Notre Dame Graduate Student Seminar

# Notre Dame Graduate Student Seminar This is a seminar organized by graduate students at the University of Notre Dame. We meet on Thursdays at 2 PM EST unless otherwise specified - modified days and times are bolded in the schedule. We are able to accomodate remote as well as in-person speakers and viewers. This is the [Zoom link](https://notredame.zoom.us/j/93888654312), while the meeting room for in-person talks will be announced via email. If you would like to be added to the email list for this seminar or contribute a talk, please contact Ilya Marchenko at <TT>imarchen at nd dot edu</TT> or Lorenzo Riva at <TT>lriva at nd dot edu</TT>. You can choose to present on any day marked with _Open slot_ in the schedule below. ## Spring 2022 Schedule |Name|Affiliation|Title|Date| | --- | --- | --- | --- | | Panel | Various | Panel on Academic Jobs | 4/21/2022 | |Eric Jovinelly|University of Notre Dame| Extreme Divisors on $M_{0,7}$ and Differences over Characteristic $2$ |**Monday 4/18/2022 4:00 PM**| | Will Dudarov | University of Washington | Colored Gelfand-Tsetlin Patterns and Symmetric Lattice Models | 4/14/2022 | | Gurutam Thockchom | University of Notre Dame | Operads and First-Order Theories| **Tuesday 4/5/2022 12:30 PM** | | Misha Gekhtman | University of Notre Dame | Five Glimpses of Cluster Algebras | **Monday 4/4/2022 2:00 PM** | | David Galvin | University of Notre Dame | Stirling numbers and the normal order problem | 3/31/2022 | | Felix Janda | University of Notre Dame | Counting covers of a torus | 3/24/2022 | | Philippe Mathieu | University of Notre Dame | Extensions of the Abelian Turaev-Viro construction and $\mathrm{U}\!\left(1\right)$ BF theory to any finite dimensional smooth oriented closed manifold | **Wednesday 3/23/2022 3:30 PM** | | Anish Chedalavada | University of Illinois Chicago | An introduction to tensor-triangular geometry | 3/17/2022 | | -- | -- | _Spring Break_ | 3/10/2022 | | Jeff Diller | University of Notre Dame | Dynamics of rational maps: a best case example and reasonable hopes | 3/3/2022 | | Minh Chieu Tran | University of Notre Dame | o-minimal method and generalized sum-product phenomena | 2/24/2022 | | Pavel Mnev | University of Notre Dame | On the Fukaya-Morse A-infinity category | 2/17/2022 | | Alex Himonas | University of Notre Dame |Analysis of the Korteweg-de Vries (KdV) equation| 2/10/2022 | | Lorenzo Riva | University of Notre Dame | You could have formulated the Cobordism Hypothesis | 1/27/2022 | ## Fall 2021 Schedule | Name| Affiliation| Title| Date| | ----------- | --- | -------- | -------- | |Nikolai Konovalov|University of Notre Dame|Polynomial functors and Steenrod algebra|11/15/2021| |Eric Riedl|University of Notre Dame|Geometric Manin's Conjecture in Fano Threefolds|11/9/2021| |Multiple Panelists|Various|Panel on non-academic jobs for Mathematicians|11/1/2021| |Nick Salter|University of Notre Dame|Life After Galois|10/11/2021| |Sarah Petersen|University of Notre Dame|The $RO(C_2)$ - Homology of $C_2$ - Equivariant Eilenberg-Maclane Spaces|10/6/2021 |Luan Minh Doan|University of Notre Dame|Segal–Bargmann spaces, coherent state transforms, and the large-$N$ limit problem|9/20/2021| |Richard Birkett|University of Notre Dame|Dynamically Stabilising Birational Surface Maps: Two Methods|9/13/2021| |Connor Malin|University of Notre Dame|Demystifying buzzwords: real homotopy theory is derived algebra|9/8/2021| ## Summer 2021 Schedule | Name| Affiliation| Title| Date| | ----------- | --- | -------- | -------- | |Randy Van Why|Northwestern University|Disk bundle plumbings, lens spaces, and continued fractions|7/12/2021| |Hari Rau-Murthy|University of Notre Dame|The matrix exponential, the Bismut Chern character, and the character of a representation|6/14/2021| ## Winter/Spring 2021 Schedule | Name| Affiliation| Title| Date| | ----------- | --- | -------- | -------- | |Wern Yeong|University of Notre Dame|Survey of techniques and progress toward the Green-Griffiths-Lang and Kobayashi conjectures on hyperbolicity|4/26/2021| |Pavel Mnev|University of Notre Dame|Topological quantum mechanics, Stasheff’s associahedra and homotopy transfer of algebraic structures|4/19/2021| |Juanita Pinzon Caicedo|University of Notre Dame|Instantons and Knot Concordance|4/12/2021| |Nikolai Konovalov|University of Notre Dame|Quillen F-isomorphism Theorem|4/5/2021| |Gabor Szekelyhidi|University of Notre Dame|Gromov-Hausdorff limits of Kahler manifolds|3/29/2021| |Ethan Reed|Universiy of Notre Dame|A Proof of Stillman's Conjecture Using Ultraproducts|2/22/2021| |Elia Portnoy|MIT|Distortions of Embedded Knots|1/25/2021| | Richard Birkett |University of Notre Dame| Berkovich Space | 1/13/2021 | | John Siratt|University of Notre Dame|The Strength of Büchi's Decidability Theorem|1/4/2021| ## Abstracts ### 4/19/2022 Eric Jovinelly The cone of effective divisors controls the rational maps from a variety. We study this important object for $M_{0,n}$, the moduli space of stable rational curves with $n$ markings. Fulton once conjectured the effective cones for each nwould follow a certain combinatorial pattern. However, this pattern holds true only for $n < 6$. Despite many subsequent attempts to describe the effective cones for all $n$, we still lack even a conjectural description. We study the simplest open case, $n=7$, and identify the first known difference between characteristic $0$ and characteristic $p$. Although a full description of the effective cone for $n=7$ remains open, our methods allowed us to compute the entire effective cones of spaces associated with other stability conditions. ### 4/14/2022 Will Dudarov In the search for a bijective version of the proof presented in Weyl Group Multiple Dirichlet Series by Brubaker, Bump, and Friedberg, which states that two definitions of the p-parts of a multiple Dirichlet series given a Type A root system are equal, our group, mentored by Ben Brubaker himself, tackled defining new combinatorial objects: colored symmetric lattice models and their corresponding colored Gelfand-Tsetlin patterns. In this talk, we will discuss the main problem, which is motivated by representation theory and number theory, how our definitions tackle this problem, and our conjecture towards a bijective proof. In particular, we conjecture a generalization of the Schützenberger involution to these new classes of combinatorial objects. ### 4/5/2022 Gurutam Thockchom Operads of algebraic topology and first-order theories of mathematical logic both give ways to describe the structure of objects. In this talk I define an operad and an algebra over an operad, and explore some correspondences between operads and theories in first-order logic. I will also go over some obstructions to a complete correspondence between these ideas, and the cost of resolving one of these obstructions. To conclude, I will discuss the role operads might play in a certain generalization of first-order logic. ### 4/4/2022 Misha Gekhtman Cluster algebras were introduced by Fomin and Zelevinsky 20 years ago and have since found exciting applications in many areas including algebraic geometry, representation theory, integrable systems and theoretical physics. I will use examples to explain a definition of a cluster algebra and then sketch several applications of the theory, including Somos-5 recursion, pentagram map and generalizations of Abel's pentagon identity. ### 3/31/2022 David Galvin The Stirling numbers of the second kind, introduced in 1730, arise in many contexts —combinatorial, analytic, algebraic, probabilistic... I'll introduce these versatile numbers, and describe some of their interpretations and applications. The standard combinatorial interpretation of the Stirling numbers involves set partitions, and this interpretation has a natural generalization to graphs. I’ll discuss an application of this generalization to a problem coming from the Weyl algebra (the algebra on alphabet $\{x, D\}$ with the single relation $Dx=xD+1$). This is joint work with J. Hilyard and J. Engbers. ### 3/24/2022 Felix Janda I will outline a connection (observed by Dijkgraaf) between counts of ramified covers of a 2-torus, Fourier expansions of modular forms, and mirror symmetry. ### 3/23/2022 Philippe Mathieu In 1992, V. Turaev and O. Viro defined an invariant of smooth oriented closed $3$-manifolds consisting of labelling the edges of a triangulation of the manifold with representations of $\mathcal{U}_{q}\!\left(\mathfrak{sl}_{2}\left(\mathbb{C}\right)\right)$ ($q$ being a root of unity), associating a (quantum) $6j$-symbol to each tetrahedron of the triangulation, taking the product of the $6j$-symbols over all the tetrahedra of the manifold, then summing over all the admissible labelling representations. It is commonly admitted that this construction is a regularization of a path integral occurring in quantum gravity, the so-called "Ponzano-Regge model", which is a kind of SU(2) BF gauge theory. A naive question is: Is it possible to define an abelian version of this invariant? If yes, is there a relation with an abelian BF gauge theory? These questions were answered positively in 2016, and the corresponding Turaev-Viro invariant is built from $\mathbb{Z}/k\mathbb{Z}$ labelling representations (the equivalent of $6j$-symbols being "modulo $k$" Kronecker symbols) while the associated gauge theory is a particular $\mathrm{U}\!\left(1\right)$ BF theory (with coupling constant $k$). This $\mathrm{U}\!\left(1\right)$ BF theory can be straightforwardly extended to any finite dimensional closed oriented manifold, and so can be the Turaev-Viro construction built from $\mathbb{Z}/k\mathbb{Z}$ labelling representations. A natural question is thus: Are these extensions still related? I will answer this question during the talk. ### 3/17/2022 Anish Chedalavada In this talk, I will discuss a perspective on tensor-triangulated categories that has evolved in the last few decades which provides a unifying theme behind the chromatic picture of the stable homotopy category, stable categories of modules in modular representation theory, and derived categories of quasicoherent sheaves on a coherent scheme. Heuristically, this is based on an analogy between tt-categories and commutative rings, with thick tensor ideals playing the mediating role between the tt-category and topological spaces. Time permitting, I will also (god forbid) attempt a computation and point to some interesting questions from algebraic geometry that are still waiting to be generalized. ### 3/3/2022 Jeff Diller A map $f:\mathbb{C}^n\to \mathbb{C}^n$ is rational if its component functions are themselves rational. A result of Gromov from the 70s suggests that the complexity of the dynamics (i.e. behavior of iterates) of $f$ should be closely tied to the way the degrees of the components grow under iteration. I'll lead with a particularly good example of this connection, present Gromov's result and, time permitting, say some things about the proof and a program for further understanding the dynamics of a rational map. ### 2/24/2022 Minh Chieu Tran I will discuss a joint work with Yifan Jing and Souktik Roy where we show that for a bivariate polynomial $P(x,y) \in \mathbb R[x,y]\setminus (\mathbb R[x] \cup \mathbb R[y])$ to exhibit small expansion on a finite set $A \subseteq \mathbb R$, we must have $$ P(x,y)=f(\gamma u(x)+\delta u(y)) \text{ or } P(x,y)=f(u^m(x)u^n(y)) $$ for some univariate $f, g, u \in \mathbb R[t]\setminus \mathbb R$, constants $\gamma, \delta \in \mathbb R^{\neq 0}$, and $m, n\in \mathbb N^{\geq 1}$. This yields an Elekes-Ronyai type structural result for symmetric nonexpanders, resolving a question mentioned by de Zeeuw. Our result uses o-minimal/semialgebro geometric techniques to replace algebraic geometric techniques, which are only applicable to more special cases. ### 2/17/2022 Pavel Mnev I will sketch the construction of the Fukaya-Morse category of a Riemannian manifold X -- an A-infinity category (a category where associativity of composition holds only "up-to-homotopy") where the higher composition maps are given in terms of numbers of embedded trees in X, with edges following the gradient trajectories of certain Morse functions. I will give simple examples and explain different approaches to understanding the structure and proving the quadratic relations on the structure maps -- (1a) via homotopy transfer, (1b) effective field theory approach, (2) topological quantum mechanics approach. (Maybe a subset of this, depending on time.) The talk is based on a joint work with O. Chekeres, A. Losev and D. Youmans, arXiv:2112.12756. ### 2/10/2022 Alex Himonas The KdV equation is one of the most ubiquitous models in mathematics and physics. It was first derived by Boussinesq in 1877 in his effort to demystify Russell's observation of what he called the "great wave of translation" in a Union Canal near Edinburgh in 1834. Korteweg and de Vries rediscovered the KdV equation in 1895 and confirmed that it has traveling wave solutions (solitons). The solving of the KdV initial value problem, with nice data, was initiated by Gardner, Greene, Kruskal, and Miura in 1967 by recognizing its remarkable integrability properties. Its solving for rough ($L^2$) data was accomplished by Bourgain in 1993 using novel ideas from classical and harmonic analysis. In 1996, Kenig, Ponce, and Vega advanced these ideas and solved the KdV with data of regularity below $L^2$. It is interesting that these ideas are also needed if one studies KdV solutions with the nicest possible data (analytic). In this talk we will try to present the key points of this remarkable KdV story. ### 1/27/2022 Lorenzo Riva In this talk we will try to study manifolds in their totality: all $n$-manifolds, or all manifolds up to a fixed dimension, or all manifolds equipped with a certain structure. It turns out that these collections form specific algebraic structures that are fully characterized by the Cobordism Hypothesis, formulated by Baez-Dolan and proven by Lurie. In particular, we will talk about building manifolds by inductively gluing cobordisms, some category theory, and the algebraic restrictions imposed by a topological field theory, which is a representation of such an "algebra of manifolds". The talk should be accessible to everyone, but in particular it will interesting for topologists, algebraists, and mathematical physicists. ### 11/15/2021 Nikolai Konovalov The Steenrod algebra is an useful and classical tool in algebraic topology which appears naturally as the algebra of cohomology operations. However, there is a completely topology-free way to construct it. Namely, one could consider the category of polynomial functors from the category of vector spaces over $\mathbb F_p$ to itself. Then this category embeds fully faithfully into the category of modules over the Steenrod algebra. In my talk, I am going to explain this result and also discuss the image of the embedding. ### 11/9/2021 Eric Riedl Manin's Conjecture predicts the number of rational points on a variety with bounded height. Like many conjectures in Number Theory, it appears far out of reach to prove in general. However, we can obtain evidence for this conjecture by considering the analogous version for varieties defined over the function field of a curve, namely, Geometric Manin's Conjecture. In this talk, we describe why Geometric Manin's Conjecture is the analogue of the number theory conjecture, then we talk about some results toward proving Geometric Manin's Conjecture for Fano Threefolds. ### 11/1/2021 Panel on Non-Academic Jobs This seminar will be a panel discussion featuring four Notre Dame PhD's in mathematics who are currently working in a job outside of academia, or in academia but not as a math professor. The panelists are: Ben Jones ( Notre Dame PhD, 2007, Senior Applied Scientist at Amazon Web Services, Portland) Katie Meixner (Notre Dame PhD, 2013, Data Scientist at Johns Hopkins Applied Physics Lab) Danny Orton (Notre Dame PhD, 2019, Physicist at Peraton in Colorado Springs) Jeremy Mann (Notre Dame PhD, 2020, Senior Data Scientist at Concert AI, Oakland, CA) The purpose of this panel discussion is to give current Notre Dame students and other interested parties a better idea of career options outside of academia after getting a math PhD. Each panelist will discuss their own career path and useful skills to develop for non-academic jobs. We hope this will be the first of several panel discussions, and that the panelists and others can serve as useful resources for Notre Dame math students who are exploring or are interested in working outside of academia. ### 10/11/2021 Nick Salter The famous Abel-Ruffini theorem asserts that there is no formula for expressing the roots of a general fifth-degree polynomial using only radicals. Rather than being the definitive end to a story, it turns out that there are many further aspects of root-finding left for contemporary (and indeed future) mathematicians to discover. In this talk, I will discuss some more modern perspectives on root-finding that showcase the roles played by topology and by dynamics. ### 10/6/2021 Sarah Petersen This talk describes work in progress computing the $H\underline{\mathbb{F}}_2$ homology of the $C_2$ - equivariant Eilenberg-Maclane spaces associated to the constant Mackey functor $\underline{\mathbb{F}}_2.$ We expand a Hopf ring argument of Ravenel-Wilson computing the mod p homology of non-equivariant Eilenberg-Maclane spaces to the $RO(C_2)$-graded setting. An important tool that arises in this equivariant context is the twisted bar spectral sequence which is quite complicated, lacking an explicit $E_2$ page and having arbitrarily long equivariant degree shifting differentials. We avoid working directly with these differentials and instead use a computational lemma of Behrens-Wilson along with norm and restriction maps to complete the computation. ### 9/20/2021 Luan Minh Doan The Segal–Bargmann transforms, or coherent state transforms, have been interesting objects of study in mathematical physics. In the operator-theoretic settings, a coherent state transform can be described as a unitary linear map from a Hilbert space of square summable functions $L^2(M^N,d\rho N)$ to a Hilbert space of entire functions $\mathcal{HL}^2(M^N_\mathbb C,d\mu^N)$, where $M^N$ is a configuration space (of dimension $N$), $M^N_\mathbb C$ is the complexified version of $M^N$, and $\rho^N$ and $\mu^N$ are appropriate heat kernel measures. We will look at the case $M^N = \mathbb R^N$ and briefly discuss the case where $M^N$ is a compact Lie group or a compact symmetric group. We will also see some examples in which there are convergence phenomena of both the Hilbert spaces and the transforms when $N$ is large. ### 9/13/2021 Richard Birkett Given a rational map $f: X \dashrightarrow X$, we have a natural pullback operator on curves $f^* : \text{Pic}(X) \to \text{Pic}(X)$, and understanding the sequence of actions $(f^n)^*$ is fundamental to understanding the dynamics of $f$. It is important to stress however that pullback need not be functorial, meaning $(f^n)^*$ may not be the same as $(f^*)^n$ for $n \ne 1$. When this does hold for every $n \in \mathbb{N}$, we call $f$ *algebraically stable*. Without algebraic stability $(f^n)^*$ is nearly intractable, furthermore it is difficult to construct certain natural invariant divisors, currents and measures of the dynamics of $f$. The situation is better when the map $f$ is *birational*. It was shown by Diller and Favre (2001) that if $f: X \dashrightarrow X$ is a birational map then there exists a surface $\hat X$ and a birational morphism $\pi : \hat X \to X$ which lifts $f$ to an algebraically stable $\hat f : \hat X \dashrightarrow \hat X$. I will discuss these interesting issues with pullbacks, then sketch an old and a new elementary method to lift a birational map to an algebraically stable one. ### 9/8/2021 Connor Malin In this talk, I aim to demystify two intimidating notions: homotopy theory and derived mathematics. After describing what these things really are, I will show their versatility and apply them to studying classical algebraic topology. ### 7/12/2021 Randy Van Why I will attempt to state and prove a theorem showing a strange connection between continued fractions, lens spaces, and disk bundle plumbings. I will start by introducing the plumbing construction and surgery and then sketch a proof of the theorem. ### 6/14/2021 Hari Rau-Murthy The matrix exponential can be used to solve the differential equation $\frac{d\gamma}{dt} = A(t)\gamma(t)$. We will discuss a cool trick involving this matrix exponential. This trick will be used to define the Bismut Chern character, which is the trace of a certain matrix exponential associated to a loop, $\gamma(t)$, in a manifold. The Bismut Chern character has striking connections to the group theoretic character of a representation, which is trace of a matrix that represents an element, $g$, of a group. The loop $\gamma(t)$ will end up corresponding to the conjugacy class of $g$. Thus we relate a differential geometric construction to an algebraic construction. ### 4/26/2021 Wern Yeong The Green-Griffiths-Lang and Kobayashi conjectures make connections between concepts in geometry and number theory, e.g. curvature, positivity, entire curves and rational points. These conjectures can be thought of as generalizations of the behavior of Riemann surfaces depending on their geometric genus, which is the starting point of this talk. Then we discuss some techniques and progress toward these conjectures, and their algebraic analogs. Aimed for a general math audience. ### 4/19/2021 Pavel Mnev I will explain the setup of topological quantum mechanics and how its natural extension to spacetimes being metric trees leads to the construction of a family of differential forms $I_n$ on the moduli space of metric trees (a.k.a. Stasheff’s associahedron). Periods of these differential forms give the Kontsevich-Soibelman sum-over-trees formula for the $A_\infty$ algebra structure on the cohomology of a differential graded algebra (e.g. Massey operations on de Rham cohomology). Higher associativity relations for the $A_\infty$ structure correspond in this construction to the factorization property of the differential forms $I_n$ on the compactification strata of the moduli space. ### 4/12/2021 Juanita Pinzon Caicedo Knot concordance can be regarded as the study of knots as boundaries of surfaces embedded in spaces of dimension 4. Specifically, two knots $K_0$ and $K_1$ are said to be smoothly concordant if there is a smooth embedding of the annulus $S^1 × [0, 1]$ into the “cylinder” $S^3 × [0, 1]$ that restricts to the given knots at each end. Smooth concordance is an equivalence relation, and the set $C$ of smooth concordance classes of knots is an abelian group with connected sum as the binary operation. The algebraic structure of $C$, the concordance class of the unknot, and the set of knots that are topologically slice but not smoothly slice are much studied objects in low-dimensional topology. Gauge theoretical results on the nonexistence of certain definite smooth 4-manifolds can be used to better understand these objects. In particular, the study of anti-self dual connections on 4-manifolds can be used to shown that (1) the group of topologically slice knots up to smooth concordance contains a subgroup isomorphic to $\mathbb Z^\infty$, and (2) satellite operations that are similar to cables are not homomorphisms on $C$. ### 4/5/2021 Nikolai Konovalov Group cohomology is notoriously hard and yet interesting invariant. In 1971, D. Quillen suggested a way to approximate (up to nilpotents) the mod p cohomology ring of a group in terms of cohomology rings of its elementary p-subgroup. In my talk, I am going to briefly discuss the proof of his theorem, possible corollaries, and if time permits, one interesting generalization in terms of the Steenrod operations. ### 3/29/2021 Gabor Szekelyhidi I will give an overview of the Cheeger-Colding theory of Gromov-Hausdorff limits of Riemannian manifolds with Ricci curvature bounds, and the more recent results of Donaldson-Sun on the additional structure one obtains in the Kahler case. ### 2/22/2021 Ethan Reed Following the work of Erman, Sam, and Snowden, we give a proof of Stillman’s conjecture. As in the work of Ananyan and Hochster, we first reduce to the "existence of small algebras". This reduces to proving a specific manifestation of the Ananyan-Hochster Principle, which we prove using ultraproducts. ### 1/25/2021 Elia Portnoy Is it true that any knot can be embedded in $\mathbb R^3$ so that the ratio between the intrinsic distance and induced distance is not too large? Pardon first showed that the answer was surprisingly, No! Later Gromov and Guth constructed knots with arbitrarily large distortion using hyperbolic geometry of 3-manifolds. I will sketch some ideas in Gromov and Guth's construction. ### 1/13/2021 Richard Birkett Berkovich spaces have proven themselves a great tool in algebraic geometry and dynamics in recent years. This talk serves as a crash course or introduction. I will begin with the opaque algebraic definition and provide a perspective of how Berkovich space is actually an incredible geometric lens for seeing many varieties at once. Only basic algebra or geometry needed to follow the talk, although familiarity with divisors on a plane provide important motivation. ### 1/4/2021 John Siratt Although second order logic is not decidable in general, there are interesting theories in fragments of second order logic that are decidable. Büchi's Decidability Theorem for the monadic second order theory of the natural numbers with the less-than-or-equal-to relation is one such decidability result. Recent work by Kolodziejczyk, Michaelewski, Pradic, and Skrzypczak, has shown that the logical strength of Buchi decidability over $\text{RCA}_0$ is equivalent to induction over $\Sigma_2$ formulas. We will present some needed background including some automata theory and reverse mathematics, then sketch a modern proof of Buchi decidability. This proof can be easily modified to establish one direction of the strength result. Finally, we will give a short overview of the authors' proof showing the other direction.