The Hilbert function is a classical invariant of a variety (with a given embedding) that is easy to compute. It determines some properties of the variety (such as degree, dimension, and arithmetic genus), but it cannot determine more sophisticated invariants. A minimal free resolution determines more sophisticated properties of the variety while still being easily computable. For example, any set of seven points in P^3 in linearly general position has the same Hilbert function, but minimal free resolutions can distinguish whether the points lie on a rational normal curve. Furthermore, a minimal free resolution retains all the information of the Hilbert function. In this talk, we will define a minimal free resolution and associated invariants. With the definitions in place, we will show that minimal free resolutions retain all the information of the Hilbert function then explain the above example. If time permits, we will additionally show that minimal free resolutions are well behaved when restricting a variety to a hypersurface.

Title: $SL_k$ character varieties and quantum cluster algebras

Date: 04/04/2019

Time: 2:00 PM - 3:00 PM

Place: C304 Wells Hall

briefly recall a combinatorial approach to the description and quantization of Teichmuller spaces of Riemann surfaces $\Sigma_{g,s}$ of genus $g$ with $s$ holes and algebras of geodesic functions on these surfaces. We describe sets of geodesic functions in W.Thurston shear coordinates based on an ideal triangle decomposition of Riemann surfaces with holes and demonstrate the polynomiality and positivity properties of the corresponding geodesic functions. In the algebraic setting, these sets are related to traces of monodromies of $SL_2$ connection on $\Sigma_{g,s}$, and Darboux-type Poisson and quantum relations on shear coordinates were proven to generate Goldman brackets on geodesic functions. I will describe these structures and their recent generalizations to $SL_2$ and $SL_n$ (decorated) character varieties on Riemann surfaces $\Sigma_{g,s,n}$ with holes and $n$ marked points on hole boundaries and how it is interlaced with cluster algebras, reflection equations, and groupoids of upper triangular matrices. [Based on work in collaboration with M.Mazzocco, V.Roubtsov, and M.Shapiro.]

Title: Sign variation and boundary measurement in projective space

Date: 04/04/2019

Time: 3:00 PM - 4:00 PM

Place: C204A Wells Hall

We are interested in the topology of some spaces obtained by relaxing total positivity in the real Grassmannian. We define two families of subsets of the Grassmannian each of which include both the totally nonnegative Grassmannian and the whole Grassmannian. In this initial study of such subsets of the Grassmannian we focus of subsets of real projective space where interesting topology already appears. We we are able to find a regular CW complex which can be leveraged to compute some invariants like the fundamental group and Euler characteristic. We also conjecture some "ball-like" properties (e.g. Cohen-Macualayness).

Speaker: Alexander Schnurr, University Siegen (Germany)

Title: Ordinal Patterns in Clusters of Extremes of Regularly Varying Time Series

Date: 04/04/2019

Time: 3:00 PM - 3:50 PM

Place: C405 Wells Hall

The purpose is to investigate temporal clusters of extremes defined as subsequent exceedances of high thresholds in a stationary time series. Two meaningful features of these clusters are the probability distribution of the cluster size and the ordinal patterns within a cluster. The latter have been introduced in order to handle data sets with several thousand data points appearing in medicine, biology, finance and computer science. Since these patterns take only the ordinal structure of consecutive data points into account, the method is robust under monotone transformations and measurement errors. We verify the existence of the corresponding limit distributions in the framework of regularly varying time series, develop non-parametric estimators and show and their asymptotic normality under appropriate mixing conditions. (This is joint work with Marco Oesting.)

Title: Uncertainty Quantification and Machine Learning of the Physical Laws Hidden Behind the Noisy Data

Date: 04/05/2019

Time: 4:10 PM - 5:00 PM

Place: 1502 Engineering Building

In this talk, I will present a new data-driven paradigm on how to quantify the structural uncertainty (model-form uncertainty) and learn the physical laws hidden behind the noisy data in the complex systems governed by partial differential equations. The key idea is to identify the terms in the underlying equations and to approximate the coefficients of the terms with error bars using Bayesian machine learning algorithms on the available noisy measurement. In particular, Bayesian sparse feature selection and parameter estimation are performed. Numerical experiments show the robustness of the learning algorithms with respect to noisy data and size, and its ability to learn various candidate equations with error bars to represent the quantified uncertainty.

Title: The equivariant stable parametrized h-cobordism theorem

Date: 04/08/2019

Time: 3:00 PM - 4:00 PM

Place: C304 Wells Hall

The stable parametrized h-cobordism theorem provides a critical link in the chain of homotopy theoretic constructions that show up in the classification of manifolds and their diffeomorphisms. For a compact smooth manifold M it gives a decomposition of Waldhausen's A(M) into QM_+ and a delooping of the stable h-cobordism space of M. I will talk about joint work with Malkiewich on this story when M is a smooth compact G-manifold.

Title: New examples of local rigidity of solvable algebraic partially hyperbolic actions

Date: 04/10/2019

Time: 3:00 PM - 4:00 PM

Place: C117 Wells Hall

We show $C^\infty$ local rigidity for a broad class of new examples of solvable algebraic partially hyperbolic actions on ${\mathbb G}=\mathbb{G}_1\times\cdots\times \mathbb{G}_k/\Gamma$, where $\mathbb{G}_1$ is of the following type: $SL(n, {\mathbb R})$, $SO_o(m,m)$, $E_{6(6)}$, $E_{7(7)}$ and $E_{8(8)}$, $n\geq3$, $m\geq 4$. These examples include rank-one partially hyperbolic actions. The method of proof is a combination of KAM type iteration scheme and representation theory. The principal difference with previous work
that used KAM scheme is very general nature of the proof: no specific information about unitary representations of ${\mathbb G}$ or ${\mathbb G}_1$ is required.
This is a continuation of the last talk.

A classical problem in knot theory is determining whether or not a given 2-dimensional diagram represents the unknot. The UNKNOTTING PROBLEM was proven to be in NP by Hass, Lagarias, and Pippenger. A generalization of this decision problem is the GENUS PROBLEM. We will discuss the basics of computational complexity, knot genus, and normal surface theory in order to present an algorithm (from HLP) to explicitly compute the genus of a knot. We will then show that this algorithm is in PSPACE and discuss more recent results and implications in the field.

We show that the three-dimensional homology cobordism group admits an infinite-rank summand. It was previously known that the homology cobordism group contains an infinite-rank subgroup and a Z-summand. The proof relies on the involutive Heegaard Floer homology package of Hendricks-Manolescu and Hendricks-Manolescu-Zemke. This is joint work with I. Dai, M. Stoffregen, and L. Truong.

There is a close analogy between function fields over finite fields and number fields. In this analogy $\text{Spec } \mathbb{Z}$ corresponds to an algebraic curve over a finite field. However, this analogy often fails. For example, $\text{Spec } \mathbb{Z} \times \text{Spec } \mathbb{Z} $ (which should correspond to a surface) is $\text{Spec } \mathbb{Z}$ (which corresponds to a curve). In many cases, the Fargues-Fontaine curve is the natural analogue for algebraic curves. In this first talk, we will give the construction of the Fargues-Fontaine curve.

Speaker: Fernando Guevara Vasquez, University of Utah

Title: Manipulation of particles in a fluid with standing acoustic waves

Date: 04/12/2019

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

Consider a collection of particles in a fluid that is subject to
a standing acoustic wave. In some situations, the particles tend to
cluster about the nodes of the wave. We study the problem of finding a
standing acoustic wave that can position particles in desired locations,
i.e. whose nodal set is as close as possible to desired curves or
surfaces. We show that in certain situations we can expect to reproduce
patterns up to the diffraction limit. For periodic particle patterns, we
show that there are limitations on the unit cell and that the possible
patterns in dimension d can be determined from an eigendecomposition of a
2d x 2d matrix.

Title: Introduction to Seiberg Witten invariants on three manifolds

Date: 04/17/2019

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

Although Seiberg Witten invariant originally introduced for four manifolds, but its three dimensional version is also interesting .After a brief discussion on the definition of the Seiberg Witten invariant on three manifolds we will see some results from literature equating this invariant to some known invariants of three manifolds.

Title: Quasi-positivity in free groups and braid groups

Date: 04/18/2019

Time: 2:00 PM - 3:00 PM

Place: C304 Wells Hall

I'll discuss joint work with Rita Gitik (UM) on the problem of recognizing quasi-positive elements of a group G defined by
a finite presentation (X ; R). An element of G is quasi-positive if it can be represented by a word that is a product of conjugates of positive powers of letters in X. The recognition problem is to determine whether or not a given word (using both positive and negative powers of letters in X) represents an element of G that is quasi-positive. This problem was solved by Orevkov when G is free with basis X or when G is the 3-strand braid group with its standard generating set. I'll present a new solution to the recognition problem for free groups and discuss some of the challenges posed by braid groups and related groups.

Title: Cluster algebras with Grassmann variables (joint with V. Ovsienko)

Date: 04/18/2019

Time: 3:00 PM - 4:00 PM

Place: C117 Wells Hall

We develop a version of cluster algebra extending the ring of Laurent polynomials by adding Grassmann variables. These algebras can be described in terms of “extended quivers” which are oriented hypergraphs. We describe mutations of such objects and deﬁne a corresponding commutative superalgebra. Our construction includes the notion of weighted quivers that has already appeared in diﬀerent contexts. This project is a step towards understanding the notion of cluster superalgebra.

Title: Flexibility in contact and symplectic geometry

Date: 04/18/2019

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

We discuss a number of $h$-principle phenomena which were recently discovered in the field of contact and symplectic geometry. In generality, an $h$-principle is a method for constructing global solutions to underdetermined PDEs on manifolds by systematically localizing boundary conditions. In symplectic and contact geometry, these strategies typically are well suited for general constructions and partial classifications. Some of the results we discuss are the characterization of smooth manifolds admitting contact structures, high dimensional overtwistedness, the symplectic classification of flexible Stein manifolds, and the construction of exotic Lagrangians in $C^n$.

Title: Hausdorff dimensions of exceptional sets in homogeneous spaces

Date: 04/23/2019

Time: 3:00 PM - 4:00 PM

Place: C304 Wells Hall

This talk is the second part of the talk that I gave last semester in the dynamics seminar. I will present sketch of the proof of the main result of the joint work with Dmitry Kleinbock in which we found an upper bound for the set of points in a homogeneous space whose orbit miss a fixed open subset.

Speaker: Nestor Guillen, University of Massachusetts at Amherst

Title: Transportation methods for Lévy measures and applications

Date: 04/24/2019

Time: 4:10 PM - 5:00 PM

Place: C517 Wells Hall

The comparison principle for second order elliptic equations is one of the cornerstone of the theory of viscosity solutions. Works of Sayah and more recently by Jakobsen-Karlsen and Barles-Imbert have expanded it to many important subfamilies of nonlocal elliptic equations. In joint work with Chenchen Mou and Andrzej Święch we show how optimal transportation methods can be used to couple Lévy measures, which encode the integro-differential part of nonlocal operators, which allow us to obtain comparison principles for new families of nonlocal equations. Our method puts all previous subfamilies of nonlocal operators (Levy-Ito operators, operators of order less than 1) in a single framework, while also yielding results for new families.

Title: More concordance homomorphisms from knot Floer homology

Date: 04/25/2019

Time: 2:00 PM - 3:00 PM

Place: C304 Wells Hall

I will describe an infinite family of integer-valued concordance homomorphisms defined using knot Floer homology. These invariants have topological applications to concordance genus, concordance unknotting number, and bridge index. This is joint work with Irving Dai, Jen Hom, and Matt Stoffregen.

Title: Recent progress on existence of minimal surfaces

Date: 04/25/2019

Time: 3:10 PM - 4:00 PM

Place: C304 Wells Hall

A long standing problem in geometry, conjectured by Yau in 1982, is that any any $3$-manifold admits an infinite number of distinct minimal surfaces. The analogous problem for geodesics on surfaces led to the discovery of deep interactions between dynamics, topology, and analysis. The last couple of years brought dramatic developments to Yau’s conjecture, which has now been settled due to the work of Marques-Neves and Song. I will survey the history of the problem and the several contributions made.

Title: Validating and Extending Methods for Extracting the Backbone of Bipartite Projections

Date: 04/29/2019

Time: 3:00 PM - 4:00 PM

Place: C517 Wells Hall

Networks are widely recognized as important for phenomena that span multiple disciplines
including the diffusion of information (communication), the contagion of illness (public health), the
passage of legislation (political science), the emergence of norms (sociology), the coordination of economic activities (economics & geography), the formation of relationships (developmental psychology), and detection of national security threats (intelligence). However, the challenges of collecting network data have limited the feasibility of network research in these areas. In this project, we will explore and refine the use of bipartite projections to validly infer otherwise unobserved or unobservable networks. Several formal methods for making such inferences have been proposed, however it remains unknown which models lead to valid inferences and under what circumstances. We use bipartite network data to systematically test, compare, and extend available models for inferring networks from bipartite projections.