Let $X$ be a 2-dimensional simplicial complex. Denote by $\text{ex}_{\hom}(n,X)$ the maximum number of 2-simplices in an $n$-vertex simplicial complex that has no sub-simplicial complex homeomorphic to $X$. The asymptotics of $\text{ex}_{\hom}(n,X)$ have recently been determined for all surfaces $X$. I will discuss these results, as well as ongoing work bounding $\text{ex}_{\hom}(n,X)$ for arbitrary 2-dimensional simplicial complexes $X$.

We show fixed-parameter tractability of the Directed Multicut problem with three terminal pairs (with a randomized algorithm). This problem, given a directed graph $G$, pairs of vertices (called terminals) $(s_1,t_1)$, $(s_2,t_2)$, and $(s_3,t_3)$, and an integer $k$, asks to find a set of at most $k$ non-terminal vertices in $G$ that intersect all $s_1t_1$-paths, all $s_2t_2$-paths, and all $s_3t_3$-paths. The parameterized complexity of this case has been open since Chitnis, Cygan, Hajiaghayi, and Marx proved fixed-parameter tractability of the 2-terminal-pairs case at SODA 2012, and Pilipczuk and Wahlström proved the W[1]-hardness of the 4-terminal-pairs case at SODA 2016. On the technical side, we use two recent developments in parameterized algorithms. Using the technique of directed flow-augmentation [Kim, Kratsch, Pilipczuk, Wahlström, STOC 2022] we cast the problem as a CSP problem with few variables and constraints over a large ordered domain. We observe that this problem can be in turn encoded as an FO model-checking task over a structure consisting of a few 0-1 matrices. We look at this problem through the lenses of twin-width, a recently introduced structural parameter [Bonnet, Kim, Thomassé, Watrigant, FOCS 2020]: By a recent characterization [Bonnet, Giocanti, Ossona de Mendez, Simon, Thomassé, Toruńczyk, STOC 2022] the said FO model-checking task can be done in FPT time if the said matrices have bounded grid rank. To complete the proof, we show an irrelevant vertex rule: If any of the matrices in the said encoding has a large grid minor, a vertex corresponding to the "middle" box in the grid minor can be proclaimed irrelevant — not contained in the sought solution — and thus reduced.

The aim of the lecture series is twofold: (1) give an overview of the “classical” theory of Breuil-Kisin modules, and (2) discuss its application to the construction of certain p-adic Galois deformation rings. In the first talk by Wansu Kim (Moday at 10am), we will give a general introduction to the p-adic Hodge theory (focusing on p-adic Galois representations). In the first half of the lecture series, Wansu Kim will explain the “classical” theory of Breuil-Kisin modules following the seminal paper of Kisin’s (Crystalline representations and F-crystals, from Drinfeld’s 50th birthday conference proceeding). In the second half of the lecture series, Chol Park will explain its application to the explicit computation of p-adic local Galois deformation rings cut out by certain p-adic Hodge-theoretic condition. Here is the time and venue for each talk: 13 Feb (Mon): 3 talks *) 10-11:30 & 14-15:30 at E6-1, Rm 1401 (최석정강의실) *) 16:00 -17:30 at E6-1, Rm 3434 14 Feb (Tue): 1 talk *) 16:15 -17:45 at E6-1, Rm 3434 15 Feb (Wed) to 16 Feb (Thu): *) 10-11:30 & 14-15:30 at E6-1, Rm 1401 (최석정강의실) *) (TBD) 16:00 -17:30 at E6-1, Rm 3434 (in case we need extra lecture) 17 Feb (Fri) *) 10-11:30 at E6-1, Rm 3438 *) (TBD) 14-15:30 at E6-1, Rm 3438 (in case we need extra lecture)

TBA

In the context of science, the well-known adage “a picture is worth a thousand words” might well be “a model is worth a thousand datasets.” In this manuscript we introduce the SciML software ecosystem as a tool for mixing the information of physical laws and scientific models with data-driven machine learning approaches. We describe a mathematical object, which we denote universal differential equations (UDEs), as the unifying framework connecting the ecosystem. We show how a wide variety of applications, from automatically discovering biological mechanisms to solving high-dimensional Hamilton-Jacobi-Bellman equations, can be phrased and efficiently handled through the UDE formalism and its tooling. We demonstrate the generality of the software tooling to handle stochasticity, delays, and implicit constraints. This funnels the wide variety of SciML applications into a core set of training mechanisms which are highly optimized, stabilized for stiff equations, and compatible with distributed parallelism and GPU accelerators.

The aim of the lecture series is twofold: (1) give an overview of the “classical” theory of Breuil-Kisin modules, and (2) discuss its application to the construction of certain p-adic Galois deformation rings. In the first talk by Wansu Kim (Moday at 10am), we will give a general introduction to the p-adic Hodge theory (focusing on p-adic Galois representations). In the first half of the lecture series, Wansu Kim will explain the “classical” theory of Breuil-Kisin modules following the seminal paper of Kisin’s (Crystalline representations and F-crystals, from Drinfeld’s 50th birthday conference proceeding). In the second half of the lecture series, Chol Park will explain its application to the explicit computation of p-adic local Galois deformation rings cut out by certain p-adic Hodge-theoretic condition. Here is the time and venue for each talk: 13 Feb (Mon): 3 talks *) 10-11:30 & 14-15:30 at E6-1, Rm 1401 (최석정강의실) *) 16:00 -17:30 at E6-1, Rm 3434 14 Feb (Tue): 1 talk *) 16:15 -17:45 at E6-1, Rm 3434 15 Feb (Wed) to 16 Feb (Thu): *) 10-11:30 & 14-15:30 at E6-1, Rm 1401 (최석정강의실) *) (TBD) 16:00 -17:30 at E6-1, Rm 3434 (in case we need extra lecture) 17 Feb (Fri) *) 10-11:30 at E6-1, Rm 3438 *) (TBD) 14-15:30 at E6-1, Rm 3438 (in case we need extra lecture)

The aim of the lecture series is twofold: (1) give an overview of the “classical” theory of Breuil-Kisin modules, and (2) discuss its application to the construction of certain p-adic Galois deformation rings. In the first talk by Wansu Kim (Moday at 10am), we will give a general introduction to the p-adic Hodge theory (focusing on p-adic Galois representations). In the first half of the lecture series, Wansu Kim will explain the “classical” theory of Breuil-Kisin modules following the seminal paper of Kisin’s (Crystalline representations and F-crystals, from Drinfeld’s 50th birthday conference proceeding). In the second half of the lecture series, Chol Park will explain its application to the explicit computation of p-adic local Galois deformation rings cut out by certain p-adic Hodge-theoretic condition. Here is the time and venue for each talk: 13 Feb (Mon): 3 talks *) 10-11:30 & 14-15:30 at E6-1, Rm 1401 (최석정강의실) *) 16:00 -17:30 at E6-1, Rm 3434 14 Feb (Tue): 1 talk *) 16:15 -17:45 at E6-1, Rm 3434 15 Feb (Wed) to 16 Feb (Thu): *) 10-11:30 & 14-15:30 at E6-1, Rm 1401 (최석정강의실) *) (TBD) 16:00 -17:30 at E6-1, Rm 3434 (in case we need extra lecture) 17 Feb (Fri) *) 10-11:30 at E6-1, Rm 3438 *) (TBD) 14-15:30 at E6-1, Rm 3438 (in case we need extra lecture)

The aim of the lecture series is twofold: (1) give an overview of the “classical” theory of Breuil-Kisin modules, and (2) discuss its application to the construction of certain p-adic Galois deformation rings. In the first talk by Wansu Kim (Moday at 10am), we will give a general introduction to the p-adic Hodge theory (focusing on p-adic Galois representations). In the first half of the lecture series, Wansu Kim will explain the “classical” theory of Breuil-Kisin modules following the seminal paper of Kisin’s (Crystalline representations and F-crystals, from Drinfeld’s 50th birthday conference proceeding). In the second half of the lecture series, Chol Park will explain its application to the explicit computation of p-adic local Galois deformation rings cut out by certain p-adic Hodge-theoretic condition. Here is the time and venue for each talk: 13 Feb (Mon): 3 talks *) 10-11:30 & 14-15:30 at E6-1, Rm 1401 (최석정강의실) *) 16:00 -17:30 at E6-1, Rm 3434 14 Feb (Tue): 1 talk *) 16:15 -17:45 at E6-1, Rm 3434 15 Feb (Wed) to 16 Feb (Thu): *) 10-11:30 & 14-15:30 at E6-1, Rm 1401 (최석정강의실) *) (TBD) 16:00 -17:30 at E6-1, Rm 3434 (in case we need extra lecture) 17 Feb (Fri) *) 10-11:30 at E6-1, Rm 3438 *) (TBD) 14-15:30 at E6-1, Rm 3438 (in case we need extra lecture)

The aim of the lecture series is twofold: (1) give an overview of the “classical” theory of Breuil-Kisin modules, and (2) discuss its application to the construction of certain p-adic Galois deformation rings. In the first talk by Wansu Kim (Moday at 10am), we will give a general introduction to the p-adic Hodge theory (focusing on p-adic Galois representations). In the first half of the lecture series, Wansu Kim will explain the “classical” theory of Breuil-Kisin modules following the seminal paper of Kisin’s (Crystalline representations and F-crystals, from Drinfeld’s 50th birthday conference proceeding). In the second half of the lecture series, Chol Park will explain its application to the explicit computation of p-adic local Galois deformation rings cut out by certain p-adic Hodge-theoretic condition. Here is the time and venue for each talk: 13 Feb (Mon): 3 talks *) 10-11:30 & 14-15:30 at E6-1, Rm 1401 (최석정강의실) *) 16:00 -17:30 at E6-1, Rm 3434 14 Feb (Tue): 1 talk *) 16:15 -17:45 at E6-1, Rm 3434 15 Feb (Wed) to 16 Feb (Thu): *) 10-11:30 & 14-15:30 at E6-1, Rm 1401 (최석정강의실) *) (TBD) 16:00 -17:30 at E6-1, Rm 3434 (in case we need extra lecture) 17 Feb (Fri) *) 10-11:30 at E6-1, Rm 3438 *) (TBD) 14-15:30 at E6-1, Rm 3438 (in case we need extra lecture)

In this talk, we derive effective dynamics governed by PDEs (nonlinear Schrodinger equation, Vlasov-Poisson equation) from the N-body Schrödinger equation with interactions in the large N limit. For the derivation, we first visit quantum mechanics. Then, using the knowledge of many-body quantum mechanics given in the talk and with the suitable conditions for each situation, we derive PDEs describing the effective motion of the system.

The aim of the lecture series is twofold: (1) give an overview of the “classical” theory of Breuil-Kisin modules, and (2) discuss its application to the construction of certain p-adic Galois deformation rings. In the first talk by Wansu Kim (Moday at 10am), we will give a general introduction to the p-adic Hodge theory (focusing on p-adic Galois representations). In the first half of the lecture series, Wansu Kim will explain the “classical” theory of Breuil-Kisin modules following the seminal paper of Kisin’s (Crystalline representations and F-crystals, from Drinfeld’s 50th birthday conference proceeding). In the second half of the lecture series, Chol Park will explain its application to the explicit computation of p-adic local Galois deformation rings cut out by certain p-adic Hodge-theoretic condition. Here is the time and venue for each talk: 13 Feb (Mon): 3 talks *) 10-11:30 & 14-15:30 at E6-1, Rm 1401 (최석정강의실) *) 16:00 -17:30 at E6-1, Rm 3434 14 Feb (Tue): 1 talk *) 16:15 -17:45 at E6-1, Rm 3434 15 Feb (Wed) to 16 Feb (Thu): *) 10-11:30 & 14-15:30 at E6-1, Rm 1401 (최석정강의실) *) (TBD) 16:00 -17:30 at E6-1, Rm 3434 (in case we need extra lecture) 17 Feb (Fri) *) 10-11:30 at E6-1, Rm 3438 *) (TBD) 14-15:30 at E6-1, Rm 3438 (in case we need extra lecture)

Hadwiger's famous coloring conjecture states that every t-chromatic graph contains a $K_t$-minor. Holroyd [Bull. London Math. Soc. 29, (1997), pp. 139-144] conjectured the following strengthening of Hadwiger's conjecture: If G is a t-chromatic graph and S⊆V(G) takes all colors in every t-coloring of G, then G contains a $K_t$-minor rooted at S. We prove this conjecture in the first open case of t=4. Notably, our result also directly implies a stronger version of Hadwiger's conjecture for 5-chromatic graphs as follows: Every 5-chromatic graph contains a $K_5$-minor with a singleton branch-set. In fact, in a 5-vertex-critical graph we may specify the singleton branch-set to be any vertex of the graph. Joint work with Anders Martinsson (ETH).

The aim of the lecture series is twofold: (1) give an overview of the “classical” theory of Breuil-Kisin modules, and (2) discuss its application to the construction of certain p-adic Galois deformation rings. In the first talk by Wansu Kim (Moday at 10am), we will give a general introduction to the p-adic Hodge theory (focusing on p-adic Galois representations). In the first half of the lecture series, Wansu Kim will explain the “classical” theory of Breuil-Kisin modules following the seminal paper of Kisin’s (Crystalline representations and F-crystals, from Drinfeld’s 50th birthday conference proceeding). In the second half of the lecture series, Chol Park will explain its application to the explicit computation of p-adic local Galois deformation rings cut out by certain p-adic Hodge-theoretic condition. Here is the time and venue for each talk: 13 Feb (Mon): 3 talks *) 10-11:30 & 14-15:30 at E6-1, Rm 1401 (최석정강의실) *) 16:00 -17:30 at E6-1, Rm 3434 14 Feb (Tue): 1 talk *) 16:15 -17:45 at E6-1, Rm 3434 15 Feb (Wed) to 16 Feb (Thu): *) 10-11:30 & 14-15:30 at E6-1, Rm 1401 (최석정강의실) *) (TBD) 16:00 -17:30 at E6-1, Rm 3434 (in case we need extra lecture) 17 Feb (Fri) *) 10-11:30 at E6-1, Rm 3438 *) (TBD) 14-15:30 at E6-1, Rm 3438 (in case we need extra lecture)

The aim of the lecture series is twofold: (1) give an overview of the “classical” theory of Breuil-Kisin modules, and (2) discuss its application to the construction of certain p-adic Galois deformation rings. In the first talk by Wansu Kim (Moday at 10am), we will give a general introduction to the p-adic Hodge theory (focusing on p-adic Galois representations). In the first half of the lecture series, Wansu Kim will explain the “classical” theory of Breuil-Kisin modules following the seminal paper of Kisin’s (Crystalline representations and F-crystals, from Drinfeld’s 50th birthday conference proceeding). In the second half of the lecture series, Chol Park will explain its application to the explicit computation of p-adic local Galois deformation rings cut out by certain p-adic Hodge-theoretic condition. Here is the time and venue for each talk: 13 Feb (Mon): 3 talks *) 10-11:30 & 14-15:30 at E6-1, Rm 1401 (최석정강의실) *) 16:00 -17:30 at E6-1, Rm 3434 14 Feb (Tue): 1 talk *) 16:15 -17:45 at E6-1, Rm 3434 15 Feb (Wed) to 16 Feb (Thu): *) 10-11:30 & 14-15:30 at E6-1, Rm 1401 (최석정강의실) *) (TBD) 16:00 -17:30 at E6-1, Rm 3434 (in case we need extra lecture) 17 Feb (Fri) *) 10-11:30 at E6-1, Rm 3438 *) (TBD) 14-15:30 at E6-1, Rm 3438 (in case we need extra lecture)

The aim of the lecture series is twofold: (1) give an overview of the “classical” theory of Breuil-Kisin modules, and (2) discuss its application to the construction of certain p-adic Galois deformation rings. In the first talk by Wansu Kim (Moday at 10am), we will give a general introduction to the p-adic Hodge theory (focusing on p-adic Galois representations). In the first half of the lecture series, Wansu Kim will explain the “classical” theory of Breuil-Kisin modules following the seminal paper of Kisin’s (Crystalline representations and F-crystals, from Drinfeld’s 50th birthday conference proceeding). In the second half of the lecture series, Chol Park will explain its application to the explicit computation of p-adic local Galois deformation rings cut out by certain p-adic Hodge-theoretic condition. Here is the time and venue for each talk: 13 Feb (Mon): 3 talks *) 10-11:30 & 14-15:30 at E6-1, Rm 1401 (최석정강의실) *) 16:00 -17:30 at E6-1, Rm 3434 14 Feb (Tue): 1 talk *) 16:15 -17:45 at E6-1, Rm 3434 15 Feb (Wed) to 16 Feb (Thu): *) 10-11:30 & 14-15:30 at E6-1, Rm 1401 (최석정강의실) *) (TBD) 16:00 -17:30 at E6-1, Rm 3434 (in case we need extra lecture) 17 Feb (Fri) *) 10-11:30 at E6-1, Rm 3438 *) (TBD) 14-15:30 at E6-1, Rm 3438 (in case we need extra lecture)

Mapping individual differences in behavior is fundamental to personalized neuroscience, but quantifying complex behavior in real world settings remains a challenge. While mobility patterns captured by smartphones have increasingly been linked to a range of psychiatric symptoms, existing research has not specifically examined whether individuals have person-specific mobility patterns. We collected over 3000 days of mobility data from a sample of 41 adolescents and young adults (age 17–30 years, 28 female) with affective instability. We extracted summary mobility metrics from GPS and accelerometer data and used their covariance structures to identify individuals and calculated the individual identification accuracy—i.e., their “footprint distinctiveness”. We found that statistical patterns of smartphone-based mobility features represented unique “footprints” that allow individual identification (p < 0.001). Critically, mobility footprints exhibited varying levels of person-specific distinctiveness (4–99%), which was associated with age and sex. Furthermore, reduced individual footprint distinctiveness was associated with instability in affect (p < 0.05) and circadian patterns (p < 0.05) as measured by environmental momentary assessment. Finally, brain functional connectivity, especially those in the somatomotor network, was linked to individual differences in mobility patterns (p < 0.05). Together, these results suggest that real-world mobility patterns may provide individual-specific signatures relevant for studies of development, sleep, and psychopathology.

The syzygies of a curve are the algebraic relation amongst the equation defining it. They are an algebraic concept but they have surprising applications to geometry. For example, the Green-Lazarsfeld secant conjecture predicts that the syzygies of a curve of sufficiently high degree are controlled by its special secants. We prove this conjecture for all curves of Clifford index at least two and not bielliptic and for all line bundles of a certain degree. Our proof is based on a classic result of Martens and Mumford on Brill-Noether varieties and on a simple vanishing criterion that comes from the interpretation of syzygies through symmetric products of curves.

Ordinary differential equation (ODE) models are widely used to describe chemical or biological processes. This article considers the estimation and assessment of such models on the basis of time-course data. Due to experimental limitations, time-course data are often noisy and some components of the system may not be observed. Furthermore, the computational demands of numerical integration have hindered the widespread adoption of time-course analysis using ODEs. To address these challenges, we explore the efficacy of the recently developed MAGI (MAnifold-constrained Gaussian process Inference) method for ODE inference. First, via a range of examples we show that MAGI is capable of inferring the parameters and system trajectories, including unobserved components, with appropriate uncertainty quantification. Second, we illustrate how MAGI can be used to assess and select different ODE models with time-course data based on MAGI’s efficient computation of model predictions. Overall, we believe MAGI is a useful method for the analysis of time-course data in the context of ODE models, which bypasses the need for any numerical integration.

The Erdős–Faber–Lovász conjecture (posed in 1972) states that the chromatic index of any linear hypergraph on n vertices is at most n. In this talk, I will sketch a proof of this conjecture for every large n. Joint work with D.Kang, T. Kelly, D. Kühn and D. Osthus.

Complexity of the cellular organization of the tumor microenvironment as an ecosystem remains to be fully appreciated. Here, for a comprehensive investigation of tumor ecosystems across a wide variety of cancer types, we performed integrative transcriptome analyses of 4.4 million single cells from 978 tumor and 474 normal samples in combination with 9,510 TCGA and 1,339 checkpoint inhibitor-treated bulk tumors. Our analysis enabled us to define 28 different epithelial cell states, some of which had prognostic effects in cancers of relevant origin. Malignant fibroblast signatures defined according to the organ of origin demonstrated prognostic significance across diverse cancer types and revealed FN1, BGN, THBS2, and CTHRC1 as common cancer-associated fibroblast genes. Novel associations were revealed between the AKR1C1+ inflammatory fibroblast and myeloid-derived PRR-induced activation states and between the CXCL10+ fibroblast and squamous/LAMP3+ DC/SPP1+ macrophage states. We discovered tumor-specific rewiring of the tertiary lymphoid structure (TLS) network, involving previously unappreciated DC1, and pDC.. Along with other TLS component states, the tumor-associated germinal center B cell state identified from adjacent normal tissues was able to predict responses to checkpoint immunotherapy. Distinct groups of pan-cancer ecosystems were identified and characterized along the axis of immunotherapy responses. Our systematic, high-resolution dissection of tumor ecosystems provides a deeper understanding of inter- and intra-tumoral heterogeneity.

The Discrete Gaussian model is a type of integer-valued random height function. In the 2D setting, it exhibits a phase transition between a localised phase and a delocalised phase. This phenomenon is also called the Kosterlitz-Thouless phase transition, whose terminology originates from its dual counterpart, the planar XY model. Motivation for studying the Discrete Gaussian model is multifold. Due to its duality relations with a number of 2D mathematical physics models, such as the XY model or the Coulomb gas, studies on integer-value height functions are capable of proving a number of conjectures usually not accessible using classical methods. Other discrete height functions also have dualities with a number of different interesting models, so it will be of vast interest to develop a general framework that deals with discrete height functions. Also, discrete height functions are considered to be appropriate test cases for recently developed techniques from probability theory. In this talk, we discuss a particular method called the renormalisation group method, which is believed to serve as a general framework for studying random fields. We also discuss briefly how the renormalisation group method can be used to prove that the scaling limit of the 2D Discrete Gaussian model is a 2D Gaussian free field.

Prismatic cohomology, which is recently developed by Bhatt and Scholze, is a p-adic cohomology theory unifying etale, de Rham, and crystalline cohomology. In this series of two talks, we will discuss its central object of study called prismatic F-crystals, and some applications to studying p-adic Galois representations. The first part will be mainly devoted to explaining motivational background on the topic. Then we will discuss the relation between prismatic F-crystals and crystalline local systems on p-adic formal scheme, and talk about applications on purity of crystalline local system and on crystalline deformation ring. If time permits, we will also discuss recent work in progress on log prismatic F-crystals and semistable local systems. A part of the results is based on joint work with Du, Liu, Shimizu.

We introduce the notion of delineation. A graph class $\mathcal C$ is said delineated by twin-width (or simply, delineated) if for every hereditary closure $\mathcal D$ of a subclass of $\mathcal C$, it holds that $\mathcal D$ has bounded twin-width if and only if $\mathcal D$ is monadically dependent. An effective strengthening of delineation for a class $\mathcal C$ implies that tractable FO model checking on $\mathcal C$ is perfectly understood: On hereditary closures of subclasses $\mathcal D$ of $\mathcal C$, FO model checking on $\mathcal D$ is fixed-parameter tractable (FPT) exactly when $\mathcal D$ has bounded twin-width. Ordered graphs [BGOdMSTT, STOC ’22] and permutation graphs [BKTW, JACM ’22] are effectively delineated, while subcubic graphs are not. On the one hand, we prove that interval graphs, and even, rooted directed path graphs are delineated. On the other hand, we observe or show that segment graphs, directed path graphs (with arbitrarily many roots), and visibility graphs of simple polygons are not delineated. In an effort to draw the delineation frontier between interval graphs (that are delineated) and axis-parallel two-lengthed segment graphs (that are not), we investigate the twin-width of restricted segment intersection classes. It was known that (triangle-free) pure axis-parallel unit segment graphs have unbounded twin-width [BGKTW, SODA ’21]. We show that $K_{t,t}$-free segment graphs, and axis-parallel $H_t$-free unit segment graphs have bounded twin-width, where $H_t$ is the half-graph or ladder of height $t$. In contrast, axis-parallel $H_4$-free two-lengthed segment graphs have unbounded twin-width. We leave as an open question whether unit segment graphs are delineated. More broadly, we explore which structures (large bicliques, half-graphs, or independent sets) are responsible for making the twin-width large on the main classes of intersection and visibility graphs. Our new results, combined with the FPT algorithm for first-order model checking on graphs given with $O(1)$-sequences [BKTW, JACM ’22], give rise to a variety of algorithmic win-win arguments. They all fall in the same framework: If $p$ is an FO definable graph parameter that effectively functionally upperbounds twin-width on a class C, then $p(G) \ge k$ can be decided in FPT time $f(k)\cdot |V (G)|O(1)$. For instance, we readily derive FPT algorithms for k-Ladder on visibility graphs of 1.5D terrains, and k-Independent Set on visibility graphs of simple polygons. This showcases that the theory of twin-width can serve outside of classes of bounded twin-width. Joint work with Édouard Bonnet, Dibyayan Chakraborty, Eun Jung Kim, Raul Lopes and Stéphan Thomassé.

Prismatic cohomology, which is recently developed by Bhatt and Scholze, is a p-adic cohomology theory unifying etale, de Rham, and crystalline cohomology. In this series of two talks, we will discuss its central object of study called prismatic F-crystals, and some applications to studying p-adic Galois representations. The first part will be mainly devoted to explaining motivational background on the topic. Then we will discuss the relation between prismatic F-crystals and crystalline local systems on p-adic formal scheme, and talk about applications on purity of crystalline local system and on crystalline deformation ring. If time permits, we will also discuss recent work in progress on log prismatic F-crystals and semistable local systems. A part of the results is based on joint work with Du, Liu, Shimizu.

Affine Deligne-Lusztig varieties show up naturally in the study of Shimura varieties, Rapoport-Zink spaces, and moduli spaces of local shtukas. Among various questions on its geometric properties, the question on the connected components turns out to be a fairly important problem. For example, Kisin, in his proof of the Langlands-Rapoport conjecture (in a weak sense) for abelian type Shimura variety with the hyperspecial level structure, crucially used the description of the set of connected components. Since then, many authors have answered this question in various restricted cases. I will first discuss what is the conjectural description of the connected components and related previous works. Then, I will explain my new result (joint work with Ian Gleason and Yujie Xu) which finishes the question in the mixed characteristic case and, if time permits, new ingredients.

Affine Deligne-Lusztig varieties are first defined by Rapoport as the (conjectural) p-part of the so-called Langlands-Rapoport conjecture. It can be understood as a p-adic generalization of the classical Deligne-Lusztig varieties. One of the most basic questions is 'when they are nonempty'. For a certain union, the nonemptiness criterion is completely known (by the so-called Mazur's inequality or B(G,μ)). However, the question about the "individual" ones is moderately open (with no general conjecture). I will discuss old and new nonemptiness results and suggest a new conjecture, for the individual ones, in the basic case. As an application, I will briefly mention a new explicit dimension formula in the rank 2 case (for which no conjectural formula was stated before).

The converse theorem for automorphic forms has a long history beginning with the work of Hecke (1936) and a work of Weil (1967): relating the automorphy relations satisfied by classical modular forms to analytic properties of their L-functions and the L-functions twisted by Dirichlet characters. The classical converse theorems were reformulated and generalised in the setting of automorphic representations for GL(2) by Jacquet and Langlands (1970). Since then, the converse theorem has been a cornerstone of the theory of automorphic representations. Venkatesh (2002), in his thesis, gave new proof of the classical converse theorem for modular forms of level 1 in the context of Langlands’ “Beyond Endoscopy”. In this talk, we extend Venkatesh’s proof of the converse theorem to forms of arbitrary levels and characters with the gamma factors of the Selberg class type. This is joint work with Andrew R. Booker and Michael Farmer.

The Graphic Travelling Salesman Problem is the problem of finding a spanning closed walk (a TSP walk) of minimum length in a given connected graph. The special case of the Graphic TSP on subcubic graphs has been studied extensively due to their worst-case behaviour in the famous $\frac{4}{3}$-integrality-gap conjecture on the "subtour elimination" linear programming relaxation of the Metric TSP. We prove that every simple 2-connected subcubic graph on $n$ vertices with $n_2$ vertices of degree 2 has a TSP walk of length at most $\frac{5n+n_2}{4}-1$, confirming a conjecture of Dvořák, Král', and Mohar. This bound is best possible and we characterize the extremal subcubic examples meeting this bound. We also give a quadratic time combinatorial algorithm to find such a TSP walk. In particular, we obtain a $\frac{5}{4}$-approximation algorithm for the Graphic TSP on cubic graphs. Joint work with Michael Wigal and Xingxing Yu.