Speaker: Panupong Vichitkunakorn, University of Illinois at Urbana-Champaign

Title: Conserved quantites of Q-systems from dimer integrable systems

Date: 11/01/2016

Time: 1:00 PM - 1:50 PM

Place: C304 Wells Hall

In 2013, Goncharov and Kenyon constructed integrable systems from a class of quivers on a torus, parametrized by integral convex polygons. Associating a Y-pattern to the quiver, the phase space coordinates of the dynamical systems are y-variables together with two extra variables. A Y-seed mutation at a vertex having two incoming and two outgoing arrows gives a change of coordinates. We study this dimer integrable system on Cluster variables, extend it to some quivers outside the class, and construct conserved quantities of Q-systems.

Title: Riemann Roch Theorem for Compact Riemann Surfaces

Date: 11/01/2016

Time: 3:00 PM - 3:50 PM

Place: C304 Wells Hall

We will introduce Riemann Surface and define related objects like meromorphic function, holomorphic mapping and meromorphic one form. We will also define divisors on a Riemann Surface, and state the Riemann Roch theorem for compact Riemann Surface.

Algebraic K-theory brings together classical invariants of rings with homotopy groups of topological spaces. In general algebraic K-theory groups are difficult to compute, but in recent years methods in equivariant stable homotopy theory have led to many important K-theory computations. I will introduce this approach to K-theory computations, and discuss some of my recent joint work with Angeltveit on the algebraic K-theory of the group ring Z[C_2].

Speaker: Thomas Sideris, University of California, Santa Barbara

Title: Affine motion of 3d incompressible ideal fluids surrounded by vacuum

Date: 11/03/2016

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

We shall discuss the existence and long-time behavior of affine (spatially linear) solutions to the initial
free boundary value problem for incompressible fluids surrounded by vacuum in 3d. The general problem is known to be
locally well-posed. For affine motion, the equations of motion reduce to a globally solvable system of ordinary differential
equations corresponding to geodesic flow in SL(3,R) viewed as a submanifold embedded in R^9 with the Euclidean metric.
The domain occupied by the fluid at each time is an ellipsoid of constant volume whose diameter grows linearly in time,
provided the pressure remains nonnegative. We shall examine the motion in several representative cases, including
swirling flow geometry where elementary phase plane analysis can be used.

Title: Threading Fáry and Milnor with Buffon's Needle

Date: 11/04/2016

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

There is a fundamental invariant of a curve in Euclidean 3-space called the total curvature which, roughly speaking, measures how 'twisted' the curve is. A beautiful result of Fáry and Milnor from the early 1950's relates this to a branch of mathematics called knot theory, and shows that if the curvature of a curve is small enough, then the curve is actually 'unknotted'; that is, it can be untied without cutting it. Their proofs were rather technical, and in this talk I'll relate their problem to an experimental method for computing the value of pi known as the Buffon Needle Problem. Using this interpretation I will give an elegant and self-contained proof of their result, due to Ari Turner. The talk should be accessible to anyone who has taken multivariable calculus.

Speaker: Tom Sideris, University of California at Santa Barbara; Thomas C. Sideris, University of California, Santa Barbara

Title: Affine motion of 3d compressible ideal fluids surrounded by vacuum

Date: 11/04/2016

Time: 4:10 PM - 5:00 PM

Place: 1502 Engineering Building

We shall discuss the existence and long-time behavior of affine (spatially linear) solutions to the initial free boundary value problem for compressible fluids surrounded by vacuum in 3d. The general problem is known to be locally well-posed. For affine motion, the equations of motion reduce to a globally solvable Hamiltonian system of ordinary differential equations on GL_+(3,R). The domain occupied by the fluid at each time is an ellipsoid whose diameter grows linearly in time. Upon rescaling to a fixed diameter, the asymptotic limit of the fluid domain is determined by a semi-definite
quadratic form of rank r = 1,2, or 3, corresponding to a collapse of the ellipsoid along 2,1, or 0 of its principal axes. When
the adiabatic index \gamma determining the pressure law lies in the interval ( 4/3 , 2 ), the rank of this form is 3 and there is a scattering operator, i.e. a bijection between states at minus and plus infinity. However, larger values of \gamma can lead to collapsed asymptotic states of rank 1 or 2.

Speaker: Tom Sideris, University of California at Santa Barbara; Thomas C. Sideris, University of California, Santa Barbara

Title: Affine motion of 3d compressible ideal fluids surrounded by vacuum

Date: 11/04/2016

Time: 4:10 PM - 5:00 PM

Place: 1502 Engineering Building

We shall discuss the existence and long-time behavior of affine (spatially linear) solutions to the initial
free boundary value problem for compressible fluids surrounded by vacuum in 3d. The general problem is known to be
locally well-posed. For affine motion, the equations of motion reduce to a globally solvable Hamiltonian system of ordinary
differential equations on GL_+(3,R). The domain occupied by the fluid at each time is an ellipsoid whose diameter grows
linearly in time. Upon rescaling to a fixed diameter, the asymptotic limit of the fluid domain is determined by a semi-definite
quadratic form of rank r = 1,2, or 3, corresponding to a collapse of the ellipsoid along 2,1, or 0 of its principal axes. When
the adiabatic index \gamma determining the pressure law lies in the interval ( 4/3 , 2 ), the rank of this form is 3 and there
is a scattering operator, i.e. a bijection between states at minus and plus infinity. However, larger values of \gamma can
lead to collapsed asymptotic states of rank 1 or 2.

Speaker: Alexander Reznikov, Vanderbilt University

Title: Covering properties of configurations optimal for discrete Chebyshev constants

Date: 11/07/2016

Time: 4:02 PM - 4:52 PM

Place: C517 Wells Hall

It is well known that, for a fixed integer N and as s goes to infinity, the optimal configurations for minimal discrete N-point s-Riesz energy on a compact set A converge to N-point configurations that solves the best-packing problem on A. We present a max-min problem that, in the limit, solves the best-covering problem, which is somewhat dual to best-packing. We will discuss other distributional properties of optimal configurations for this new problem, as well as applications to numerical integration and convex geometry.

Our idea of constructing a cofinite graph starts by defining a uniform topological graph Gamma. This idea is motivated by cofinite groups structure, due to B. Hartley.
We define and establish a theory of cofinite connectedness of a cofinite graph. We found that if G is a cofinite group and Gamma = Gamma(G;X) is the Cayley graph with respect to a generating set X of G, then Gamma can be given a suitable cofinite uniform topological structure so that X generates G topologically if and only if Gamma is cofinitely connected.
Next we develop group actions on cofinite graphs. Defining the action of an abstract group over a cofinite graph in the most natural way we characterize a unique way of uniformizing an abstract group with a cofinite structure, obtained from the cofinite structure of the graph in the underlying action, so that the aforesaid action becomes uniformly continuous. We then apply the general theory to additional structure such as groupoids,
thus leading to the notions of cofinite groupoids.
This is a joint work with Dr. J.M.Corson (University of Alabama, Department of Mathematics) and Dr. B. Das (University of North Georgia, Department of Mathematics).

Speaker: Gabriel Angelini-Knoll, Wayne State University

Title: Periodicity in iterated algebraic K-theory

Date: 11/10/2016

Time: 2:00 PM - 2:50 PM

Place: C304 Wells Hall

Periodicity is a highly studied phenomenon in homotopy theory. For example, R. Bott showed that the homotopy groups of the classifying space of the infinite orthogonal group are periodic with period eight. This periodicity is also reflected in the stable homotopy groups of spheres using the J homomorphism, a map from the homotopy groups of the infinite orthogonal group to the stable homotopy groups of spheres. Using this map J.F. Adams produced a “height one” periodic family in the homotopy groups of spheres.
The image of J can be realized as a topological space and at odd primes this space is equivalent to algebraic K-theory of certain finite fields after p-completion. Therefore, a periodic family of height one in the language of chromatic homotopy theory is detected by algebraic K-theory of finite fields. In my talk, I will describe a higher height version of this phenomenon that I prove in my thesis. In particular, I demonstrate that a periodic family of height two is detected in mod (p, v_1)-homotopy of iterated algebraic K-theory of a finite field of order q, where q is a generator of the units in the p-adic integers. This result gives some evidence for a red-shift type phenomenon, which loosely states that algebraic K-theory increases the wavelength of periodicity.

In many practical imaging scenarios, including computed tomography and magnetic resonance imaging (MRI), the goal is to reconstruct an image from few of its Fourier domain samples. Many state-of-the-art reconstruction techniques, such as total variation minimization, focus on discrete 'on-the-grid' modelling of the problem both in spatial domain and Fourier domain. While such discrete-to-discrete models allow for fast algorithms, they can also result in sub-optimal sampling rates and reconstruction artifacts due to model mismatch. Instead, we present a framework that allows for the recovery of a continuous domain 'off-the-grid' representation of piecewise constant images from the optimal number of Fourier samples. The main idea is to model the edge set of the image as the level-set curve of a continuous domain band-limited function. Sampling guarantees can be derived for this framework by investigating the algebraic geometry of these curves. Finally, we show how this model can be put into a robust and efficient optimization framework by posing signal recovery entirely in Fourier domain as a structured low-rank matrix completion problem, and demonstrate the benefits of this approach over standard discrete methods in the context of undersampled MRI reconstruction.

We will prove Theorem 3.8 of Preiss's paper: roughly
speaking, given a uniformly distributed measure and a point in its
support, it is possible to find a nearby point for which an
appropriate scaling of the measure looks like Lebesgue measure on a
hyperplane. The proof will involve a Taylor expansion of the Gaussian
moments studied a few weeks ago.

Special Lagrangian cones play a central role in the understanding of the SYZ conjecture, an important conjecture in mathematics based upon mirror symmetry and certain string theory models in physics. According to string theory, our universe is a product of the standard Minkowski space with a Calabi-Yau 3-fold. Strominger, Yau, and Zaslov conjectured that Calabi-Yau 3-folds can be fibered by special Lagrangian 3-tori with singular fibers. To make this idea rigorous one needs control over the singularities, which can be modeled by special Lagrangian cones. In this talk, we discuss special Lagrangian cones, the difficulties involved in defining and computing invariants of them, and the hope that these invariants may offer in understanding the SYZ conjecture.

Title: Anomalous blow-up with vanishing energy in 1-D Perona-Malik diffusion

Date: 11/14/2016

Time: 4:02 PM - 4:52 PM

Place: C517 Wells Hall

We consider the initial-Neumann boundary value problem of non-parabolic equations in one space dimension. In particular, we mainly focus on the problem with a diffusion flux function of the Perona-Malik type in image processing, which is a well-known example of forward-backward parabolic problems. For this problem, we will discuss the existence of weak solutions that converge uniformly to the initial mean value as time approaches a certain final value while the spatial derivative blows up and the associated energy vanishes in some sense. The method is a combination of a classical parabolic theory, the convex integration method in Baire's category setup and the almost transition gauge invariance. This is a joint work with Baisheng Yan(MSU).

Speaker: Scott Baldridge, Louisiana State University

Title: Math Curriculum Designed for Instructors and Students

Date: 11/15/2016

Time: 10:00 AM - 11:00 AM

Place: C304 Wells Hall

In designing new courses and curricula, it is essential to begin with a
good sense of four features: the purpose, the content, the audience, and the
narrative. In the usual approach, students are assumed to be the audience.
However, research on pedagogical content knowledge suggests that instructors
should equally be considered an audience of the content. In fact, instructors
are often the more important audience due to the critical role they play in engaging
students with the mathematics. In this talk, I will present two projects
that focused on instructor’s mathematical knowledge for teaching from the
onset. I will end by discussing a new paper on instructor-level knowledge of
quantities and rates, and its implications for the creation of instructor-student
curricula for beginning-level university math courses.

Our idea of constructing a cofinite graph starts by defining a uniform topological graph Gamma. This idea is motivated by cofinite groups structure, due to B. Hartley.
We define and establish a theory of cofinite connectedness of a cofinite graph. We found that if G is a cofinite group and Gamma = Gamma(G;X) is the Cayley graph with respect to a generating set X of G, then Gamma can be given a suitable cofinite uniform topological structure so that X generates G topologically if and only if Gamma is cofinitely connected.
Next we develop group actions on cofinite graphs. Defining the action of an abstract group over a cofinite graph in the most natural way we characterize a unique way of uniformizing an abstract group with a cofinite structure, obtained from the cofinite structure of the graph in the underlying action, so that the aforesaid action becomes uniformly continuous. We then apply the general theory to additional structure such as groupoids,
thus leading to the notions of cofinite groupoids.
This is a joint work with Dr. J.M.Corson (University of Alabama, Department of Mathematics) and Dr. B. Das (University of North Georgia, Department of Mathematics).

Speaker: Ricardo Nemirovsky, Manchester Metropolitan University, UK

Title: Affects and Emergent Learning

Date: 11/16/2016

Time: 3:30 PM - 5:00 PM

Place: 252 EH

In this talk I will distinguish ‘emergent learning’ from ‘teleological learning,’ which is learning for the sake of passing predefined tests and goals. While teleological learning may succeed or fail, emergent learning is always going on in ways that are under-determined and largely unpredictable. Emergent learning involves getting engaged in events and places and participating in the circulation of affects imbuing them. I will describe ongoing work towards a new theory of emergent learning as self-sustained development in the thick of circulating affects. My talk will be divided in two parts: 1) Articulation of a theoretical/philosophical framework about the nature of affects; and, 2) Exploration of the socio-historical and material dynamics of affects and emergent learning in the course of a videotaped 10-minute episode during a field trip by 5th graders to a science museum.

We discuss optimal insurance contract design where an individual's preference is of
the rank-dependent expected utility (RDU) type. Although this problem has been studied
in the literature, their contracts suffer from a problem of moral hazard for paying
more compensation for a smaller loss. Our project addresses this setback by exogenously
imposing the constraint that both the indemnity function and the insured's retention
function be increasing with respect to the loss. We characterize the optimal
solutions via calculus of variations, and then apply the result to obtain explicitly expressed
contracts for problems with Yaari's dual criterion and general RDU.

The theory of complex projective plane curves has a long history. However, curves of higher genus are rarely studied. It turns out that Heegaard-Floer theory can be effectively used to obtain constraints on possible cusp types of such curves. In fact, restricting ourselves to the case of curves with one cusp having a torus knot link, one can obtain an almost complete classification of possible torus knot types for infinitely many curve genera. The proof is a nice interplay of the theory of numerical semigroups, generalized Pell equations and birational transformations.
These results were obtained in a joint work with Daniele Celoria and Marco Golla. Independently, similar work was done by Maciej Borodzik, Matthew Hedden and Charles Livingston.

Currently there is a new generation of large astronomical telescope under construction, e.g. the European Extremely Large Telescope (E-ELT) of the European Southern Observatory (ESO) with a mirror diameter of 39 meters or the Thirty Meter Telescope (TMT), build by a consortium headed by Caltech. The operation of those huge telescopes require new mathematical methods in particular for the Adaptive Optics systems of the telescopes.
The image quality of ground based astronomical telescopes suffers from turbulences in the atmosphere. Adaptive Optics (AO) systems use wavefront sensor measurements of incoming light from guide stars to determine an optimal shape of deformable mirrors (DM) such that the image of the scientific object is corrected after reflection on the DM. The solution of this task involves several inverse problems: First, the incoming wavefronts have to be reconstructed from wavefront sensor measurements. The next step involves the solution of the Atmospheric Tomography problem, i.e., the reconstruction of the turbulence profile in the atmosphere. Finally, the optimal shape of the mirrors has to be determined. As the atmosphere changes frequently, these computations have to be done in real time. In the talk we introduce mathematical models for the elements of different Adaptive Optics system such as Single Conjugate Adaptive Optics (SCAO) or Multi Conjugate Adaptive Optics (MCAO) and present fast reconstruction algorithms as well as related numerical results for each of the sub-tasks that achieve the accuracy and speed required for the operation of ELTs.

Title: Computational Modeling in Studying Blood Clot Formation

Date: 11/18/2016

Time: 4:10 PM - 5:00 PM

Place: 1502 Engineering Building

Blood clotting is a multiscale process involving blood cells, fibrinogen polymerization, coagulation reactions, ligand-receptor interactions and blood plasma flow. Detailed multiscale models of blood clotting to cover all aspects of clotting are, if not possible, extremely difficult to develop. Models focusing on specific events across one or two spatial-temporal scales seem to be plausible. In this talk, ligand-receptor binding kinetics model, computational model of fluid-structure interaction (FSI) for simulating flow-elastic membrane with mass and a continuum model for studying the structural stability of clots will be presented. The binding kinetics model revealed that platelet αIIbβ3 integrin and fibrin interacts through a two-step mechanism. The new FSI model is derived by using the energy law and distributed-Lagrange-multiplier/fictitious-domain (DLM/FD) formulation. The continuum model for studying the structural stability of clots utilized the phase field and energetic variational approaches. Simulation results show that rheological response of the blood clot to the flow is determined by mechanical and structural properties of its components. Two main mechanisms are shown to significantly affect volume of the already formed clot: dynamic balance between platelet adhesion and platelet removal by the flow on the blood clot surface and removal of parts of the clot through rupture.

Title: Tangents of Uniformly Distributed Measures - II

Date: 11/21/2016

Time: 1:40 PM - 3:00 PM

Place: C304 Wells Hall

We will prove Theorem 3.8 of Preiss's paper: roughly
speaking, given a uniformly distributed measure and a point in its
support, it is possible to find a nearby point for which an
appropriate scaling of the measure looks like Lebesgue measure on a
hyperplane. The proof will involve a Taylor expansion of the Gaussian
moments studied a few weeks ago.

Title: Risk sharing and risk aggregation via risk measures

Date: 11/21/2016

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

In this talk, we discuss two problems in risk management using the tools of risk
measures.
In the first part of the talk, we address the problem of risk sharing among agents using
a two-parameter class of quantile-based risk measures, the so-called RangeValue-at-Risk
(RVaR), as their preferences. We first establish an inequality for RVaRbased
risk aggregation, showing that RVaR satisfies a special form of subadditivity.
Then, the risk sharing problem is solved through explicit construction.
Comonotonicty and robustness of the optimal allocations are investigated. We
show that, in general, a robust optimal allocation exists if and only if none of the risk
measures is a VaR. Practical implications of our main results for risk management
and policy makers will be discussed.
In the second part of the talk, we study the aggregation of inhomogeneous risks with
a special type of model uncertainty, called dependence uncertainty, evaluated by a
generic risk measure. We establish general asymptotic equivalence results for the
classes of distortion risk measures and convex risk measures under different mild
conditions. The results implicitly suggest that it is only reasonable to implement a
coherent risk measure for the aggregation of a large number of risks with uncertainty
in the dependence structure, a relevant situation for risk management practice.

This talk will be a follow up for the previous talk during the the cluster algebra seminar. We will explore specific cases of the amplituhedron when m=4. The goal will be to work through few examples of this specific case, and see if we can state some of the main results in the area.

Title: Restricted Stirling and Lah numbers, and their inverses

Date: 11/22/2016

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

The matrix of Stirling numbers of the second kind (counting partitions of a set into non-empty blocks) is lower triangular with integer entries and 1’s down the diagonal, so its inverse shares the same properties. It’s well-known that the entries in the inverse matrix have a nice combinatorial meaning — they are Stirling numbers of the first kind (counting permutations by number of cycles).
We explore restricted Stirling numbers of the second kind, in which the block sizes are required to lie in some specified set. As long as this set contains 1, the matrix of these restricted Stirling numbers has an inverse with integer entries, so it is natural to ask: do these integers count things?
In many cases, we find that they do. This is joint work with John Engbers (Marquette) and Cliff Smyth (UNC Greensboro).

Title: A priori upper bounds for the inhomogeneous Landau equation

Date: 11/28/2016

Time: 4:02 PM - 4:52 PM

Place: C517 Wells Hall

We consider the Landau equation, an integro-differential kinetic model from plasma physics that describes the evolution of a particle density in phase space. It arises as the limit of the Boltzmann equation when grazing collisions predominate. I will give an overview of prior work on the regularity theory of the Landau equation, and describe how to prove a priori upper bounds that decay polynomially in the velocity variable. The key ingredients are precise upper and lower bounds on the coefficients, and tracking how local estimates scale as the velocity grows. I will also explain why the polynomial decay cannot be improved to exponential decay. This talk is based on joint work with Stephen Cameron and Luis Silvestre.

Title: Nonembeddability and Tverberg-type Theorems in Combinatorics

Date: 11/29/2016

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

It is well-known that there are exactly two 'minimal' non-planar graphs, that is, graphs which cannot be embedded in the plane. It should then come as no surprise that one would like to generalize this combinatorial result to the case of embeddability of simplicial complexes. Topological methods have proven to be very useful here, with one of the landmark theorems being the topological Tverberg theorem of 1966. In this sequence of talks, we survey nonembeddabililty results, a surprising recent development of the topological Tverberg conjecture, and notions related to these topics.

We will discuss the 'group extension' problem, which asks: given groups G and N, in what ways may we construct a group E which has N as a normal subgroup and quotient E/N isomorphic to G? A brief introduction to group cohomology will be given, along with its connections to the extension problem. Time permitting, we will also mention the analogous theory for Lie algebras (i.e. Lie algebra extensions and Lie algebra cohomology).