Sayan Banerjee : Singular Reflected Diffusions
- Probability ( 342 Views )I will talk about some models coming from Physics and Queueing Theory that give rise to singular reflected processes in their diffusion limit. Such diffusions are characterized by non-elliptic generators (which are not even hypoelliptic) in the interior, and ergodicity arises from non-trivial interactions between the diffusion, drift and reflection. I will introduce a regenerative process approach which identifies renewal times in diffusion paths and analyzes excursions between successive renewal times. This provides a detailed description of the stationary distribution even when closed form expressions are unavailable. Based on joint works with Chris Burdzy, Brendan Brown, Mauricio Duarte and Debankur Mukherjee.
Bruce Donald : Some mathematical and computational challenges arising in structural molecular biology
- Applied Math and Analysis ( 280 Views )Computational protein design is a transformative field with exciting prospects for advancing both basic science and translational medical research. New algorithms blend discrete and continuous mathematics to address the challenges of creating designer proteins. I will discuss recent progress in this area and some interesting open problems. I will motivate this talk by discussing how, by using continuous geometric representations within a discrete optimization framework, broadly-neutralizing anti-HIV-1 antibodies were computationally designed that are now being tested in humans - the designed antibodies are currently in eight clinical trials (See https://clinicaltrials.gov/ct2/results?cond=&term=VRC07&cntry=&state=&city=&dist= ), one of which is Phase 2a (NCT03721510). These continuous representations model the flexibility and dynamics of biological macromolecules, which are an important structural determinant of function. However, reconstruction of biomolecular dynamics from experimental observables requires the determination of a conformational probability distribution. These distributions are not fully constrained by the limited information from experiments, making the problem ill-posed in the sense of Hadamard. The ill-posed nature of the problem comes from the fact that it has no unique solution. Multiple or even an infinite number of solutions may exist. To avoid the ill-posed nature, the problem must be regularized by making (hopefully reasonable) assumptions. I will present new ways to both represent and visualize correlated inter-domain protein motions (See Figure). We use Bingham distributions, based on a quaternion fit to circular moments of a physics-based quadratic form. To find the optimal solution for the distribution, we designed an efficient, provable branch-and-bound algorithm that exploits the structure of analytical solutions to the trigonometric moment problem. Hence, continuous conformational PDFs can be determined directly from NMR measurements. The representation works especially well for multi-domain systems with broad conformational distributions. Ultimately, this method has parallels to other branches of applied mathematics that balance discrete and continuous representations, including physical geometric algorithms, robotics, computer vision, and robust optimization. I will advocate for using continuous distributions for protein modeling, and describe future work and open problems.
Yiannis Sakellaridis : Moment map and orbital integrals
- Algebraic Geometry ( 255 Views )In the Langlands program, it is essential to understand spaces of Schwartz measures on quotient stacks like the (twisted) adjoint quotient of a reductive group. The generalization of this problem to spherical varieties calls for an understanding of the double quotient H\G/H, where H is a spherical subgroup of G. This has been studied by Richardson for symmetric spaces. In this talk, I will present a new approach, for spherical varieties "of rank one", based on Friedrich Knop's theory of the moment map and the invariant collective motion.
Jonathan P. Wang : Derived Satake equivalence for Godement-Jacquet monoids
- Number Theory ( 251 Views )Godement-Jacquet use the Schwartz space of n-by-n matrices to construct the standard L-function for GL_n. Ben-Zvi, Sakellaridis and Venkatesh conjecture that the local unramified part of this theory can be categorified to an equivalence between an 'analytic' category of constructible sheaves and a 'spectral' category of dg modules. In this talk I will explain the proof of this equivalence and some of its properties. I will also discuss connections to conjectures of Braverman-Kazhdan on constructions of general automorphic L-functions. This is joint work with Tsao-Hsien Chen (in preparation).
Nelia Charalambous : On the $L^p$ Spectrum of the Hodge Laplacian on Non-Compact Manifolds
- Geometry and Topology ( 237 Views )One of the central questions in Geometric Analysis is the interplay between the curvature of the manifold and the spectrum of an operator. In this talk, we will be considering the Hodge Laplacian on differential forms of any order $k$ in the Banach Space $L^p$. In particular, under sufficient curvature conditions, it will be demonstrated that the $L^p\,$ spectrum is independent of $p$ for $1\!\leq\!p\!\leq\! \infty.$ The underlying space is a $C^{\infty}$-smooth non-compact manifold $M^n$ with a lower bound on its Ricci Curvature and the Weitzenb\"ock Tensor. The further assumption on subexponential growth of the manifold is also necessary. We will see that in the case of Hyperbolic space the $L^p$ spectrum does in fact depend on $p.$ As an application, we will show that the spectrum of the Laplacian on one-forms has no gaps on certain manifolds with a pole and on manifolds that are in a warped product form. This will be done under weaker curvature restrictions than what have been used previously; it will be achieved by finding the $L^1$ spectrum of the Laplacian.
Tony Feng : Steenrod operations and the Artin-Tate pairing
- Number Theory ( 215 Views )In 1966 Artin and Tate constructed a canonical pairing on the Brauer group of a surface over a finite field, and conjectured it to be alternating. This duality has analogous incarnations across arithmetic and topology, namely the Cassels-Tate pairing for a Jacobian variety, and the linking form on a 5-manifold. I will explain a proof of the conjecture, which is based on a surprising connection to Steenrod operations.
Robert Karp : A convex optimization problem in string theory
- String Theory ( 212 Views )For nearly 25 years Calabi-Yau spaces have played a central role in string theory, yet no explicit metric was known. I will outline ideas pioneered by Donaldson and Yau that lead to such metrics numerically; then extend this approach to solving the hermitian Yang-Mills equation, and also obtain metrics on moduli spaces. Knowledge of these quantities is unavoidable for physical predictions.
Dmitry Vagner : Higher Dimensional Algebra in Topology
- Graduate/Faculty Seminar ( 210 Views )In his letter, "Pursuing Stacks," Grothendieck advocated to Quillen for the use of "higher" categories to encode the higher homotopy of spaces. In particular, Grothendieck dreamt of realizing homotopy n-types as n-groupoids. This powerful idea both opened the field of higher dimensional algebra but also informed a paradigm in which the distinction between topology and algebra is blurred. Since then, work by Baez and Dolan among others further surveyed the landscape of higher categories and their relationship to topology. In this talk, we will explore this story, beginning with some definitions and examples of higher categories. We will then proceed to explain "the periodic table of higher categories" and the four central hypotheses of higher category theory. In particular, these give purely algebraic characterizations of homotopy types, manifolds, and generalized knots; and account for the general phenomena of stabilization in topology. No prerequisites beyond basic ideas in algebraic topology will be expected.
Lan-Hsuan Huang : Positive mass theorems and scalar curvature problems
- Presentations ( 205 Views )More than 30 years ago, Schoen-Yau and later Witten made major breakthroughs in proving the positive mass theorem. It has become one of the most important theorems in general relativity and differential geometry. In the first part of the talk, I will introduce the positive mass theorem and present our recent work that extends the classical three-dimensional results to higher dimensions. In the second part, I will discuss how the observation from general relativity enables us to solve classical geometric problems related to the scalar curvature.
Phillip Andreae : Spectral geometry and topology; Euler characteristic and analytic torsion
- Graduate/Faculty Seminar ( 199 Views )What do eigenvalues have to do with geometry and topology? The first part of the talk will provide a few answers to that very broad question, including a discussion of the Euler characteristic from a spectral theory perspective. The second part of the talk will be a brief introduction to my research in analytic torsion, a topological invariant defined in terms of eigenvalues. In particular I'll explain some similarities and differences between analytic torsion and Euler characteristic.
Firas Rassoul-Agha : On the almost-sure invariance principle for random walk in random environment
- Probability ( 198 Views )Consider a crystal formed of two types of atoms placed at the nodes of the integer lattice. The type of each atom is chosen at random, but the crystal is statistically shift-invariant. Consider next an electron hopping from atom to atom. This electron performs a random walk on the integer lattice with randomly chosen transition probabilities (since the configuration seen by the electron is different at each lattice site). This process is highly non-Markovian, due to the interaction between the walk and the environment. We will present a martingale approach to proving the invariance principle (i.e. Gaussian fluctuations from the mean) for (irreversible) Markov chains and show how this can be transferred to a result for the above process (called random walk in random environment). This is joint work with Timo Sepp\"al\"ainen.
Curtis Porter : Straightening out degeneracy in CR geometry: When can it be done?
- Geometry and Topology ( 197 Views )CR geometry studies boundaries of domains in C^n and their generalizations. A central role is played by the Levi form L of a CR manifold M, which measures the failure of the CR bundle to be integrable, so that when L has a nontrivial kernel of constant rank, M is foliated by complex manifolds. If the local transverse structure to this foliation still determines a CR manifold N, then we say M is CR-straightenable, and the Tanaka-Chern-Moser classification of CR hypersurfaces with nondegenerate Levi form can be applied to N. It remains to classify those M for which L is degenerate and no such straightening exists. This was accomplished in dimension 5 by Ebenfelt, Isaev-Zaitzev, and Medori-Spiro. I will discuss their results as well as my recent progress on the problem in dimension 7 (http://arxiv.org/abs/1511.04019).
Ravindra Girivaru : Lefschetz type theorems for algebraic cycles and vector bundles.
- Algebraic Geometry ( 189 Views )The Weak Lefschetz theorem (or the Lefschetz hyperplane theorem) states that for a smooth, projective variety Y and a smooth hyperplane section X in Y, the restriction map of cohomologies H^i(Y) to H^i(X) is an isomorphism for i less than dim{X}, and an injection when i equal to dim{X}. Analogues of this theorem have been conjectured for algebraic cycles. We will talk about some results in this area. We will also talk about such questions for vector bundles.
Aubrey HB : Persistent Homology
- Graduate/Faculty Seminar ( 169 Views )Persistent Homology is an emerging field of Computational Topology that is developing tools to discover the underlying structure in high-dimensional data sets. I will discuss the origins and main concepts involved in Persistent Homology in an accessible way, with illustrations and comprehensive examples. If time allows, I will also describe some current, as well as, future applications of Persistent Homology.
Shrawan Kumar : Topology of Lie groups
- Graduate/Faculty Seminar ( 163 Views )I will give an overview of some of the classical results on the topology of Lie groups, including Hopf's theorem which fully determines the cohomology algebra over the real numbers of any Lie group. We will also discuss how the deRham cohomology of a compact Lie group can be represented by bi-invariant forms. In addition, we will discuss first and the second homotopy groups of Lie groups.
Lorenzo Foscolo : New G2-holonomy cones and exotic nearly Kähler structures on the 6-sphere and the product of two 3-spheres.
- Geometry and Topology ( 162 Views )Compact 6-dimensional nearly Kähler manifolds are the cross-sections of Riemannian cones with holonomy G2. A long-standing problem has been the question of existence of complete nearly Kähler 6-manifolds besides the four known homogeneous ones. We resolve this problem by proving the existence of exotic (inhomogeneous) nearly Kähler structures on the 6-sphere and on the product of two 3-spheres. This is joint work with Mark Haskins, Imperial College London.
Michael Harris : L-functions and the local Langlands correspondence
- Number Theory ( 161 Views )Henniart derived the following theorem from his numerical local Langlands correspondence: If $F$ is a non-archimedean local field and if $\pi$ is an irreducible representation of $GL(n,F)$, then, after a finite series of cyclic base changes, the image of $\pi$ contains a fixed vector under an Iwahori subgroup. This result was indispensable in all demonstrations of the local correspondence. Scholze gave a different proof, based on the analysis of nearby cycles in the cohomology of the Lubin-Tate tower (and this result also appears, in a somewhat different form, in proofs based on the global correspondence for function fields). An analogous theorem should be valid for every reductive group, but the known proofs only work for GL(n). I will sketch a different proof, based on properties of L-functions and assuming the existence of cyclic base change, that also applies to classical groups; I will also explain how the analogous result for a general reductive group is related to the local parametrization of Genestier-Lafforgue.
Remy van Dobben de Bruyn : A variety that cannot be dominated by one that lifts.
- Algebraic Geometry ( 161 Views )Abstract: In the sixties, Serre constructed a smooth projective variety in characteristic p that cannot be lifted to characteristic 0. If a variety does not lift, a natural question is whether some variety related to it does. We construct a smooth projective variety that cannot be rationally dominated by a smooth projective variety that lifts.
Andrew Cooper : Singularities of Lagrangian Mean Curvature Flow
- Geometry and Topology ( 147 Views )In a Calabi-Yau manifold, mean curvature flow--the downward gradient for the area functional--preserves the Lagrangian condition. Thus Lagrangian mean curvature flow suggests a way to find minimal Lagrangian submanifolds of a CY manifold, provided the flow lasts for all time. However, finite-time singularities are expected along the flow; in fact, ill-behaved singularities are generic in some sense. In this talk we will discuss two main results: one, that type I (mild) finite-time singularities can be predicted by looking the cohomology of the initial Lagrangian submanifold, and two, that type II (ill-behaved) singularities can be modeled as unions of special Lagrangian cones. We will also discuss what these results say about using mean curvature flow to understand the topology of Lagrangian submanifolds.
Ziva Myer : Product Structures for Legendrian Submanifolds with Generating Families
- Geometry and Topology ( 146 Views )In contact topology, invariants of Legendrian submanifolds in 1-jet spaces have been obtained through a variety of techniques. I will discuss how I am enriching one Morse-theoretic invariant, Generating Family Cohomology, to an A-infinity algebra by constructing product maps. The construction uses moduli spaces of Morse flow trees: spaces of intersecting gradient trajectories of functions whose critical points encode Reeb chords of the Legendrian submanifold. I will focus my talk on the construction of a 2-to-1 product and discuss how it lays the foundation for the A-infinity algebra.
Laura DeMarco : Complex dynamics and potential theory
- Presentations ( 146 Views )I will begin with the basics of the two subjects, with the goal of explaining how each has been used as a method to obtain results in the other. The first half will be devoted to foundational results, dating to the 1930s for potential theory and the 1980s for complex dynamics. The second half will be devoted to more recent developments.
Jason Lotay : Hyperkaehler metrics on a 4-manifold with boundary
- Geometry and Topology ( 142 Views )An oriented hypersurface in a hyperkaehler 4-manifold naturally inherits a coclosed coframing. Bryant showed that, in the real analytic case, any oriented 3-manifold with a coclosed coframing can always be locally thickened to a hyperkaehler 4-manifold, in an essentially unique way. This raises the natural question: when can these 3-manifolds with this structure arise as the boundary of a hyperkaehler 4-manifold? In particular, starting from a compact hyperkaehler 4-manifold with boundary, which deformations of the boundary structure can be extended to a hyperkaehler deformation of the interior? I will discuss recent progress on this problem, which is joint work with Joel Fine and Michael Singer.
Dave Penneys : Planar algebras and evaluation algorithms
- Geometry and Topology ( 120 Views )Jones' planar algebras are a useful tool for studying and constructing fusion categories, which generalize the representation categories of (quantum) groups. Thus we think of fusion categories and planar algebras as encoding quantum symmetries. I will give an overview of Jones' planar algebras with attention to specific examples. Along the way, we will discuss several evaluation algorithms which give quantum invariants, including the Jones polynomial.
Seth Baldwin : Positivity in T-equivariant K-theory of flag varieties associated to Kac-Moody groups
- Algebraic Geometry ( 118 Views )The cohomology rings of flag varieties have long been known to exhibit positivity properties. One such property is that the structure constants of the Schubert basis with respect to the cup product are non-negative. Brion (2002) and Anderson-Griffeth-Miller (2011) have shown that positivity extends to K-theory and T-equivariant K-theory, respectively. In this talk I will discuss recent work (joint with Shrawan Kumar) which generalizes these results to the case of Kac-Moody groups.
Igor Zelenko (Texas A&M U) : Gromovs h-principle for corank two distribution of odd rank with maximal first Kronecker index
- Geometry and Topology ( 113 Views )While establishing various versions of the h-principle for contact distributions (Eliashberg (1989) in dimension 3, Borman-Eliashberg-Murphy (2015) in arbitrary dimension, and even-contact contact (D. McDuff, 1987) distributions are among the most remarkable advances in differential topology in the last four decades, very little is known about analogous results for other classes of distributions, e.g. generic distributions of corank 2 or higher. The smallest dimensional nontrivial case of corank 2 distributions is Engel distributions, i.e. the maximally nonholonomic rank 2 distributions on $4$-manifolds. This case is highly nontrivial and was treated recently by Casals-Pérez-del Pino-Presas (2017) and Casals-Pérez-Presas (2017). In my talk, I will show how to use the method of convex integration in order to establish all versions of the h-principle for corank 2 distributions of arbitrary odd rank satisfying a natural generic assumption on the associated pencil of skew-symmetric forms. During the talk, I will try to give all the necessary background related to the method of convex integration in principle. This is the joint work with Milan Jovanovic, Javier Martinez-Aguinaga, and Alvaro del Pino.
Mu-Tao Wang : A variational problem for isometric embeddings and its applications in general relativity
- Geometry and Topology ( 112 Views )I shall discuss a variational problem arising from the study of quasilocal energy in general relativity. Given a spacelike 2-surface in spacetime, the Euler-Lagrange equation for the quasilocal energy is the isometric embedding equation into the Minkowski space coupled with a fourth order nonlinear elliptic equation for the time function. This equation is important in that it gives the ground configuration in GR. In joint work with PoNing Chen and Shing-Tung Yau, we solved this system in the cases of large and small sphere limits.