Turkish Math Society proudly presents a monthly Distinguished Colloquium Series in pure and applied mathematics. With the wide usage of online talks in the post-pandemic era, we aim to host renowned mathematicians from all over the world to promote the latest developments in their fields.

**Organizing Team**

Erhan Bayraktar (U. Michigan), İlker Birbil (U. Amsterdam), Kazım Büyükboduk (UC-Dublin), İzzet Coşkun (U. Illinois – Chicago), Barış Coşkunüzer (UT Dallas), Burak Erdoğan (U. Illinois – Urbana-Champaign), Sinan Güntürk (NYU-Courant), Özlem İmamoğlu (ETH Zürich), Gizem Karaali (Pomona C.), Ekin Özman (Boğaziçi U.).

#### Talk 1 – Jordan Ellenberg – U. Wisconsin

**Date** : 15 / 09 / 2021 Wednesday, 18h00 (GMT+3)

**Title** : Upper Bounds for Rational Points

**Abstract** : The question “what are the solutions in rational numbers to an algebraic equation?” is the one that drives the subject of Diophantine geometry, and has for centuries. It is much, much too hard. So instead one might ask: “how many solutions does an algebraic equation have?” Still too hard. One might thus be willing to settle for “Are there good upper bounds for the number of solutions an algebraic equation have?” and here at last there are some good general results. I’ll talk about what is known, with special attention to the distinction between uniform results (those with no dependence, or minimal dependence, on the particular equation at issue) and non-uniform results (which depend strongly on the arithmetic properties of the individual equation), and will close with a new result (joint with Brian Lawrence and Akshay Venkatesh) showing that there are in a sense “very few” hypersurfaces in projective space whose determinant takes a fixed integer value — a non-uniform bound which uses in a critical way the existence of uniform bounds developed in the last twenty years.

**Video recording** : https://youtu.be/3_PTQ5oIeXY

#### Talk 2 – Rahul Pandharipande – ETH Zurich

**Date** : 13 / 10 / 2021 Wednesday, 18h00 (GMT +3)

**Title** : Algebraic curves, Hurwitz covers and meromorphic differentials

**Abstract** : Hurwitz’s paper ”Ueber die Anzahl der Riemannischen Flächen mit gegebenen Verzweigungspunkten” (1901) started the study of the enumeration of branched coverings of the Riemann sphere. Though more than a century has passed now, there have been many recent developments in the subject that Hurwitz opened. I will explain new results and perspectives on Hurwitz numbers, Hurwitz moduli spaces, and related constructions concerning meromorphic differentials.

** Video recording **: https://youtu.be/_DMpiMaZ9Vs

**Talk 3 – Nizar Touzi – Ecole Polytechnique**

**Date** : 10 / 11 / 2021 Wednesday, 19h00 (GMT +3)

**Title** : The propagation of chaos for the multiple optimal stopping problem

**Abstract** : The optimal stopping problem of $N$ particles deriven by interacting diffusion processes can be characterized by a cascade of obstacle Cauchy problems. The limiting problem is an optimal stopping problem of a McKean-Vlasov diffusion with criterion defined as a function of the law of the stopped process. The corresponding dynamic programming equation is an obstacle problem on the Wasserstein space, and is obtained by means of a general Itô formula for flows of marginal laws of càdlàg semimartingales. We provide a verification result which characterizes the nature of optimal stopping policies, highlighting the crucial need to randomized stopping. We also introduce a notion of viscosity solutions on the Wassertsein space which allows to characterize the value function, and we prove a result of propagation of chaos by adapting the monotone scheme convergence argument.

** Video recording :** https://youtu.be/PgMJp2P8AMM

**Talk 4 – Wilhelm Schlag – Yale**

**Date** : 15 / 12 / 2021 Wednesday, 19h00 (GMT +3)

**Title** : Asymptotic stability for the sine-Gordon kink under odd perturbations via super-symmetry

**Abstract** : Kinks are examples of topological solitons in classical field theory. They have been studied for decades, mostly by methods of complete integrability such as the inverse scattering transform. One of the most basic models, known as phi^4, is not accessible to these techniques and much less is known even about the most basic object of nonzero charge: the kink in one spatial dimension. I will describe the recent asymptotic analysis with Jonas Luehrmann (TAMU) of the sine-Gordon evolution of odd data near the kink. While sine-Gordon is completely integrable, we do not rely on this property. The talk will present some background on classical fields and the history of the problem.

**Video recording** : https://youtu.be/99ytmBMW0WA

**Talk 5 – Steph van Willigenburg** – UBC

**Date** : 12 / 01 / 2022 Wednesday, 19h00 (GMT +3)

**Title** : The (3+1)-free conjecture of chromatic symmetric functions

**Abstract** : The chromatic symmetric function, dating from 1995, is a generalization of the chromatic polynomial. A famed conjecture on it, called the Stanley-Stembridge (3+1)-free conjecture, has been the focus of much research lately. In this talk we will be introduced to the chromatic symmetric function, the (3+1)-free conjecture, new cases and tools for resolving it, and answer another question of Stanley of whether the (3+1)-free conjecture can be widened. This talk requires no prior knowledge.

**Video Recording** : https://youtu.be/ldlqpOp_auA

**Talk 6 – Joel Tropp – CalTech**

**Date** : 09 / 02 / 2022 Wednesday, 20h00 (GMT +3)

**Title** : Universality Laws for Randomized Dimension Reduction

**Abstract** : Dimension reduction is the process of embedding high-dimensional data into a lower dimensional space to facilitate its analysis. In the Euclidean setting, one fundamental technique for dimension reduction is to apply a random linear map to the data. The question is how large the embedding dimension must be to ensure that randomized dimension reduction succeeds with high probability.

This talk describes a phase transition in the behavior of the dimension reduction map as the embedding dimension increases. The location of this phase transition is universal for a large class of datasets and random dimension reduction maps. Furthermore, the stability properties of randomized dimension reduction are also universal. These results have many applications in numerical analysis, signal processing, and statistics.

Joint work with Samet Oymak (UCR). For more information, see https://arxiv.org/abs/1511.09433.

**Video Recording** : https://youtu.be/LzXAhVNiV6w

Poster

**Talk 7 – Alessio Figalli – ETH Zurich**

**Date** : 23 / 03 / 2022 Wednesday, 11am NYT, 4pm London, 6pm Istanbul (No daylight savings)

**Title** : Quantitative Stability in Geometric and Functional Inequalities

**Abstract** : Geometric and functional inequalities play a crucial role in several problems arising in analysis and geometry. Proving the validity of such inequalities, and understanding the structure of minimizers, is a classical and important question. In this talk, I will overview this beautiful topic and discuss some recent results.

**Video Recording** : https://youtu.be/fd8f-NkrZJA

** Talk 8 – Koray Kavukçuoglu – DeepMind** CANCELLED

** Date** :

**Title** : to be announced

**Abstract** : to be announced

**Connection info** : to be announced

**Talk 9 – Karen Uhlenbeck – UT Austin (emerita) – IAS (visitor)**

**Date** : 18 / 05 / 2022 Wednesday, 11am NYT, 4pm London, 6pm Istanbul

**Title** : The Noether Theorems: Then and Now

**Abstract** : The 1918 Noether theorems were a product of the general search for energy and momentum conservation in Einstein’s newly formulated theory of general relativity. Although widely referred to as the connection between symmetry and conservation laws, the theorems themselves are often not understood properly and hence have not been as widely used as they might be. In the first part of the talk, I outline a brief history of the theorems, explain a bit of the language, translate the first theorem into coordinate invariant language and give a few examples. I will briefly mention their historical importance in physics and integrable systems. In the second part of the talk, I describe why they are still relevant: why George Daskalopoulos and I came to be interested in them for our investigation into the best Lipschitz maps of surfaces of Bill Thurston and the open problems in higher dimensions. I will finish by mentioning two recent papers, one in math and the other in physics, which greatly simplify the derivations of important identities by using the theorems.

**Video Recording: **https://youtu.be/gl6IpPRhmcU

**Talk 10 – Amie Wilkinson – U. Chicago**

**Date** : 15 / 06 / 2022 Wednesday, 11am NYT, 4pm London, 6pm Istanbul

**Title** : Asymmetry in dynamics

**Abstract** : The origins of the subject of dynamical systems lie in classical mechanics, in the study of such fundamental problems as the stability of the solar system. A theme that traces back to Noether’s theorem is that symmetries in such physical systems must occur for a reason: for example, if the motion of a system does not depend on position in space, then there must be a conserved quantity, such as angular momentum. I will discuss, in the broader contexts of modern dynamics, how this theme expands and reoccurs in

**Video Recording:** https://www.youtube.com/watch?v=-LMGL_co1y8