from the works of Erdos, Yau, Schlein with coauthors, significant progress in understanding universal behavior of many random graph and random matrix models were achieved. However for random matrices with a spatial structure, our understanding is still very limited.  In this talk I am going to overview applications of another approach to the study of the local eigenvalue statistics in random matrix theory based on so-called supersymmetry techniques (SUSY). The SUSY approach is based on the representation of the determinant as an integral over the Grassmann (anticommuting) variables. Combining this representation with the representation of an inverse determinant as an integral over the Gaussian complex field, SUSY allows to obtain an integral representation for the main spectral characteristics of random matrices such as limiting density, correlation functions, the resolvent's elements, etc. This method is widely (and successfully) used in the physics literature and is potentially very powerful but the rigorous control of the integral representations, which can be obtained by this method, is quite difficult, and it requires powerful analytic and statistical mechanics tools. In this talk we will discuss some recent progress in application of SUSY  to the analysis of local spectral characteristics of the prominent ensemble of random band matrices, i.e. random matrices whose entries become negligible if their distance from the main diagonal exceeds a certain parameter called the band width. 
 

Involutive Heegaard Floer homology

Series
Geometry Topology Student Seminar
Time
Wednesday, December 11, 2019 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Sally Collins – Georgia Tech

Introduced by Hendricks and Manolescu in 2015, Involutive Heegaard Floer homology is a variation of the 3-manifold invariant Heegaard Floer homology which makes use of the conjugation symmetry of the Heegaard Floer complexes. This theory can be used to obtain two new invariants of homology cobordism. This talk will involve a brief overview of general Heegaard Floer homology, followed by a discussion of the involutive theory and some computations of the homology cobordism invariants. 

Classifying contact structures on hyperbolic 3-manifolds

Series
Geometry Topology Seminar
Time
Monday, December 9, 2019 - 14:30 for 1 hour (actually 50 minutes)
Location
Skiles 202
Speaker
James Conway – UC, Berkeley

Please Note: Note time and place of seminar

Two of the most basic questions in contact topology are which manifolds admit tight contact structures, and on those that do, can we classify such structures. In dimension 3, these questions have been answered for large classes of manifolds, but with a notable absence of hyperbolic manifolds. In this talk, we will see a new classification of contact structures on an family of hyperbolic 3-manifolds arising from Dehn surgery on the figure-eight knot, and see how it suggests some structural results about tight contact structures. This is joint work with Hyunki Min.

Ordered groups and n-dimensional dynamics

Series
School of Mathematics Colloquium
Time
Friday, December 6, 2019 - 16:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Dale Rolfsen – UBC

A group is said to be torsion-free if it has no elements of finite order.  An example is the group, under composition, of self-homeomorphisms (continuous maps with continuous inverses) of the interval I = [0, 1] fixed on the boundary {0, 1}.  In fact this group has the stronger property of being left-orderable, meaning that the elements of the group can be ordered in a way that is nvariant under left-multiplication.  If one restricts to piecewise-linear (PL) homeomorphisms, there exists a two-sided (bi-)ordering, an even stronger property of groups.

I will discuss joint work with Danny Calegari concerning groups of homeomorphisms of the cube [0, 1]^n fixed on the boundary.  In the PL category, this group is left-orderable, but not bi-orderable, for all n>1.  Also I will report on recent work of James Hyde showing that left-orderability fails for n>1 in the topological category.  

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