Zeta and L-functions have been classically associated, first to schemes of finite type over Z, and then to the l-adic cohomology of smooth projective varieties over a global field. In the latter definition, independence from l and actual existence remain partly conjectural in characteristic 0. In these lectures, I will explain how to associate unconditionally an L-function to objects in triangulated categories of motives. For the motive of a smooth projective variety this definition differs from the one above in general, but only up to a finite number of Euler factors. This might be useful to tackle the Beilinson conjectures.

In critical PDEs, finite-time blow-up may occur while the scaling-critical norm remains bounded, with singularity formation arising from concentration and loss of compactness. In this first talk, we consider the radial focusing energy-critical wave equation in three dimensions and discuss the role of concentration-compactness and profile decomposition in the study of Type II blow-up. We then introduce radiation for the radial free wave equation and the channel of energy method. These ideas will provide the starting point for the rigidity analysis in the second talk. [References] 1. H. Bahouri, P. Gérard, High frequency approximation of solutions to critical nonlinear wave equations, Amer. J. Math. 121 (1999), 131–175. 2. C. E. Kenig, A. Lawrie, B. Liu, W. Schlag, Channels of energy for the linear radial wave equation, Adv. Math. 285 (2015), 877–936.

Hadwiger famously conjectured that every $K_h$-minor-free graph is properly $(h-1)$-colourable. This talk will present the following improper analogue of Hadwiger’s Conjecture: for fixed $h$, every $K_h$-minor-free graph is $(h-1)$-colourable with monochromatic components of bounded size. The number of colours is best possible regardless of the size of monochromatic components. This solves an open problem of Edwards, Kang, Kim, Oum and Seymour [SIAM J. Disc. Math. 2015], and concludes a line of research initiated in 2007. Similarly, for fixed $t\geqslant s$, we show that every $K_{s,t}$-minor-free graph is $(s+1)$-colourable with monochromatic components of bounded size. The number of colours is best possible, solving an open problem of van de Heuvel and Wood [J. London Math. Soc. 2018]. We actually prove a single theorem from which both of the above results are immediate corollaries. For an excluded apex minor, the result is strengthened  as follows: for fixed $t \geqslant s \geqslant 3$, and for any fixed apex graph $X$, every $K_{s,t}$-subgraph-free $X$-minor-free graph is $(s+1)$-colourable with monochromatic components of bounded size. The number of colours is again best possible. This is joint work with Vida Dujmović, Louis Esperet and Pat Morin [arXiv:2306.06224].

ML/DL method are widely used for prediction in biomedical research, but predictive performance alone often provides limited scientific insight. Explainable artificial intelligence (XAI) and visualization methods can help identify important variables, characterize complex relationships, and generate interpretable findings beyond prediction accuracy. In this seminar, I will introduce practical XAI and visualization approaches and demonstrate their applications using two biomedical examples: drug-related expression prediction in human and mouse liver data, and analysis of hypnotic medication timing and use patterns. These examples illustrate how predictive modeling can be extended toward interpretation, hypothesis generation, and data-driven scientific discovery.

A classic problem in topology is determining how many smooth structures a given manifold can admit. We say that a manifold admits exotic smooth structures if there are at least two distinct smooth structures on the manifold. Understanding these exotic phenomena has been a central topic in both geometric and algebraic topology. Ever since Milnor discovered exotic 7-spheres, there has been extensive research searching for exotic smooth structures on various manifolds. Furthermore, the work of Donaldson and Freedman opened a new world for exotic phenomena in dimension 4. In this talk, we will provide an overview of exotic smooth structures on topological manifolds, review their history, and explore cutting-edge developments in the field.

I discuss ‘almost counterexamples’ to Seymour’s second neighbourhood conjecture. In what we call Seymour-tight orientations, the size of the first neighbourhood of each vertex equals the size of its second neighbourhood. We give several examples and constructions. Specifically, we prove that the class of Seymour-tight orientations is closed under taking (generalized) lexicographic products. Moreover, the lexicographic product of a putative counterexample to Seymour’s second neighbourhood conjecture and a Seymour-tight orientation is again a counterexample. Using lexicographic products, we show that if the conjecture is false, then there exist counterexamples that are close to regular tournaments, and moreover that any digraph occurs as an induced subgraph of a counterexample. We then use this same machinery to construct special putative counterexamples to Sullivan’s conjecture. The inherent symmetry of these orientations give access to an algebraic perspective. Seymour-tight orientations that are also Cayley digraphs correspond to special pairs of critical sets in groups, which connects potentially to additive combinatorics. We use Kemperman’s theorem to characterize those Seymour-tight orientations that are the Cayley digraph of an abelian group.

We discuss lower bounds on Lyapunov exponents for linear PDEs driven by random velocity fields, including the advection–diffusion equation and the linearized stochastic Navier–Stokes equations. In particular, we show that the exponential rate of decay of the L^2-norm for solutions to the advection–diffusion equation is optimal. A key ingredient in the proof is high-frequency stochastic instability, arising from the non-degeneracy of the driving noise. This talk is based on joint work with Martin Hairer, Tommaso Rosati, and Sam Punshon-Smith.

Elliptic curves are fundamental objects in modern number theory, connecting algebraic geometry and arithmetic. In this talk, I will give an introductory overview of elliptic curves and their arithmetic, comparing the number field and function field settings and highlighting their similarities and differences. I will then present recent results on elliptic curves of analytic rank one and their ranks over certain ring class fields, based on joint work with Seokhyun Choi and Bo-Hae Im.

We present some basics of L-functions usually (to be) covered in the undergraduate and graduate entry level number theory classes.

We introduce the concept of the saturation of a (bi)graph: the union closure after inductively adding its virtual elements, which are weighted ε-good (respectively ε-excellent sets) as in the Stable Regularity Lemma. In the Littlestone class and stable graph case, we show that if the saturation has bounded Littlestone dimension, then it is the smallest ε-saturated object containing the initial one. We show that for certain values of ε, the saturations of Littlestone classes are Littlestone, although not necessarily of the same dimension. For ε large enough, we find examples to show that VC and Littlestone dimensions may grow arbitrarily. For certain ε, we bound Littlestone dimension of the saturation by a finite value depending on VC dimension, by using techniques including the Fundamental Theorem of Statistical Learning and the Littlestone Minimax Theorem. We will focus on the class (or bigraph) case and time permitting, we will discuss the stable graph case. Joint work with Maryanthe Malliaris and Shay Moran.

We present some basics of L-functions usually (to be) covered in the undergraduate and graduate entry level number theory classes.

The stability theory of entropy solutions to the compressible Euler system on a half-line with prescribed boundary conditions, such as inflow, outflow, and impermeable boundary conditions, remains largely open. It is worth emphasizing that establishing the stability of even a single reflected shock at the boundary has been a highly challenging problem. In this talk, I will briefly review the classical stability theory for small-BV entropy solutions in the whole-space setting and then turn to the half-line problem, where I will present recent progress on the stability of entropy solutions in the presence of a boundary.

Let $P$ be a set of points in $PG(n+d,q)$, and let $L$ be a set of $n$-flats. Here, $n$-flat is a short name for $n$-dimensional projective subspaces. A classical bound of Haemers, rediscovered in an influential paper of Vinh, gives an upper bound on the difference between the number of incidences between $P$ and $L$ and the expected number of incidences for random sets of points and flats with the same cardinalities as $P$ and $L$. Haemers’ bound is tight as a function of $|P|$ times $|L|$. Recent work of Kong and Tamo improves the bound under the assumption that $|L|$ is not too large. I will discuss recent work, joint with Tao Zhang, that improves the bound of Kong and Tamo. The proof depends on an independently interesting upper bound on the number of pairs $(l_1,l_2)$ of flats in $L$ such that $\dim(l_1 \cap l_2)=j$, for $0 \leq j \leq n$.

The “Convexity Conjecture” by Talagrand asks, roughly speaking, whether one can “create convexity” in a bounded number of steps regardless of the dimension of the ambient space. Talagrand also proposed a discrete version of this conjecture, calling it his “lifetime favorite problem” and offering a $1,000 prize for its solution. While the continuous version of the conjecture was recently proven by Hua, Song, and Tudose, the discrete analogue remains wide open. In this talk, we introduce a reformulation of the discrete convexity conjecture using the new notion of “k-thresholds,” an extension of the traditional definition of thresholds. Using this framework, we establish the conjecture for several special cases, focusing primarily on graph properties.

(This is a reading seminar presented by two graduate students.) In this reading seminar, we will present an introduction to étale cohomology and the Weil conjectures based on Milne's lecture notes. Beginning with the étale topology and the theory of sheaves on étale sites, we will develop the basic constructions and properties of étale cohomology. We will then explain several fundamental results of the theory including the purity theorem, the base change theorems, and the comparison theorem. Finally, we will discuss how these results culminate in the proof of the Weil conjectures.

Let $R_k(3)$ denote the smallest integer $N$ such that every $k$-edge-coloring of the complete graph $K_N$ contains a monochromatic triangle. A simple inductive argument gives the classical factorial upper bound $R_k(3)\leq k!=k^{O(k)}$, whereas the best previously known lower bound was only exponential in $k$, namely, $R_k(3)\geq 2^{\Omega(k)}$. It was a longstanding open problem of Erd\H{o}s whether $R_k(3)$ grows exponentially or super-exponentially in $k$. On August 1, 2026, OpenAI, using an internal AI model, discovered a construction establishing the super-exponential lower bound $R_k(3)\geq k^{\Omega(k)}$, thereby resolving Erdős’ longstanding question. In this talk, I will explain the construction and discuss possible directions for further research, some of which may already have been explored by other researchers.

The first major step towards the graph minor structure theorem by Robertson and Seymour was the grid theorem, a result describing that every graph of large treewidth contains a grid as minor. In 2014 Wollan gave a definition for a tree-like decomposition and a width parameter tree-cutwidth with respect to immersions, a different graph containment relation. He provided results linking this parameter to immersions of large walls. This talk presents a version of this parameter for directed graphs, the directed tree-cutwidth. The main result is a grid theorem for directed tree-cutwidth establishing that it is linked to directed immersions of large cylindrical walls. The presented work is joined with Marcin Briański, Karolina Okrasa, and Michał Pilipczuk.

We study restricted-link augmentation to 2-vertex-connectivity. An instance consists of a graph G, possibly disconnected, a set L of admissible links on its vertices, integer link costs in {1, …, W}, and an integer k; the task is to add at most k links of minimum total cost so that the resulting multigraph is 2-vertex-connected. Recent work gives $O^*(k^{O(k)})$-time algorithms for unweighted λ-vertex-connectivity augmentation for every λ ≤ 4 [Carmesin and Ramanujan, SODA 2026], and an $O^*((k + λ)^{O(k)})$-time algorithm for arbitrary λ [Korhonen and Thorup, FOCS 2026]. We give a deterministic algorithm with running time $O^*(36^k W)$. Thus, for λ = 2, the unweighted running time improves from $O^*(k^{O(k)})$ to $O^*(36^k)$, and the algorithm also handles link costs with pseudo-polynomial dependence on W. We reduce the problem to a boundary-pair variant of 2-vertex-connected spanning subgraph, where each vertex is assigned a pair of incident edges with an associated pair cost. We solve this variant using a cancellation identity, inspired by Cut&Count [Cygan et al., TALG 2022], obtained by applying Möbius inversion to decompositions along cut vertices: the identity cancels every connected spanning graph with more than one block and keeps exactly the 2-vertex-connected spanning graphs.