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Mathematical Physics and Gauge Theory
- New Results and New Conjectures on the Monomer-Dimer Problem
- 12/01/2016
- 11:00 AM - 11:50 AM
- C304 Wells Hall
- Paul Federbush, U. Michigan
The number of configurations of dimers on a large periodic lattice cube that cover a fraction p of the vertices is asymptotically of the form of the exponential of a function of p times the volume V, as V goes to infinity. This function is commonly called the entropy. For hyper-cubical lattices in d dimensions we establish an expression for this function of p, for small p, involving a (convergent) power series in p. With the conjecture that all the terms in this series are positive, confirmed by computation for the first 20 terms in all dimensions, the series is valid for 0<p<1. In fact all analogous terms so far computed for any bipartite lattices have turned out to be positive. Similar results hold for the Virial series, the expansion of the pressure as a power series in p, for a dimer lattice gas, on any lattice. Here there is no restriction to bipartite lattices. By either a spark of creative incite, or a fortuitous fumbling about, one was lead to consider for the monomer-dimer problem on connected regular bipartite graphs the quantity d(i)=ln(N(i)/r^i)-ln(N'(i)/(v-1)^i) where v is the number of vertices, r is the degree, and N(i) and N'(i) are the number of i-dimer configurations on the graph and on the complete graph with the same set of vertices, respectively. This quantity for a sequence of graphs converging to the lattice converges to a multiple of the sum in the entropy expression, if we set p ~ (2i/v). Then the fact that all derivatives of the sum in the entropy are positive (the conjecture holding) leads one to conjecture that all finite difference derivatives of d(i) are positive. The Virial series conjecture leads to a putative upper bound for the same finite difference derivatives. In fact it seems to be true that 'almost all' graphs satisfy these lower and upper finite difference bounds! Looking just at the 0th derivative case one gets new upper and lower conjectured bounds on N(i), higher derivatives yielding a rich set of new bounds to study.
Colloquium
- Loop erased random walk, uniform spanning tree and bi-Laplacian Gaussian field in the critical dimension.
- 12/01/2016
- 4:10 PM - 5:00 PM
- C304 Wells Hall
- Wei Wu, Courant
Critical lattice models are believed to converge to a free field in the scaling limit, at or above their critical dimension. This has been (partially) established for Ising and Phi^4 models for d \geq 4. We describe a spin model from uniform spanning forests in $\Z^d$ whose critical dimension is 4 and prove that the scaling limit is the bi-Laplacian Gaussian field for $d\ge 4$. At dimension 4, there is a logarithmic correction for the spin-spin correlation and the bi-Laplacian Gaussian field is a log correlated field. The proof also improves the known mean field picture of LERW in d=4, by showing that the renormalized escape probability (and arm events) of 4D LERW converge to some 'continuum escaping probability'. Based on joint works with Greg Lawler and Xin Sun.
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Colloquium
- General Insurance Deductible Ratemaking
- 12/02/2016
- 3:10 PM - 4:00 PM
- C304 Wells Hall
- Gee Lee, University of Wisconsin-Madison
Insurance claims have deductibles, which must be considered when pricing for insurance premium. Deductibles may cause censoring and truncation to the observed insurance claims. For this type of data, the regression approach is often used with deductible amount included as an explanatory variable inside a frequency-severity model, so that the resulting coefficient can be used for an assessment of the relativities for deductibles. This approach has the advantage of incorporating the selection effect into deductible ratemaking. On the other hand, standard actuarial textbooks recommend the maximum likelihood approach for estimating parametric loss models, which can be used for calculating the coverage modification amounts due to the deductibles. In this paper, a comprehensive overview of deductible ratemaking is provided, and the pros and cons of various approaches under different parametric models are compared. The regression approach proves to have an advantage in predicting aggregate claims when deductible choices influence the frequency and severity distributions. The maximum likelihood approach becomes necessary for calculating theoretically correct relativities for deductible levels beyond those observed, for each policyholder. For demonstration, loss models are fit to the Wisconsin Local Government Property Insurance Fund data, and examples are provided for the ratemaking of per-loss deductibles offered by the fund. Selected parametric models from the generalized beta family distributions are compared. Models for specific peril types can be combined to improve the ratemaking, and estimation issues for such models under truncation and censoring are discussed.
Applied Mathematics
- Modeling Molecular Interactions in Biomolecular Systems
- 12/02/2016
- 4:10 PM - 5:00 PM
- 1502 Engineering Building
- Pengyu Ren, University of Texas at Austin
Noncovalent interactions, electrostatic in nature, are essential in biomolecular processes such as protein/RNA folding and binding. Recently we have been systematically investigating the fundamental electrostatic forces including short-range induction and penetration effects by using ab initio Symmetry Adapted Perturbation Theory (SAPT), in order to advance the accuracy and transferability in physics-driven classical mechanics model. In this talk, I will present the development of AMOEBA polarizable multipole based force field along with its applications in molecular dynamics simulations towards understanding protein-ligand binding thermodynamics.
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CoIntegrate Mathematics
- Successes of Project-based Learning in Undergraduate Quantitative Literacy Courses
- 12/05/2016
- 12:00 PM - 12:50 PM
- 133G Erickson Hall
- Andy Krause, Jeffrey Craig, MSU
During Summer 2016, we implemented a course-long project in MTH 101: Quantitative Literacy I. Students individually researched a topic of their interest and choosing, and used some of the analytical perspectives we developed elsewhere in the course. Students analyzed media, examined data related to their issue, created info-graphics to communicate their findings, and wrote reflections about their learning. The work that students completed for their projects represents some of the highest levels of thinking on Bloom's taxonomy, and it was clear to us that many students both enjoyed the course project and found it meaningful for their lives in general. We will lead a discussion about the philosophy behind the design of the project, present prompts for the course project, share examples of student work, and explain how the project continues to evolve as an important piece of the QL courses.
Geometric Measure Theory Read
- On the dimension of harmonic measure on self-similar sets
- 12/05/2016
- 1:40 PM - 3:00 PM
- C304 Wells Hall
- Alexander Volberg, MSU
We will try to reduce an old problem concerning the harmonic measure and its Hausdorff dimension
(still a largely open problem on self-similar sets) to purely analytic problem
about two harmonic functions and their boundary Harnack principle.
Analysis and PDE
- Weighted bounds for the Variational Carleson Operator - capturing locality using embedding maps into time-frequency space.
- 12/05/2016
- 4:02 PM - 4:52 PM
- C517 Wells Hall
- Gena Uraltsev, University of Bonn, Germany
We will introduce the Variational Carleson Operator and present a novel
proof of its boundedness using embedding maps. The correct functional
framework for dealing with embedding maps seems to be given by the
iterated outer measure L^p spaces.
We will show how iterated outer measure L^p spaces capture the locality
of the operator and thus allow to obtain sparse domination results.
Sharp Muckenhoupt weighted bounds then follow by well established
methods.
Topology
- Divergence of CAT(0) groups
- 12/05/2016
- 4:10 PM - 5:00 PM
- C304 Wells Hall
- Robert Bell, MSU
The divergence of a geodesic metric space X is, roughly speaking, the rate of growth of the maximum length of geodesics in X - B(r) that join pairs of points on the r-sphere \partial B(r). For example, in the Euclidean plane, the divergence is linear since this quantity is half the circumference of a circle; but in the hyperbolic plane the divergence is exponential. The divergence of a finitely generated group is defined to be that of a Cayley graph; this is well-defined and therefore an invariant of the group. In this introductory talk, I will survey what is known about the divergence of groups which are the fundamental group of a compact non-positively curved cube complex.
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Mathematical Physics and Gauge Theory
- The Batalin-Vilkovisky formalism of the spinning particle
- 12/06/2016
- 11:00 AM - 11:50 AM
- C304 Wells Hall
- Ezra Getzler, Northwestern
We show that the axiom of Felder and Kazhdan on the vanishing of the cohomology groups in negative degree associated to solutions of the classical master equation in the Batalin-Vilkovisky formalism is violated by the spinning particle in a flat background coupled to D=1 supergravity: there are nontrivial cohomology groups in all negative degrees. We will develop all of the background in the BV formalism that we need: no prior experience necessary!
Seminar in Cluster algebras
- Generalizations of the theorem of Moore and Seiberg in 2 dimensional topological field theory
- 12/06/2016
- 1:00 PM - 1:50 PM
- C304 Wells Hall
- Ezra Getzler, Northwestern University
The Teichmüller space of a compact Riemann surface has a natural bordification: that is, there is a real-analytic manifold with corners containing Teichmüller space as its top-dimensional stratum, such that the action of the mapping class-group extends to the boundary. This construction is due to Harvey. In this talk, I discuss generalizations of this construction in the presence of cusps and orbifold points: this leads to a generalization of the Moore-Seiberg theorem, which may be formulated as the statement that the 2-skeleton of this bordification is simply connected. We also give analogues for real Riemann surfaces, which allows us to extend these results to open/closed topological field theories in 2 dimensions.
Colloquium
- A glimpse into the world of cluster algebras
- 12/06/2016
- 4:10 PM - 5:00 PM
- C304 Wells Hall
- Gregory Muller, University of Michigan
Cluster algebras were introduced by Fomin and Zelevinsky to axiomatize a remarkable relation they observed among special coordinate systems on Lie groups. Since then, this same pattern has been discovered in many other algebras in many other contexts. This talk will explore the fundamental examples of this phenomenon, and demonstrate how the general definition unifies and generalizes these examples.
This naturally leads to the consideration of cluster algebras in the abstract, where many surprises await. One surprise is the existence of pathological cluster algebras, with such horrors as non-Noetherian singularities. I will survey the theory of `locally acyclic cluster algebras', a sub-class of cluster algebras which eliminates the horrible examples while containing all known well-behaved examples.
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Algebra
- Exotic cluster algebras
- 12/07/2016
- 3:00 PM - 3:50 PM
- C304 Wells Hall
- Idan Eisner, Technion, Haifa, Israel
Using the notion of compatibility between Poisson brackets and cluster algebras in the coordinate rings of simple complex Lie groups, Gekhtman Shapiro and Vainshtein conjectured a correspondence between the two. Poisson Lie groups are classified by the Belavin-Drinfeld classification of solutions to the classical Yang Baxter equation. For a simple complex Lie group G and a Belavin-Drinfeld class, one can define a corresponding Poisson bracket on the ring of regular functions on G. For some of these classes a compatible cluster structure can be constructed. We will describe some of these for G=SLn. In some cases, the compatible structure is a generalized cluster algebra, where the exchange relations are polynomial rather than binomial. We will show this for G=SP6.
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Geometry and Topology
- Real hyperbolic hyperplane complements in the complex hyperbolic plane
- 12/08/2016
- 2:00 PM - 2:50 PM
- C304 Wells Hall
- Barry Minemyer, The Ohio State University
In the late 80's Gromov and Thurston constructed examples of manifolds which do not admit a hyperbolic metric but do admit metrics whose sectional curvature is pinched arbitrarily close to -1. Their construction involves taking cyclic branched covers of hyperbolic manifolds over 'nice' codimension two totally geodesic submanifolds. In a joint project with J.F. Lafont, J. Meyer, and B. Tshishiku we have extended this construction to 4-manifolds whose metric is modeled on the complex hyperbolic plane. This will be the content of the first half of the talk.
For this group project, I needed to understand the metric in the complex hyperbolic plane expressed in polar coordinates about a copy of the real hyperbolic plane. This research will make up the second half of the talk.
Probability
- Liouville quantum gravity and peanosphere
- 12/08/2016
- 3:00 PM - 3:50 PM
- C405 Wells Hall
- Xin Sun, MIT
We will discuss Liouville quantum gravity as a scaling limit theory for random planar maps. In particular, we will focus on a recent approach called peanosphere or mating of trees and provide several applications of this framework.
Colloquium
- Surfaces of Low Entropy
- 12/08/2016
- 4:10 PM - 5:00 PM
- C304 Wells Hall
- Jacob Bernstein, Johns Hopkins
Following Colding and Minicozzi, we consider the entropy of (hyper)-surfaces in Euclidean space. This is a numerical measure of the geometric complexity of the surface. In addition, this quantity is intimately tied to to the singularity formation of the mean curvature flow — a natural geometric heat flow of submanifolds. In the talk, I will discuss several results that show that closed surfaces for which the entropy is small are simple in various senses.
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Special Lecture
- Actuarial Science Research Seminar
- 12/09/2016
- 1:00 PM - 2:00 PM
- B122 Wells Hall
- Asaf Cohen, University of Michigan
We study a queueing model with weakly interacting strategic servers and with reflecting boundaries under heavy-traffic.
We use mean-field game techniques in order to find an asymptotic Nash equilibrium. We also provide a numerical scheme for solving the mean-field game.
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