Title: Double Bott-Samelson cell and the moduli space of microlocal rank-1 sheaves

Date: 10/01/2019

Time: 3:00 PM - 4:00 PM

Place: C204A Wells Hall

Shende, Treumann, and Zaslow gave a combinatorial description of the moduli space of microlocal rank-1 sheaves in their paper “Legendrian Knots and Constructible sheaves”. Following a result of Guillermou, Kashiwara, and Schapira, this moduli space is an invariant of Legendrian links. In this talk, I will review the definition and the cluster structure on the (undecorated) double Bott-Samelson cells, and show that in the cases of positive braids of Dynkin type A_r, the undecorated double Bott-Samelson cells are isomorphic to moduli spaces of microlocal rank-1 sheaves associated to the corresponding braid closures. As a corollary, the undecorated double Bott-Samelson cells of Dynkin type A_r are also Legendrian link invariants for positive braid closures. If time allows, I will also talk about how to count F_q points on the undecorated double Bott-Samelson cells.

Speaker: Victoria Hand , University of Colorado, Boulder; Elizabeth Mendoza, University of California, Irvine; Justin TenEyck, “I have a Dream” Foundation of Boulder County

Title: Toward Re-humanizing Mathematics Education: Participatory Approaches to Noticing in Mathematics Classrooms

Date: 10/02/2019

Time: 3:00 PM - 4:30 PM

Place: 252 EH

Researchers are increasingly calling for participatory approaches to educational research that center the voices, experiences, and participation of minoritized communities. This talk will report on the Co-Attend research project, which is grounded in a participatory approach to mathematics teacher noticing. The project involves mathematics teachers, leaders of local community-based organizations and university researchers in collectively understanding expansive, multisensory noticing that supports re-humanizing practices in mathematics classrooms. All participants are positioned as researchers and co-analyze project data in video club meetings and summer institutes. We will describe emerging findings from the project, both in terms of the noticing framework, as well as the participatory process.

Title: On Geodesic Triangles in the Hyperbolic Plane

Date: 10/03/2019

Time: 2:00 PM - 2:50 PM

Place: C304 Wells Hall

Let M be an orientable hyperbolic surface without boundary and
let c be a closed geodesic in M. We prove that any side of any triangle formed by distinct lifts of c in the hyperbolic plane is shorter than c. The talk will be presented for advanced undergraduate and beginning graduate students.

An important invariant of a path-connected topological space X is the number of homomorphisms from the fundamental group of X to a finite, non-abelian, simple group G. Kuperberg and Samperton proved that, although these invariants can be powerful, they are often computationally intractable, particularly when X is an integral homology 3-sphere. More specifically, they prove that the problem of counting such homomorphisms is #P-complete via a reduction from a known #P-complete circuit satisfiability problem. Their model constructs X from a well-chosen Heegaard surface and a mapping class in its Torelli group. We will introduce the basics of complexity for counting problems, summarize the reduction used by K-S to bound the problem of counting homomorphisms, and discuss some of the topological and quantum computing implications of their results.

Title: Recent results of GCD problems on almost $S$-units and recurrences

Date: 10/07/2019

Time: 4:30 PM - 5:30 PM

Place: C517 Wells Hall

The GCD problem is one of the major problems in Diophantine Geometry. Corvaja, Zannier and Bugeaud first gave a fundamental result on GCD of integers powers and then generalized to rational numbers and algebraic numbers by many mathematicians. In this talk I will introduce recent GCD results on $S$-units due to Levin and generalize to almost $S$-units. I will give the definition of almost units and present the main theorem of GCD on multivariable polynomials, which is lead to a result about recurrence sequences. If time allows, I will also introduce Silverman’s generalized GCD along the blow up of a closed subscheme and apply to abelian surface case and its connection to Vojta’s conjecture.

Title: Double Bott-Samelson cell and the moduli space of microlocal rank-1 sheaves

Date: 10/08/2019

Time: 3:00 PM - 4:00 PM

Place: C204A Wells Hall

Shende, Treumann, and Zaslow gave a combinatorial description of the moduli space of microlocal rank-1 sheaves in their paper “Legendrian Knots and Constructible sheaves”. Following a result of Guillermou, Kashiwara, and Schapira, this moduli space is an invariant of Legendrian links. In this talk, I will review the definition and the cluster structure on the (undecorated) double Bott-Samelson cells, and show that in the cases of positive braids of Dynkin type A_r, the undecorated double Bott-Samelson cells are isomorphic to moduli spaces of microlocal rank-1 sheaves associated to the corresponding braid closures. As a corollary, the undecorated double Bott-Samelson cells of Dynkin type A_r are also Legendrian link invariants for positive braid closures. If time allows, I will also talk about how to count F_q points on the undecorated double Bott-Samelson cells.

I will describe joint work in progress with Akshay Venkatesh on the construction of 1-cocycles on $\mathrm{GL}_2(\mathbb{Z})$ valued in a limit of second motivic cohomology groups of open subschemes of the square of (1) the multiplicative group over the rationals and (2) a universal elliptic curve. I’ll explain how these cocycles specialize to homomorphisms taking modular symbols to special elements in second cohomology groups of cyclotomic fields and modular curves in the respective cases.

Speaker: Jeffrey Schenker, Michigan State University

Title: An ergodic theorem for homogeneously distributed quantum channels with applications to matrix product states

Date: 10/10/2019

Time: 11:30 AM - 12:30 PM

Place: C304 Wells Hall

Quantum channels represent the most general physical evolution of a quantum system through unitary evolution and a measurement process. Mathematically, a quantum channel is a completely positive and trace preserving linear map on the space of $D\times D$ matrices. We consider ergodic sequences of channels, obtained by sampling channel valued maps along the trajectories of an ergodic dynamical system. The repeated composition of these maps along such a sequence could represent the result of repeated application of a given quantum channel subject to arbitrary correlated noise. It is physically natural to assume that such repeated compositions are eventually strictly positive, since this is true whenever any amount of decoherence is present in the quantum evolution. Under such an hypothesis, we obtain a general ergodic theorem showing that the composition of maps converges exponentially fast to a rank-one -- “entanglement breaking’’ – channel. We apply this result to describe the thermodynamic limit of ergodic matrix product states and prove that correlations of observables in such states decay exponentially in the bulk. (Joint work with Ramis Movassagh)

Title: Modular symbols and the arithmetic of cyclotomic fields

Date: 10/10/2019

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

The arithmetic of cyclotomic fields, and the structure of their class groups, has been studied since the time of Kummer in connection with Fermat’s Last Theorem. The work of Ribet in 1976 uncovered a subtle influence of the geometry of modular curves on this structure. I’ll discuss how this connection goes even deeper and define a surprisingly explicit map from the homology group of a modular curve to a K-group related to the class group of a cyclotomic field. I’ll then indicate how this is turning out to be just one instance of a more general phenomenon, touching briefly on joint work with Takako Fukaya and Kazuya Kato and separate joint work with Akshay Venkatesh.

Speaker: Arvind Krishna Saibaba, North Carolina State University

Title: Randomized algorithms for low-rank tensor decompositions

Date: 10/11/2019

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

Many applications in data science and scientific computing require the working with large-scale datasets that are expensive to store and manipulate. These datasets have inherent multidimensional structure that can be exploited in order to efficiently compress and
store them in an appropriate tensor format. In recent years, randomized matrix methods have been used to efficiently and accurately compute low-rank matrix decompositions. Motivated by this success, we develop several randomized algorithms for compressing
tensor datasets in the Tucker format. We present probabilistic error analysis for our algorithms and numerical results on several datasets: synthetic test tensors, and realistic applications including the compression of facial image samples in the Olivetti database, and word counts in the Enron email dataset.
Joint work with Rachel Minster (NC State) and Misha Kilmer (Tufts)

Title: An introduction to intersection forms: Taking K3 surface as an example

Date: 10/14/2019

Time: 3:00 PM - 3:50 PM

Place: C304 Wells Hall

I’ll define intersection product both on 4 manifolds and in the algebraic geometry setting, then introduce the blow up technique and give some easy examples. After that I will jump to K3 surface, give definition and constructions, and talk a little bit about the elliptic fibrations of K3. If I still have time, I will talk about the relation between intersection form and characteristic classes.

Title: The Isomorphism Theorems in an Abelian Category

Date: 10/14/2019

Time: 4:30 PM - 5:30 PM

Place: C304 Wells Hall

It is often said that abelian categories are where homology can "naturally" occur. As the notion of an isomorphism is indispensable to the study of homology---and an innate aspect of a category---, one would hope that there are analogues to the usual three isomorphism theorems of algebra in an arbitrary abelian category. In this [talk] we show that there are indeed such analogues, and we spend time developing the machinery to implement them

Title: Fenchel--Nielsen coordinates on Riemann surfaces and cluster algebras

Date: 10/15/2019

Time: 1:05 PM - 2:05 PM

Place: C304 Wells Hall

It is a 30(at least)-year old subject: it is known since long that both the standard Fenchel--NIelsen (lengths--twists) coordinates and (Y-)cluster coordinates (if we have holes) result in the same Goldman bracket on the set of geodesic functions on Riemann surfaces. The proof (of "local" nature in the first case and of "global" in the second) implies that these two sets of coordinates realise the same Poisson algebra. Nevertheless, constructing a direct transition between these two sets was elusive mainly due to complexity of the transition. For a sphere with 4 holes and torus with one hole, the corresponding formulas were obtained by Nekrasov, Rosly and Shatashvili in 2011. I present some preliminary results on the corresponding algebras in the general case and discuss possible relations to objects called Yang--Yang functionals.

Title: New and Old Combinatorial Identities Part I

Date: 10/16/2019

Time: 3:00 PM - 3:50 PM

Place: C304 Wells Hall

Binomial coefficients $n \choose k$ appear in different areas of mathematics (in Pascal's triangle, counting problems and computing probabilities to name few). There are also many identities that involve binomial coefficients. In this talk we will discuss new and old identities that represent positive integers and in some cases real numbers. These identities are derived from studying the asymptotic behavior of the roots of a generalized Fibonacci polynomial sequence
given by $F_{j}(x)=x^{j}-...-x-1$.

Title: Bubble Tree Convergence of Parametrized Associative Submanifolds

Date: 10/17/2019

Time: 2:00 PM - 3:00 PM

Place: C304 Wells Hall

In symplectic geometry, part of Gromov's Compactness Theorem asserts that sequences of holomorphic curves with bounded energy have subsequences that converge to bubble trees, and that both energy and homotopy are preserved in this "bubble tree limit." In $G_2$ geometry, the analogues of holomorphic maps are the "associative Smith maps." In this talk, we'll see that familiar analytic features of holomorphic maps also hold for associative Smith maps. In particular, we'll describe how sequences of associative Smith maps give rise to bubble trees, and how energy and homotopy are again preserved in the limit. This is joint work with Da Rong Cheng and Spiro Karigiannis.

Title: Higher convexity for complements of tropical objects

Date: 10/17/2019

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

Gromov generalized the notion of convexity for open subsets
of $\mathbf{R}^n$ with hypersurface boundary, defining $k$-convexity, or
higher convexity and Henriques applied the same notion to
complements of amoebas. He conjectured that the complement
of an amoeba of a variety of codimension $k+1$ is $k$-convex.
I will discuss work with Mounir Nisse in which we study the
higher convexity of complements of coamoebas and of tropical
varieties, proving Henriques' conjecture for coamoebas and
establishing a form of Henriques' conjecture for tropical varieties in some cases.

Title: When the Game Changes: The Development of Student Agency and Autonomy in Challenging Undergraduate Mathematics

Date: 10/21/2019

Time: 12:00 PM - 1:00 PM

Place: 115 Erickson Hall

In their pre-college and introductory collegiate mathematics coursework, students learn that mathematics centrally, if not exclusively involves computation. But many who pursue STEM disciplines routinely could encounter a quite different kind of mathematical work: The composition and evaluation of formal mathematical arguments, including proofs. The locus of this shift in mathematical activity on the MSU campus is MTH 299, Transitions, which introduces students to the basics of proof and argument. In the talk, we will present our current work conceptualizing agency and autonomy, the students who take the course, the challenges they face, and what we are learning about their experience in the course. We hope that these lessons will prove useful to all efforts to enrich introduction to proof mathematics courses.

When studying knots, we can often get a lot of information by removing the knot from space, and looking at the knot complement. It's pretty natural to ask, then, what happens to the area close to the removed knot? We call these areas cusps, and, in the case of hyperbolic knots, the cusp alone can tell us quite a lot. In this talk, we will give an introduction to these cusps, including their uses in topology, as well as how to find invariants from them.

Serre duality was first proved by Serre in 1950s. It is a very useful tool in algebraic and complex geometry. In this lecture, I will use Čech cohomology to prove Serre duality of projective varieties. If time permits, I would like to talk about some applications of it.

Title: Fenchel--Nielsen coordinates on Riemann surfaces and cluster algebras

Date: 10/22/2019

Time: 3:00 PM - 4:00 PM

Place: C204A Wells Hall

It is a 30(at least)-year old subject: it is known since long that both the standard Fenchel--NIelsen (lengths--twists) coordinates and (Y-)cluster coordinates (if we have holes) result in the same Goldman bracket on the set of geodesic functions on Riemann surfaces. The proof (of "local" nature in the first case and of "global" in the second) implies that these two sets of coordinates realise the same Poisson algebra. Nevertheless, constructing a direct transition between these two sets was elusive mainly due to complexity of the transition. For a sphere with 4 holes and torus with one hole, the corresponding formulas were obtained by Nekrasov, Rosly and Shatashvili in 2011. I present some preliminary results on the corresponding algebras in the general case and discuss possible relations to objects called Yang--Yang functionals.

Title: Local-global principle for norms over semi-global fields

Date: 10/23/2019

Time: 3:00 PM - 3:50 PM

Place: C304 Wells Hall

Let $K$ be a complete discretely valued field with
residue field $\kappa$.
Let $F$ be a function field in one variable over $K$
and $\mathscr{X}$ a regular proper model of $F$
with reduced special fibre $X$ a union of regular curves
with normal crossings.
Suppose that the graph associated to
$\mathscr{X}$ is a tree (e.g. $F = K(t)$).
Let $L/F$ be a Galois extension of degree $n$ with Galois group $G$
and $n$ coprime to char$(\kappa)$.
Suppose that $\kappa$ is algebraically closed field or
a finite field containing a primitive $n^{\rm th}$ root of unity.
Then we show that an element in $F^*$ is a norm
from the extension $L/F$ if it is a norm from the
extensions $L\otimes_F F_\nu/F_\nu$
for all discrete valuations $\nu$ of $F$.

We consider the XXZ chain in the Ising phase. The particle number conservation property is used to write the Hamiltonian in a hard-core particles formulation over the $N$-symmetric product of graphs, where $N\in\mathbb{N}_0$ is the number of conserved particle. The droplet regime corresponds to a band at the bottom of the spectrum of the model consisting of a connected set (a droplet) of down-spins, up to an exponential error. It is interesting to know that in the formulation over the $N$-symmetric product graphs, with a fixed $N\geq 1$, the XXZ chain can be seen as a one-dimensional model only when it is restricted to droplet states. This justifies the recent many-body localization indicators proved in the droplet regime by Elgart/Klein/Stolz and Beaud/Warzel for the disordered model, including an area law of arbitrary states in that localized phase. As a first step beyond the droplet regime, we show that the entanglement of arbitrary states above the droplet regime (associated with multiple droplets/clusters) does not follow area laws, and instead, it follows a logarithmically corrected (enhanced) area law. We will comment on the effects of disorder on entanglement, and show how our results hint a phase transition.
(joint work with C. Fischbacher and G. Stolz, arXiv1907.11420)

Title: Obstructing Lagrangian link cobordisms via Heegaard Floer homology.

Date: 10/24/2019

Time: 2:00 PM - 3:00 PM

Place: C304 Wells Hall

I'll explain how an invariant of Legendrian links in knot Floer homology can be used to obstruct the existence of decomposable Lagrangian link cobordisms in a very general setting. The argument involves braiding the ends of the cobordism about open books and appealing to an algebraic property of the Legendrian invariant called comultiplication. Much of the talk will be spent describing the topological and contact geometric ingredients.

Title: Geodesic X-ray Transforms and Boundary Rigidity

Date: 10/24/2019

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

This talk will introduce the problem of injectivity and
inversion of geodesic X-ray transforms in various geometric settings. The
associated nonlinear boundary rigidity problem, which consists of
determining a Riemannian metric on a compact manifold-with-boundary from
the lengths of its geodesics joining boundary points, will also be
discussed. Classical results and recent progess will be described,
including current research on the analogous questions in the setting of
asymptotically hyperbolic manifolds.

The Yang-Mills equation is a celebrated topic that is studied in differential geometry and particle physics. We will motivate the equation as a generalization of Maxwell's equations, define the relevant geometrical objects and discuss their properties.

Title: Partial differential equations from evolutionary ecology

Date: 10/28/2019

Time: 4:00 PM - 5:00 PM

Place: C304 Wells Hall

In collaboration with the AWM Student Chapter, we are most happy to welcome Professor Turanova to MSU! The abstract of her talk is:
I will describe the analysis of some PDEs that arise as models of ecological and evolutionary processes. There will be (some discussion of) poisonous toads!

Serre duality was first proved by Serre in 1950s. It is a very useful tool in algebraic and complex geometry. In this lecture, I will use Čech cohomology to prove Serre duality of projective varieties. If time permits, I would like to talk about some applications of it.

Title: Character varieties, Coulomb branches, and clusters.

Date: 10/29/2019

Time: 3:00 PM - 4:00 PM

Place: C204A Wells Hall

Quantum groups admit two different geometric realizations: as quantized character varieties and as quantized Coulomb branches of certain gauge theories. These realizations endow a quantum group with two, a priori different, cluster structures. In this talk I will show these structures, explain why they coincide, and say what they have to do with Gelfand-Tsetlin subalgebras, higher rank Fenchel–Nielsen coordinates, and modular functor from higher Teichmüller theory. This talk will be based on joint works with Gus Schrader.

Title: New and Old Combinatorial Identities Part II

Date: 10/30/2019

Time: 3:00 PM - 3:50 PM

Place: C304 Wells Hall

Using a probabilistic approach, we derive some interesting identities involving beta functions. These results generalize certain well-known combinatorial identities involving binomial coefficients and gamma functions.

Title: Classification of links with Khovanov homology of minimal rank

Date: 10/31/2019

Time: 2:00 PM - 3:00 PM

Place: C304 Wells Hall

In this talk, I will present a classification of links whose Khovanov homology has minimal rank, which answers a question asked by Batson and Seed. The proof is based on an excision formula for singular instanton Floer homology that allows the excision surface to intersect the singularity. We will use the excision theorem to define an instanton Floer homology for tangles on sutured manifolds, and show that its gradings detect the generalized Thurston norm for punctured surfaces. This is joint work with Yi Xie.

Title: Dynamics in models of coagulation and fragmentation

Date: 10/31/2019

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

Coaglation-fragmentation equations are simple, nonlocal models for evolution of the size distribution of clusters, appearing widely in science and technology. But few general analytical results characterize their dynamics. Solutions can exhibit self-similar growth, singular mass transport, and weak or slow approach to equilibrium. I will review some recent results in this vein, discussing: the cutoff phenomenon (as in card shuffling) for Becker-Doering equilibration; stationary and spreading profiles in a data-driven model of fish school size; and temporal oscillations recently found in models lacking detailed balance. A special role is played by Bernstein transforms and complex function theory for Pick or Herglotz functions.