Abstract: We consider an elliptic operator, double divergence form operator L*, which is the formal adjoint of the elliptic operator in non-divergence from L. An important example of a double divergence form equation is the stationary Kolmogorov equation for invariant measures of a diffusion process. We are concerned with the regularity of weak solutions of L*u=0 and show that Schauder type estimates are available when the coefficients are of Dini mean oscillation and belong to certain function spaces. We will also discuss some applications and parabolic counterparts.

“Symmetry of solutions to a class of PDEs on hyperbolic spaces and sharp functional and geometric inequalities”

Abstract: The theory of quantitative rectifiability for Ahlfors regular subsets of Euclidean space was developed extensively by David and Semmes in the early 1990s, partly motivated by questions arising in harmonic analysis. They proved, among many other things, the equivalence of Uniform Rectifiability (UR) and the Bi-lateral Weak Geometric Lemma (BWGL). The first condition being a natural quantitative version of rectifiability, the second, a quantitative condition measuring local Hausdorff approximations by affine subspaces. Their result can be seen as quantification of the equivalence between rectifiability and the almost everywhere existence of approximate tangent planes. In this talk we discuss the equivalence of UR and BWGL for Ahlfors regular metric spaces. While the definition of UR makes sense in this context, BWGL does not. Instead, the BWGL condition is stated in terms of local Gromov-Hausdorff approximations by n-dimensional Banach spaces.

This is based on joint work with David Bate and Raanan Schul.

To better understand mutation-invariant and hereditary properties of quivers (and more generally skew-symmetrizable matrices), my coauthor and I have constructed a topology on the set of all mutation classes of quivers which we call the mutation class topology. This topology is the Alexandrov topology induced by the poset structure on the set of mutation classes of quivers. The partial order is given by quiver embedding. The closed sets of our topology—the down sets of the poset—are in bijective correspondence with mutation-invariant and hereditary properties of quivers. The mutation class space described in this presentation is the unique topological space with this property. This space is strictly \(T_0\), connected, non-Noetherian, and every open set is dense. I’ll end the talk by providing open questions from cluster algebra theory in the setting of the mutation class topology and some directions for future research.

Join us in the Logic Colloquium!

Andrew Tedder (Vienna)

“Relevant Logics as Topical Logics”

There is a simple way of reading a structure of topics into the matrix models of a given logic, namely by taking the topics of a given matrix model to be represented by subalgebras of the algebra reduct of the matrix, and then considering assignments of subalgebras to formulas. The resulting topic-enriched matrix models bear suggestive similarities to the two-component frame models developed by Berto et. al. in Topics of Thought. In this talk I’ll show how this reading of topics can be applied to the relevant logic R, and its algebraic characterisation in terms of De Morgan monoids, and indicate how we can, using this machinery and the fact that R satisfies the variable sharing property, read R as a topic-sensitive logic. I’ll then suggest how this approach to modeling topics can be applied to a broader range of logics/classes of matrices, and gesture at some avenues of research.

All welcome!

https://logic.uconn.edu/calendar/

If you are considering graduate school in mathematics or related areas after college, come to this panel discussion where you will hear from members of the UConn math department about their experiences planning for and applying to graduate school. The discussion will then be opened to answer your questions. A packet containing a suggested reading list and some general advice will be distributed too.

We consider a thin-film limit for a three-dimensional phenomenological energy functional modeling the effects of a constant electric field on a bent-core liquid crystal in the ferroelectric smectic A-type phase. We show that similarly to what is seen in micromagnetics under a proper rescaling, the electric self-interaction terms give rise to boundary terms in the two-dimensional reduction. We also present numerical simulations based on the reduced model. This is a joint work with Carlos García-Cervera and Sookyung Joo.

Speaker: Hugo Duminil-Copin (IHES, Fields Medalist 2022)

Title: Critical phenomena through the lens of the Ising model

Abstract: The Ising model is one of the most classical lattice models of statistical physics undergoing a phase transition. Initially imagined as a model for ferromagnetism, it revealed itself as a very rich mathematical object and a powerful theoretical tool to understand cooperative phenomena. Over one hundred years of its history, a profound understanding of its critical phase has been obtained. While integrability and mean-field behavior led to extraordinary breakthroughs in the two-dimensional and high-dimensional cases respectively, the model in three and four dimensions remained mysterious for years. In this talk, we will present recent progress in these dimensions.

Join Zoom Meeting
https://uconn-edu.zoom.us/j/3106002217?pwd=OVd2L0g5c0NMM3o4MjRGRWNsNVFJdz09

Meeting ID: 310 600 2217
Passcode: DG3fAA

Given a graph \(G\), a continuous-time quantum walk is defined as the time-varying unitary matrix \(U(t) = e^\) where \(A(G)\) is the adjacency matrix of \(G\) and \(t\) varies over the reals. We say \(G\) has perfect state transfer from vertex \(a\) to vertex \(b\) at time \(t\) if the \(|U(t)_|=1\). This notion was introduced by Bose (2003) and had been useful for studying quantum communication in spin networks.

In this talk, we describe speed limits for perfect state transfer in continuous-time quantum walks. We focus on weighted paths as explicit bounds are known for perfect state transfer (due to Yung and Kay). Then, within the legal bounds of quantum physics, we discuss violations of these speed limits.
Along the way, we mention a few other related observations (paradoxical or otherwise) and conclude with some open questions.

This talk is based on joint work with Alastair Kay and Weichen Xie.

Webex meeting link: https://uconn-cmr.webex.com/uconn-cmr/j.php?MTID=m3c732d1c9942c261db062d207e46dc6d

Meeting number: 2634 583 5977 Password: mSUKp9vaG64

Abstract: In this talk, we discuss a general multidimensional linear convex stochastic control problem with nondifferentiable objective function, control constraints, and random coefficients. We formulate an equivalent dual problem and prove the dual stochastic maximum principle. We also prove the primal-dual relation of the optimal control, optimal state, and adjoint processes. Finally, some examples are presented to illustrate the usefulness of the dual approach.

Speaker’s short bio: Engel is a postdoc at the Department of Statistics and Actuarial Science, University of Hong Kong (HKU), working with Prof. Tim Boonen. He obtained his PhD from the University of South Australia under the supervision of Prof. Robert Elliott and was at Imperial before joining HKU. See https://saasweb.hku.hk/staff/ejdv/.

Abstract: We develop a metastability theory for a class of stochastic reaction-diffusion equations exposed to small multiplicative noise. We consider the case where the unperturbed reaction-diffusion equation features multiple asymptotically stable equilibria. When the system is exposed to small stochastic perturbations, it is likely to stay near one equilibrium for a long period of time, but will eventually transition to the neighborhood of another equilibrium. We are interested in studying the exit time from the full domain of attraction (in a function space) surrounding an equilibrium and therefore do not assume that the domain of attraction features uniform attraction to the equilibrium. This means that the boundary of the domain of attraction is allowed to contain saddles and limit cycles. Our method of proof is purely infinite dimensional, i.e., we do not go through finite dimensional approximations. In addition, we address the multiplicative noise case and we do not impose gradient type of assumptions on the nonlinearity. We prove large deviations logarithmic asymptotics for the exit time and for the exit shape, also characterizing the most probable set of shapes of solutions at the time of exit from the domain of attraction. This is joint work with Michael Salins.

For the last math club meeting this year, we will be tackling some exercises in game theory.

Game theory is the study of strategic interactions. The fate of each person in a “game” usually depends on the decisions of all “players”. Nash Equilibrium, developed by Nobel Prize winner John Nash, says that in any finite game with a finite number of players, each player has a best strategy to win the game.

This week, we will explore different types of games, discuss possible strategies, and try some problems to start thinking about these interactions like a game theorist.

Note: Free refreshments.

The error-correcting code is a technique for removing the noise which occurred in the process of transmitting information and is widely used in our everyday life, e.g. CD players or QR codes. By means of the codes based on algebraic curves originally introduced by V.D.Goppa, Tsfasman-Vladut-Zink has shown the existence of codes with very good asymptotic properties which improved the Gilbert-Varshamov bound.

In this lecture, we discuss the parameters of algebraic geometric codes obtained from higher dimensional varieties. Especially we give estimates for the minimum distances of the codes on certain fibered varieties over a curve. By means of the Seshadri constants, we also explain some results concerning the codes defined from vector bundles of higher rank.

In this talk, we will go over what it means for an extension to be Galois. We will look at Galois and non-Galois extensions with examples. Further, we will discuss the Kronecker-Weber Theorem, which states that every finite abelian extension of \(\mathbb\) is contained in a cyclotomic field. Time permitting, we will see what happens when we look at \(\mathbb\) adjoin the division points of a curve as a field extension of \(\mathbb\).

Join us in the Logic Colloquium for a talk by

Jonas Raab (Trinity College Dublin)

“Modal QUARC and Barcan”

I develop a modal extension of the Quantified Argument Calculus (QUARC)—a novel logical system introduced by Hanoch Ben-Yami. QUARC is meant to better capture the logic of natural language. The purpose of this paper is to develop a variable domain semantics for modal QUARC (M-QUARC), and to show that even if the usual restrictions are imposed on models with variable domains, M-QUARC-analogues of the Barcan and Converse Barcan formulas still are not validated. I introduce new restrictions—restrictions on the extension of the predicates—and show that with these in place, the Barcan and Converse Barcan formulas are valid. The upshot is that M-QUARC sheds light on the in-/validity of such formulas.

https://logic.uconn.edu/calendar/

This is a joint event with the UConn Control and Optimization Seminar.

Abstract. We propose efficient and reliable simulation schemes for pricing options under the stochastic-alpha-beta-rho (SABR) model. The standard two-step simulation procedures involve (i) simulation of the integrated variance conditional on terminal variance and (ii) simulation of the terminal forward price conditional on terminal variance and integrated variance. Most simulation schemes rely on the Islah approximation formula of the conditional distribution of the terminal asset price, which is seen to fail the martingale condition in general. We embed three enhanced features in our proposed simulation schemes. Firstly, we approximate the terminal forward price as the constant elasticity of variance (CEV) process that satisfies the martingale condition, an important property that precludes arbitrage. Secondly, we adopt the displaced Poisson-mixture Gamma distribution for the exact simulation of the underlying CEV process in the simulation of the terminal forward price conditional on integrated variance and terminal variance. Thirdly, we use the shifted lognormal approximation of the integrated variance to compute the integrated variance. Our enhanced procedures avoid the tedious Laplace inversion algorithm in integrated variance calculations and non-efficient inverse transform in the forward price calculations in earlier simulation schemes. Numerical results demonstrate our simulation schemes to be highly efficient, accurate, and reliable.

Speaker’s short bio: Dr. Choi is an associate professor with tenure at Peking University HSBC Business School in Shenzhen, China. He obtained his PhD in applied mathematics from Massachusetts Institute of Technology in 2005 and BS in mathematics from the Korean Advanced Institute of Science Technology in 2000. After his PhD, he worked in the industry as a quant for several years in BNP Paribas and Goldman Sachs. His research interests include quantitative finance, mathematical modeling, numerical methods, and data science. Please visit his website https://english.phbs.pku.edu.cn/2016/fulltime_0826/81.html for more information.

Speaker: Gordon Slade (UBC)

Title:
Convergence of the lace expansion

Abstract:
The lace expansion is a flexible method that has been used since the
1980s to analyse the critical behaviour of high-dimensional random
systems, including self-avoiding walk, percolation, and spin systems. It
originated in work of Brydges and Spencer on weakly self-avoiding walk
in dimensions above 4, and since then several different approaches have
been developed to prove convergence of the expansion. I will explain
what the lace expansion for self-avoiding walk is, and will present a
new and relatively simple method for proving convergence of the lace
expansion for weakly self-avoiding walk. The talk is based on Ann. Inst.
H. Poincaré Probab. Statist., 58:26-33, (2022). Extensions of the
method have subsequently been obtained in joint work with Yucheng Liu:
arXiv:2310.07635 and arXiv:2310.07640.

Join Zoom Meeting
https://uconn-edu.zoom.us/j/3106002217?pwd=OVd2L0g5c0NMM3o4MjRGRWNsNVFJdz09

Meeting ID: 310 600 2217
Passcode: DG3fAA

In theory, proof assistants are a useful tool for validating mathematical claims. We will use simplicial topology as a case study to look at what goes into formalizing mathematics from pen and paper to the digital world. This is based on previous work with Stefan Friedl focused on formalizing stellar subdivisions of simplicial complexes. I will present a few anecdotal stories from our project that demonstrate some of the challenges involved. Topics include (but are not limited to) what changes when we introduce types to topology, decidability and computational complexity concerns, and what to do when the proof is left as an exercise for the reader.

Join us in the Logic Colloquium!

Xinhe Wu (NCSU):

“Vague Identity: A Uniform Approach”

There are numerous apparent examples of vague identity, i.e. examples where two objects appear to be neither determinately identical nor determinately distinct. Philosophers disagree on whether the source of vagueness in identity is semantic or ontic/metaphysical. In this talk, I explore the use of Boolean-valued models as a many-valued semantic framework for identity. I argue that this semantics works well with both a semantic and ontic conception of vague identity. I also discuss, in the context of Boolean-valued logic, responses to the Evans’ argument under the two conceptions.

All welcome!

https://logic.uconn.edu/calendar/

https://svenhirsch.com/